feat(libmeanfield): variadic refactor

also added normaliztion operator

CMakeLists.txt

@@ -103,7 +103,7 @@ target_sources(mean_field
         libmeanfield/impl/solver/preconditioning_diagnostics.cpp
         libmeanfield/impl/preconditioning/gravity_field.cpp
         libmeanfield/impl/operators/prepared_mass_normalization.cpp
-        libmeanfield/impl/operators/prepared_central_density_stellar_equilibrium.cpp
+        libmeanfield/impl/operators/prepared_angular_momentum.cpp
         libmeanfield/impl/operators/prepared_stellar_equilibrium.cpp
 )
 
@@ -156,6 +156,11 @@ target_sources(mean_field
         libmeanfield/interface/preconditioning/specification_border.cppm
         libmeanfield/interface/preconditioning/equilibrium_coordinates.cppm
         libmeanfield/interface/preconditioning/preconditioning.cppm
+        libmeanfield/interface/normalization/plan.cppm
+        libmeanfield/interface/normalization/physical_riesz.cppm
+        libmeanfield/interface/normalization/operators.cppm
+        libmeanfield/interface/normalization/stellar_equilibrium.cppm
+        libmeanfield/interface/normalization/normalization.cppm
         libmeanfield/interface/operators/gravity_field.cppm
         libmeanfield/interface/operators/gravity_field_jacobian.cppm
         libmeanfield/interface/operators/kernels/gravity_kernels.cppm
@@ -197,6 +202,7 @@ target_sources(mean_field
         libmeanfield/interface/models/specifications.cppm
         libmeanfield/interface/models/typed_stellar_model.cppm
         libmeanfield/interface/models/compiled_fixed_mass.cppm
+        libmeanfield/interface/models/compiled_fixed_angular_momentum.cppm
         libmeanfield/interface/models/compiled_fixed_central_density.cppm
         libmeanfield/interface/surface/constant.cppm
         libmeanfield/interface/surface/dependencies.cppm
@@ -214,11 +220,13 @@ target_sources(mean_field
         libmeanfield/interface/operators/root_manifest.cppm
         libmeanfield/interface/operators/prepared_constraint.cppm
         libmeanfield/interface/operators/prepared_mass_normalization.cppm
+        libmeanfield/interface/operators/prepared_angular_momentum.cppm
         libmeanfield/interface/operators/prepared_central_density.cppm
         libmeanfield/interface/operators/prepared_centering_constraint.cppm
         libmeanfield/interface/operators/prepared_surface_constraint.cppm
         libmeanfield/interface/operators/prepared_stellar_equilibrium.cppm
-        libmeanfield/interface/operators/prepared_central_density_stellar_equilibrium.cppm
+        libmeanfield/interface/operators/stellar_equilibrium_compiler.cppm
+        libmeanfield/interface/operators/prepared_variadic_stellar_equilibrium.cppm
         libmeanfield/interface/equilibrium/stellar_discretization.cppm
         libmeanfield/interface/operators/stellar_equilibrium_problem.cppm
         libmeanfield/interface/seed/stellar_equilibrium_projection.cppm
@@ -303,10 +311,12 @@ add_executable(tests
         tests/operators/prepared_rotation_displacement_force_affine_deformation.cpp
         tests/operators/prepared_displacement_operator.cpp
         tests/operators/root_manifest.cpp
+        tests/operators/stellar_equilibrium_compiler.cpp
         tests/operators/prepared_central_density.cpp
         tests/operators/prepared_central_density_stellar_equilibrium.cpp
         tests/models/model_specifications.cpp
         tests/models/typed_stellar_model.cpp
+        tests/models/physics_specification_frontend.cpp
         tests/models/stellar_model.cpp
         tests/operators/stellar_equilibrium_system.cpp
         tests/deformation/contracts.cpp
@@ -315,6 +325,7 @@ add_executable(tests
         tests/deformation/radial_extensions.cpp
         tests/deformation/domain_deformation.cpp
         tests/operators/prepared_mass_normalization.cpp
+        tests/operators/prepared_angular_momentum.cpp
         tests/operators/prepared_stellar_equilibrium.cpp
         tests/utils/domain.cpp
         tests/field/field_base.cpp
@@ -327,8 +338,12 @@ add_executable(tests
         tests/preconditioning/material_surface.cpp
         tests/preconditioning/stellar_structure.cpp
         tests/preconditioning/specification_border.cpp
+        tests/extensions/fixed_magnetic_specific_energy.cpp
         tests/preconditioning/equilibrium_coordinates.cpp
         tests/preconditioning/stellar_equilibrium.cpp
+        tests/normalization/plan.cpp
+        tests/normalization/physical_riesz.cpp
+        tests/normalization/stellar_equilibrium.cpp
         tests/user-api/stellar_equilibrium.cpp
         tests/solver/preconditioning_diagnostics.cpp
 )
@@ -429,3 +444,8 @@ add_custom_target(
         DEPENDS mpi_tests
         USES_TERMINAL
 )
+
+# A deliberately separate, physics-developer-facing example.  Its targets
+# depend on MeanField, but none of its sources are part of the mean_field
+# library or the main regression-test executable.
+add_subdirectory(extension_example)

experiments/preconditioning_diagnostics.cpp

@@ -228,8 +228,7 @@ TEST_CASE(
     announce(communicator, "P0 extended baseline: preparing the complete equilibrium operator");
 
     const Clock::time_point operatorPreparationStart = Clock::now();
-    const operators::PreparedCentralDensityStellarEquilibriumReport preparation =
+    const auto preparation = problem.Prepare(projected.values, make_dependencies(), make_zero_rotation());
-        problem.Prepare(projected.values, make_dependencies(), make_zero_rotation());
     REQUIRE(preparation.assembledResidual);
     const double operatorPreparationSeconds = maximum_rank_seconds(operatorPreparationStart, communicator);
 

extension_example/CMakeLists.txt

@@ -0,0 +1,50 @@
+add_library(mean_field_extension_example)
+
+target_sources(
+        mean_field_extension_example
+        PUBLIC
+        FILE_SET CXX_MODULES FILES
+        ideal_gas_radiation.cppm
+        rotating_stellar_model.cppm
+)
+
+target_link_libraries(mean_field_extension_example PUBLIC mean_field)
+
+add_executable(extension_example_demo demo.cpp)
+target_link_libraries(extension_example_demo PRIVATE mean_field_extension_example)
+
+add_executable(
+        extension_example_tests
+        tests/ideal_gas_radiation.cpp
+        tests/rotating_stellar_model.cpp
+)
+target_link_libraries(
+        extension_example_tests
+        PRIVATE
+        mean_field_extension_example
+        Catch2::Catch2WithMain
+)
+
+catch_discover_tests(
+        extension_example_tests
+        TEST_PREFIX "extension_example::"
+        PROPERTIES LABELS "extension-example"
+)
+
+find_program(LATEXMK_EXECUTABLE latexmk)
+if (LATEXMK_EXECUTABLE)
+    add_custom_target(
+            extension_example_manual
+            COMMAND ${CMAKE_COMMAND} -E make_directory "${CMAKE_CURRENT_BINARY_DIR}/manual"
+            COMMAND
+            ${LATEXMK_EXECUTABLE}
+            -pdf
+            -interaction=nonstopmode
+            -halt-on-error
+            -outdir=${CMAKE_CURRENT_BINARY_DIR}/manual
+            "${CMAKE_CURRENT_SOURCE_DIR}/manual/physics_developer_manual.tex"
+            WORKING_DIRECTORY "${CMAKE_CURRENT_SOURCE_DIR}/manual"
+            COMMENT "Compiling the MeanField physics developer manual"
+            VERBATIM
+    )
+endif ()

extension_example/README.md

@@ -0,0 +1,55 @@
+# MeanField physics extension example
+
+This directory is a small, isolated example for physicists who want to extend
+MeanField without first learning its internal block-matrix machinery.
+
+Start in this order:
+
+1. Read `ideal_gas_radiation.cppm`. It implements a monatomic ideal gas plus
+   equilibrium radiation using the public EOS relation protocol.
+2. Read `rotating_stellar_model.cppm`. It composes that EOS with the existing
+   isobaric surface, fixed-total-mass invariant, and fixed-angular-momentum
+   invariant.
+3. Read and run `demo.cpp`.
+4. Read the tests. They show which claims should be compile-time contracts and
+   which claims require physical or numerical checks.
+5. Use `manual/physics_developer_manual.pdf` as the detailed guide. Its LaTeX
+   source is beside it.
+
+## The important boundary
+
+`makeRotatingStellarModel(...)` produces a valid, strongly typed stellar-model
+specification. The current equilibrium numerical core is still barotropic: it
+expects density to be closed by specific enthalpy alone. An ideal-gas plus
+radiation EOS depends independently on density and temperature, so a complete
+thermal equilibrium solve also needs a temperature or entropy field and its
+governing equation.
+
+The example therefore proves at compile time that model composition succeeds
+and that the present discretizer rejects this model. It does not disguise the
+thermal EOS as a polytrope or claim that a missing energy equation exists.
+
+## Build only this example
+
+From the repository root, configure as usual, then build only these targets:
+
+```sh
+cmake --build cmake-build-profile-homebrew-llvm \
+  --target extension_example_demo extension_example_tests
+```
+
+Run only the extension tests:
+
+```sh
+./cmake-build-profile-homebrew-llvm/extension_example/extension_example_tests
+```
+
+Compile a fresh manual into the build directory:
+
+```sh
+cmake --build cmake-build-profile-homebrew-llvm \
+  --target extension_example_manual
+```
+
+No source under `libmeanfield/` belongs to this example, and the extension test
+executable is separate from the main MeanField regression suite.

extension_example/demo.cpp

@@ -0,0 +1,43 @@
+#include <iomanip>
+#include <iostream>
+
+import mean_field;
+import mean_field_extension_example.rotating_stellar_model;
+
+int main() {
+    using namespace mean_field;
+    using namespace mean_field::extension_example;
+
+    const IdealGasRadiation equationOfState({
+        .meanMolecularWeight = 0.61,
+        .boltzmannConstant = 1.380649e-16,
+        .atomicMassUnit = 1.66053906660e-24,
+        .radiationConstant = 7.5657e-15
+    });
+
+    const dimensions::DensityValue density{10.0};       // g cm^-3
+    const dimensions::TemperatureValue temperature{1.5e7}; // K
+    const auto pressure = eos::evaluate<dimensions::quantity::Pressure>(
+        equationOfState,
+        density,
+        temperature
+    );
+
+    const auto model = makeRotatingStellarModel({
+        .equationOfState = equationOfState.parameters(),
+        .surfacePressure = dimensions::PressureValue{0.0},
+        .totalMass = dimensions::MassValue{1.0},
+        .totalAngularMomentum = dimensions::AngularMomentumValue{0.2}
+    });
+
+    std::cout << std::scientific
+              << "P(rho = 10 g cm^-3, T = 1.5e7 K) = "
+              << pressure.value() << " dyn cm^-2\n"
+              << "Compiled specification count = "
+              << model.specificationCount << '\n'
+              << "Current barotropic backend accepts this thermal model = "
+              << std::boolalpha
+              << currentEquilibriumBackendSupportsIdealGasRadiation << '\n';
+
+    return 0;
+}

extension_example/ideal_gas_radiation.cppm

@@ -0,0 +1,403 @@
+module;
+
+#include <cmath>
+#include <stdexcept>
+#include <string>
+
+export module mean_field_extension_example.ideal_gas_radiation;
+
+import mean_field;
+
+/*
+ * This file is intended to be read from top to bottom by a physicist who is
+ * adding an equation of state (EOS).  The comments explain the small amount
+ * of type-system vocabulary required by MeanField; the thermodynamics remain
+ * visible as ordinary equations.
+ */
+export namespace mean_field::extension_example {
+    namespace eos_quantity = mean_field::dimensions::quantity;
+
+    /*
+     * A relation is only a compile-time sentence:
+     *
+     *     output = f(input 1, input 2, ...).
+     *
+     * Input order is significant.  These declarations say that density is
+     * the first argument and temperature is the second argument.  They do not
+     * allocate data and have no runtime cost.
+     */
+    using PressureFromDensityAndTemperature = mean_field::eos::Relation<
+        eos_quantity::Pressure,
+        eos_quantity::Density,
+        eos_quantity::Temperature>;
+
+    using SpecificInternalEnergyFromDensityAndTemperature = mean_field::eos::Relation<
+        eos_quantity::SpecificInternalEnergy,
+        eos_quantity::Density,
+        eos_quantity::Temperature>;
+
+    using SpecificEnthalpyFromDensityAndTemperature = mean_field::eos::Relation<
+        eos_quantity::SpecificEnthalpy,
+        eos_quantity::Density,
+        eos_quantity::Temperature>;
+
+    /*
+     * A monatomic ideal gas plus equilibrium radiation:
+     *
+     *   R       = k_B / (mu m_u)
+     *   P_gas   = rho R T
+     *   P_rad   = a T^4 / 3
+     *   u       = (3/2) R T + a T^4 / rho
+     *   h       = u + P/rho
+     *           = (5/2) R T + 4 a T^4 / (3 rho)
+     *
+     * The scalar QuantityValue wrappers identify what a number means.  They
+     * intentionally do not perform unit conversion.  Every number supplied
+     * here must therefore use one coherent unit system.
+     */
+    class IdealGasRadiation final {
+    public:
+        struct Parameters final {
+            /* Mean particle mass in atomic-mass units. */
+            double meanMolecularWeight{0.61};
+
+            /* CGS defaults: erg K^-1, g, and erg cm^-3 K^-4. */
+            double boltzmannConstant{1.380649e-16};
+            double atomicMassUnit{1.66053906660e-24};
+            double radiationConstant{7.5657e-15};
+        };
+
+        /*
+         * This one alias makes the EOS a constitutive-law specification that
+         * can be placed directly in model::StellarModel(...).  There is no
+         * registry edit and no central list of EOS combinations to maintain.
+         */
+        using ModelDefinition = mean_field::eos::ConstitutiveLaw<IdealGasRadiation,"IdealGasRadiation">;
+
+        /*
+         * The catalog is the complete public claim made by this EOS.  If an
+         * evaluate overload below is missing or has the wrong argument order,
+         * eos::EquationOfStateModel<IdealGasRadiation> becomes false at
+         * compile time.
+         */
+        using Relations = mean_field::eos::RelationCatalog<
+            PressureFromDensityAndTemperature,
+            SpecificInternalEnergyFromDensityAndTemperature,
+            SpecificEnthalpyFromDensityAndTemperature
+        >;
+
+        struct PressureContributions final {
+            mean_field::dimensions::PressureValue gas;
+            mean_field::dimensions::PressureValue radiation;
+
+            [[nodiscard]] mean_field::dimensions::PressureValue total() const noexcept {
+                return gas + radiation;
+            }
+        };
+
+        explicit IdealGasRadiation(const Parameters parameters)
+            : m_parameters(validatedParameters(parameters)),
+              m_specificGasConstant(
+                  m_parameters.boltzmannConstant /(m_parameters.meanMolecularWeight * m_parameters.atomicMassUnit)
+              ) {}
+
+        [[nodiscard]] const Parameters &parameters() const noexcept {
+            return m_parameters;
+        }
+
+        [[nodiscard]] double specificGasConstant() const noexcept {
+            return m_specificGasConstant;
+        }
+
+        /*
+         * Named component functions are not required by the EOS protocol.
+         * They are provided because they make diagnostics and physics tests
+         * easier to read than repeated algebra in client code.
+         */
+        [[nodiscard]] PressureContributions pressureContributions(
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateMaterialState(density, temperature);
+
+            const double rho = density.value();
+            const double T   = temperature.value();
+            return PressureContributions{
+                .gas = mean_field::dimensions::PressureValue{rho * m_specificGasConstant * T},
+                .radiation = mean_field::dimensions::PressureValue{
+                    m_parameters.radiationConstant * fourthPower(T) / 3.0
+                }
+            };
+        }
+
+        [[nodiscard]] mean_field::dimensions::SpecificInternalEnergyValue gasSpecificInternalEnergy(
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateTemperature(temperature);
+            return mean_field::dimensions::SpecificInternalEnergyValue{
+                1.5 * m_specificGasConstant * temperature.value()
+            };
+        }
+
+        [[nodiscard]] mean_field::dimensions::SpecificInternalEnergyValue radiationSpecificInternalEnergy(
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateMaterialState(density, temperature);
+            return mean_field::dimensions::SpecificInternalEnergyValue{
+                m_parameters.radiationConstant * fourthPower(temperature.value()) / density.value()
+            };
+        }
+
+        /* The evaluate overloads implement the three declared relations. */
+        [[nodiscard]] mean_field::dimensions::PressureValue evaluate(
+            PressureFromDensityAndTemperature,
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            return pressureContributions(density, temperature).total();
+        }
+
+        [[nodiscard]] mean_field::dimensions::SpecificInternalEnergyValue evaluate(
+            SpecificInternalEnergyFromDensityAndTemperature,
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            const auto gas       = gasSpecificInternalEnergy(temperature);
+            const auto radiation = radiationSpecificInternalEnergy(density, temperature);
+            return gas + radiation;
+        }
+
+        [[nodiscard]] mean_field::dimensions::SpecificEnthalpyValue evaluate(
+            SpecificEnthalpyFromDensityAndTemperature,
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateMaterialState(density, temperature);
+
+            const double rho = density.value();
+            const double T   = temperature.value();
+            return mean_field::dimensions::SpecificEnthalpyValue{
+                2.5 * m_specificGasConstant * T +
+                4.0 * m_parameters.radiationConstant * fourthPower(T) / (3.0 * rho)
+            };
+        }
+
+        /*
+         * Jacobian entries are ordinary analytic partial derivatives.  The
+         * WithRespectTo tag prevents accidentally returning dP/dT from the
+         * overload that promised dP/drho.
+         */
+        [[nodiscard]] mean_field::eos::PartialDerivative<
+            eos_quantity::Pressure,
+            eos_quantity::Density>
+        partialDerivative(
+            PressureFromDensityAndTemperature,
+            mean_field::eos::WithRespectTo<eos_quantity::Density>,
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateMaterialState(density, temperature);
+            return mean_field::eos::PartialDerivative<
+                eos_quantity::Pressure,
+                eos_quantity::Density>{m_specificGasConstant * temperature.value()};
+        }
+
+        [[nodiscard]] mean_field::eos::PartialDerivative<
+            eos_quantity::Pressure,
+            eos_quantity::Temperature>
+        partialDerivative(
+            PressureFromDensityAndTemperature,
+            mean_field::eos::WithRespectTo<eos_quantity::Temperature>,
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateMaterialState(density, temperature);
+            const double T = temperature.value();
+            return mean_field::eos::PartialDerivative<
+                eos_quantity::Pressure,
+                eos_quantity::Temperature>{
+                density.value() * m_specificGasConstant +
+                4.0 * m_parameters.radiationConstant * cube(T) / 3.0
+            };
+        }
+
+        [[nodiscard]] mean_field::eos::PartialDerivative<
+            eos_quantity::SpecificInternalEnergy,
+            eos_quantity::Density>
+        partialDerivative(
+            SpecificInternalEnergyFromDensityAndTemperature,
+            mean_field::eos::WithRespectTo<eos_quantity::Density>,
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateMaterialState(density, temperature);
+            return mean_field::eos::PartialDerivative<
+                eos_quantity::SpecificInternalEnergy,
+                eos_quantity::Density>{
+                -m_parameters.radiationConstant * fourthPower(temperature.value()) /
+                square(density.value())
+            };
+        }
+
+        [[nodiscard]] mean_field::eos::PartialDerivative<
+            eos_quantity::SpecificInternalEnergy,
+            eos_quantity::Temperature>
+        partialDerivative(
+            SpecificInternalEnergyFromDensityAndTemperature,
+            mean_field::eos::WithRespectTo<eos_quantity::Temperature>,
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateMaterialState(density, temperature);
+            return mean_field::eos::PartialDerivative<
+                eos_quantity::SpecificInternalEnergy,
+                eos_quantity::Temperature>{
+                1.5 * m_specificGasConstant +
+                4.0 * m_parameters.radiationConstant * cube(temperature.value()) / density.value()
+            };
+        }
+
+        [[nodiscard]] mean_field::eos::PartialDerivative<
+            eos_quantity::SpecificEnthalpy,
+            eos_quantity::Density>
+        partialDerivative(
+            SpecificEnthalpyFromDensityAndTemperature,
+            mean_field::eos::WithRespectTo<eos_quantity::Density>,
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateMaterialState(density, temperature);
+            return mean_field::eos::PartialDerivative<
+                eos_quantity::SpecificEnthalpy,
+                eos_quantity::Density>{
+                -4.0 * m_parameters.radiationConstant * fourthPower(temperature.value()) /
+                (3.0 * square(density.value()))
+            };
+        }
+
+        [[nodiscard]] mean_field::eos::PartialDerivative<
+            eos_quantity::SpecificEnthalpy,
+            eos_quantity::Temperature>
+        partialDerivative(
+            SpecificEnthalpyFromDensityAndTemperature,
+            mean_field::eos::WithRespectTo<eos_quantity::Temperature>,
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) const {
+            validateMaterialState(density, temperature);
+            return mean_field::eos::PartialDerivative<
+                eos_quantity::SpecificEnthalpy,
+                eos_quantity::Temperature>{
+                2.5 * m_specificGasConstant +
+                16.0 * m_parameters.radiationConstant * cube(temperature.value()) /
+                    (3.0 * density.value())
+            };
+        }
+
+    private:
+        [[nodiscard]] static Parameters validatedParameters(const Parameters parameters) {
+            requirePositiveFinite(parameters.meanMolecularWeight, "mean molecular weight");
+            requirePositiveFinite(parameters.boltzmannConstant, "Boltzmann constant");
+            requirePositiveFinite(parameters.atomicMassUnit, "atomic mass unit");
+            requireNonnegativeFinite(parameters.radiationConstant, "radiation constant");
+            return parameters;
+        }
+
+        static void requirePositiveFinite(const double value, const char *name) {
+            if (!std::isfinite(value) || value <= 0.0) {
+                throw std::invalid_argument(
+                    std::string{"IdealGasRadiation requires a finite, positive "} + name + "."
+                );
+            }
+        }
+
+        static void requireNonnegativeFinite(const double value, const char *name) {
+            if (!std::isfinite(value) || value < 0.0) {
+                throw std::invalid_argument(
+                    std::string{"IdealGasRadiation requires a finite, nonnegative "} + name + "."
+                );
+            }
+        }
+
+        static void validateMaterialState(
+            const mean_field::dimensions::DensityValue density,
+            const mean_field::dimensions::TemperatureValue temperature
+        ) {
+            if (!std::isfinite(density.value()) || !std::isfinite(temperature.value())) {
+                throw mean_field::eos::EvaluationError{
+                    mean_field::eos::EvaluationErrorCode::nonfinite_input,
+                    "IdealGasRadiation requires finite density and temperature."
+                };
+            }
+            if (density.value() <= 0.0 || temperature.value() < 0.0) {
+                throw mean_field::eos::EvaluationError{
+                    mean_field::eos::EvaluationErrorCode::outside_domain,
+                    "IdealGasRadiation requires rho > 0 and T >= 0."
+                };
+            }
+        }
+
+        static void validateTemperature(const mean_field::dimensions::TemperatureValue temperature) {
+            if (!std::isfinite(temperature.value())) {
+                throw mean_field::eos::EvaluationError{
+                    mean_field::eos::EvaluationErrorCode::nonfinite_input,
+                    "IdealGasRadiation requires finite temperature."
+                };
+            }
+            if (temperature.value() < 0.0) {
+                throw mean_field::eos::EvaluationError{
+                    mean_field::eos::EvaluationErrorCode::outside_domain,
+                    "IdealGasRadiation requires T >= 0."
+                };
+            }
+        }
+
+        [[nodiscard]] static double square(const double value) noexcept {
+            return value * value;
+        }
+
+        [[nodiscard]] static double cube(const double value) noexcept {
+            return value * value * value;
+        }
+
+        [[nodiscard]] static double fourthPower(const double value) noexcept {
+            const double squared = square(value);
+            return squared * squared;
+        }
+
+        Parameters m_parameters;
+        double m_specificGasConstant;
+    };
+
+    /*
+     * These assertions are executable documentation.  They prove that the
+     * class and every derivative satisfy the public extension protocol.
+     */
+    static_assert(mean_field::models::SelfDescribingModelSpecification<IdealGasRadiation>);
+    static_assert(mean_field::eos::EquationOfStateModel<IdealGasRadiation>);
+    static_assert(mean_field::eos::SupportsPartialDerivative<
+                  IdealGasRadiation,
+                  PressureFromDensityAndTemperature,
+                  eos_quantity::Density>);
+    static_assert(mean_field::eos::SupportsPartialDerivative<
+                  IdealGasRadiation,
+                  PressureFromDensityAndTemperature,
+                  eos_quantity::Temperature>);
+    static_assert(mean_field::eos::SupportsPartialDerivative<
+                  IdealGasRadiation,
+                  SpecificInternalEnergyFromDensityAndTemperature,
+                  eos_quantity::Density>);
+    static_assert(mean_field::eos::SupportsPartialDerivative<
+                  IdealGasRadiation,
+                  SpecificInternalEnergyFromDensityAndTemperature,
+                  eos_quantity::Temperature>);
+    static_assert(mean_field::eos::SupportsPartialDerivative<
+                  IdealGasRadiation,
+                  SpecificEnthalpyFromDensityAndTemperature,
+                  eos_quantity::Density>);
+    static_assert(mean_field::eos::SupportsPartialDerivative<
+                  IdealGasRadiation,
+                  SpecificEnthalpyFromDensityAndTemperature,
+                  eos_quantity::Temperature>);
+} // namespace mean_field::extension_example

extension_example/rotating_stellar_model.cppm

@@ -0,0 +1,61 @@
+module;
+
+#include <array>
+#include <utility>
+
+export module mean_field_extension_example.rotating_stellar_model;
+
+export import mean_field_extension_example.ideal_gas_radiation;
+import mean_field;
+
+/*
+ * This file is the physics-facing composition layer.  It contains no block
+ * matrices, generated residual types, Jacobian indices, or preconditioner
+ * plumbing.  StellarModel infers those structural types from the four
+ * physical specifications passed to it.
+ */
+export namespace mean_field::extension_example {
+    struct RotatingStellarModelParameters final {
+        IdealGasRadiation::Parameters equationOfState;
+        mean_field::dimensions::PressureValue surfacePressure;
+        mean_field::dimensions::MassValue totalMass;
+        mean_field::dimensions::AngularMomentumValue totalAngularMomentum;
+        std::array<double, 3> rotationAxis{0.0, 0.0, 1.0};
+        std::array<double, 3> rotationCenter{0.0, 0.0, 0.0};
+    };
+
+    [[nodiscard]] auto makeRotatingStellarModel(const RotatingStellarModelParameters &parameters) {
+        return mean_field::model::StellarModel(
+            IdealGasRadiation(parameters.equationOfState),
+            mean_field::surface::Isobaric({.Psurf = parameters.surfacePressure}),
+            mean_field::integral::FixedTotalMass({.Mtotal = parameters.totalMass}),
+            mean_field::integral::FixedAngularMomentum({
+                .Jtotal = parameters.totalAngularMomentum,
+                .axis = parameters.rotationAxis,
+                .center = parameters.rotationCenter
+            })
+        );
+    }
+
+    using RotatingStellarModel = decltype(
+        makeRotatingStellarModel(std::declval<const RotatingStellarModelParameters &>())
+    );
+
+    static_assert(mean_field::model::StellarModelType<RotatingStellarModel>);
+    static_assert(RotatingStellarModel::symbolicallySquare);
+
+    /*
+     * Deliberate capability boundary:
+     *
+     * The specification above is a valid, strongly typed stellar model.  The
+     * current numerical equilibrium core, however, closes density through a
+     * barotropic relation rho(h).  This EOS instead needs an independent
+     * temperature or entropy field and its governing equation.  Keeping this
+     * assertion false prevents an example from suggesting that discretize()
+     * already implements thermal equilibrium when it does not.
+     */
+    inline constexpr bool currentEquilibriumBackendSupportsIdealGasRadiation =
+        mean_field::equilibrium::StellarEquilibriumModel<RotatingStellarModel>;
+
+    static_assert(!currentEquilibriumBackendSupportsIdealGasRadiation);
+} // namespace mean_field::extension_example

extension_example/tests/ideal_gas_radiation.cpp

@@ -0,0 +1,292 @@
+#include <algorithm>
+#include <array>
+#include <cmath>
+#include <concepts>
+#include <limits>
+#include <type_traits>
+#include <utility>
+
+#include <catch2/catch_approx.hpp>
+#include <catch2/catch_test_macros.hpp>
+
+import mean_field;
+import mean_field_extension_example.ideal_gas_radiation;
+
+namespace {
+    namespace dimensions = mean_field::dimensions;
+    namespace eos        = mean_field::eos;
+    namespace example    = mean_field::extension_example;
+
+    [[nodiscard]] example::IdealGasRadiation makeSimpleEquationOfState() {
+        /* R = k_B / (mu m_u) = 12 / (2 * 3) = 2. */
+        return example::IdealGasRadiation({
+            .meanMolecularWeight = 2.0,
+            .boltzmannConstant = 12.0,
+            .atomicMassUnit = 3.0,
+            .radiationConstant = 9.0
+        });
+    }
+
+    template <typename Function>
+    [[nodiscard]] double centeredDifference(
+        Function function,
+        const double point
+    ) {
+        const double step = std::cbrt(std::numeric_limits<double>::epsilon()) *
+                            std::max(1.0, std::abs(point));
+        return (function(point + step) - function(point - step)) / (2.0 * step);
+    }
+
+    template <typename EquationOfState>
+    concept CanEvaluatePressureWithReversedInputs = requires(
+        const EquationOfState &equationOfState,
+        const dimensions::TemperatureValue temperature,
+        const dimensions::DensityValue density
+    ) {
+        eos::evaluate<dimensions::quantity::Pressure>(equationOfState, temperature, density);
+    };
+} // namespace
+
+TEST_CASE("The extension satisfies the EOS protocol at compile time", "[extension-example][eos][type]") {
+    using EquationOfState = example::IdealGasRadiation;
+
+    STATIC_CHECK(mean_field::models::SelfDescribingModelSpecification<EquationOfState>);
+    STATIC_CHECK(eos::EquationOfStateModel<EquationOfState>);
+    STATIC_CHECK(eos::SupportsRelation<EquationOfState, example::PressureFromDensityAndTemperature>);
+    STATIC_CHECK(eos::SupportsRelation<EquationOfState, example::SpecificInternalEnergyFromDensityAndTemperature>);
+    STATIC_CHECK(eos::SupportsRelation<EquationOfState, example::SpecificEnthalpyFromDensityAndTemperature>);
+    STATIC_CHECK_FALSE(eos::BarotropicClosureEquationOfState<EquationOfState>);
+    STATIC_CHECK_FALSE(CanEvaluatePressureWithReversedInputs<EquationOfState>);
+
+    using PressureResult = decltype(eos::evaluate<dimensions::quantity::Pressure>(
+        std::declval<const EquationOfState &>(),
+        dimensions::DensityValue{1.0},
+        dimensions::TemperatureValue{1.0}
+    ));
+    STATIC_CHECK(std::same_as<PressureResult, dimensions::PressureValue>);
+}
+
+TEST_CASE("Gas and radiation terms reproduce the defining thermodynamics", "[extension-example][eos][physics]") {
+    const auto equationOfState = makeSimpleEquationOfState();
+    const dimensions::DensityValue density{4.0};
+    const dimensions::TemperatureValue temperature{2.0};
+
+    const auto pressureContributions = equationOfState.pressureContributions(density, temperature);
+    const auto pressure = eos::evaluate<dimensions::quantity::Pressure>(
+        equationOfState,
+        density,
+        temperature
+    );
+    const auto internalEnergy = eos::evaluate<dimensions::quantity::SpecificInternalEnergy>(
+        equationOfState,
+        density,
+        temperature
+    );
+    const auto enthalpy = eos::evaluate<dimensions::quantity::SpecificEnthalpy>(
+        equationOfState,
+        density,
+        temperature
+    );
+
+    CHECK(equationOfState.specificGasConstant() == Catch::Approx(2.0));
+    CHECK(pressureContributions.gas.value() == Catch::Approx(16.0));
+    CHECK(pressureContributions.radiation.value() == Catch::Approx(48.0));
+    CHECK(pressure.value() == Catch::Approx(64.0));
+    CHECK(internalEnergy.value() == Catch::Approx(42.0));
+    CHECK(enthalpy.value() == Catch::Approx(58.0));
+
+    /* This is the thermodynamic identity h = u + P/rho. */
+    CHECK(enthalpy.value() == Catch::Approx(internalEnergy.value() + pressure.value() / density.value()));
+}
+
+TEST_CASE("The gas and photon terms have their expected scaling laws", "[extension-example][eos][physics]") {
+    const auto equationOfState = makeSimpleEquationOfState();
+    const dimensions::DensityValue density{3.5};
+    const dimensions::TemperatureValue temperature{1.25};
+
+    const auto baseline = equationOfState.pressureContributions(density, temperature);
+    const auto doubledDensity = equationOfState.pressureContributions(
+        dimensions::DensityValue{2.0 * density.value()},
+        temperature
+    );
+    const auto doubledTemperature = equationOfState.pressureContributions(
+        density,
+        dimensions::TemperatureValue{2.0 * temperature.value()}
+    );
+
+    CHECK(doubledDensity.gas.value() == Catch::Approx(2.0 * baseline.gas.value()));
+    CHECK(doubledDensity.radiation.value() == Catch::Approx(baseline.radiation.value()));
+    CHECK(doubledTemperature.gas.value() == Catch::Approx(2.0 * baseline.gas.value()));
+    CHECK(doubledTemperature.radiation.value() == Catch::Approx(16.0 * baseline.radiation.value()));
+
+    const double crossoverTemperature = std::cbrt(
+        3.0 * density.value() * equationOfState.specificGasConstant() /
+        equationOfState.parameters().radiationConstant
+    );
+    const auto crossover = equationOfState.pressureContributions(
+        density,
+        dimensions::TemperatureValue{crossoverTemperature}
+    );
+    CHECK(crossover.gas.value() == Catch::Approx(crossover.radiation.value()).epsilon(2.0e-14));
+}
+
+TEST_CASE("All declared Jacobian entries match centered numerical derivatives",
+          "[extension-example][eos][derivative][numerical]") {
+    const auto equationOfState = example::IdealGasRadiation({
+        .meanMolecularWeight = 1.25,
+        .boltzmannConstant = 2.75,
+        .atomicMassUnit = 0.8,
+        .radiationConstant = 0.35
+    });
+
+    struct State final {
+        double density;
+        double temperature;
+    };
+    const std::array states{
+        State{.density = 0.4, .temperature = 0.7},
+        State{.density = 2.0, .temperature = 1.5},
+        State{.density = 11.0, .temperature = 3.0}
+    };
+
+    for (const State state : states) {
+        const dimensions::DensityValue density{state.density};
+        const dimensions::TemperatureValue temperature{state.temperature};
+
+        const auto pressureDensity = eos::partialDerivative<
+            dimensions::quantity::Pressure,
+            dimensions::quantity::Density>(equationOfState, density, temperature);
+        const auto pressureTemperature = eos::partialDerivative<
+            dimensions::quantity::Pressure,
+            dimensions::quantity::Temperature>(equationOfState, density, temperature);
+        const auto energyDensity = eos::partialDerivative<
+            dimensions::quantity::SpecificInternalEnergy,
+            dimensions::quantity::Density>(equationOfState, density, temperature);
+        const auto energyTemperature = eos::partialDerivative<
+            dimensions::quantity::SpecificInternalEnergy,
+            dimensions::quantity::Temperature>(equationOfState, density, temperature);
+        const auto enthalpyDensity = eos::partialDerivative<
+            dimensions::quantity::SpecificEnthalpy,
+            dimensions::quantity::Density>(equationOfState, density, temperature);
+        const auto enthalpyTemperature = eos::partialDerivative<
+            dimensions::quantity::SpecificEnthalpy,
+            dimensions::quantity::Temperature>(equationOfState, density, temperature);
+
+        const double numericalPressureDensity = centeredDifference(
+            [&](const double rho) {
+                return eos::evaluate<dimensions::quantity::Pressure>(
+                    equationOfState,
+                    dimensions::DensityValue{rho},
+                    temperature
+                ).value();
+            },
+            state.density
+        );
+        const double numericalPressureTemperature = centeredDifference(
+            [&](const double T) {
+                return eos::evaluate<dimensions::quantity::Pressure>(
+                    equationOfState,
+                    density,
+                    dimensions::TemperatureValue{T}
+                ).value();
+            },
+            state.temperature
+        );
+        const double numericalEnergyDensity = centeredDifference(
+            [&](const double rho) {
+                return eos::evaluate<dimensions::quantity::SpecificInternalEnergy>(
+                    equationOfState,
+                    dimensions::DensityValue{rho},
+                    temperature
+                ).value();
+            },
+            state.density
+        );
+        const double numericalEnergyTemperature = centeredDifference(
+            [&](const double T) {
+                return eos::evaluate<dimensions::quantity::SpecificInternalEnergy>(
+                    equationOfState,
+                    density,
+                    dimensions::TemperatureValue{T}
+                ).value();
+            },
+            state.temperature
+        );
+        const double numericalEnthalpyDensity = centeredDifference(
+            [&](const double rho) {
+                return eos::evaluate<dimensions::quantity::SpecificEnthalpy>(
+                    equationOfState,
+                    dimensions::DensityValue{rho},
+                    temperature
+                ).value();
+            },
+            state.density
+        );
+        const double numericalEnthalpyTemperature = centeredDifference(
+            [&](const double T) {
+                return eos::evaluate<dimensions::quantity::SpecificEnthalpy>(
+                    equationOfState,
+                    density,
+                    dimensions::TemperatureValue{T}
+                ).value();
+            },
+            state.temperature
+        );
+
+        constexpr double tolerance = 3.0e-9;
+        CHECK(pressureDensity.value() == Catch::Approx(numericalPressureDensity).epsilon(tolerance));
+        CHECK(pressureTemperature.value() == Catch::Approx(numericalPressureTemperature).epsilon(tolerance));
+        CHECK(energyDensity.value() == Catch::Approx(numericalEnergyDensity).epsilon(tolerance));
+        CHECK(energyTemperature.value() == Catch::Approx(numericalEnergyTemperature).epsilon(tolerance));
+        CHECK(enthalpyDensity.value() == Catch::Approx(numericalEnthalpyDensity).epsilon(tolerance));
+        CHECK(enthalpyTemperature.value() == Catch::Approx(numericalEnthalpyTemperature).epsilon(tolerance));
+    }
+}
+
+TEST_CASE("The physical domain is checked at the EOS boundary", "[extension-example][eos][domain]") {
+    const auto equationOfState = makeSimpleEquationOfState();
+    const double nan = std::numeric_limits<double>::quiet_NaN();
+
+    CHECK_THROWS_AS(
+        example::IdealGasRadiation({
+            .meanMolecularWeight = 0.0,
+            .boltzmannConstant = 1.0,
+            .atomicMassUnit = 1.0,
+            .radiationConstant = 1.0
+        }),
+        std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        example::IdealGasRadiation({
+            .meanMolecularWeight = 1.0,
+            .boltzmannConstant = 1.0,
+            .atomicMassUnit = 1.0,
+            .radiationConstant = -1.0
+        }),
+        std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        eos::evaluate<dimensions::quantity::Pressure>(
+            equationOfState,
+            dimensions::DensityValue{0.0},
+            dimensions::TemperatureValue{1.0}
+        ),
+        eos::EvaluationError
+    );
+    CHECK_THROWS_AS(
+        eos::evaluate<dimensions::quantity::Pressure>(
+            equationOfState,
+            dimensions::DensityValue{1.0},
+            dimensions::TemperatureValue{-1.0}
+        ),
+        eos::EvaluationError
+    );
+    CHECK_THROWS_AS(
+        eos::evaluate<dimensions::quantity::Pressure>(
+            equationOfState,
+            dimensions::DensityValue{nan},
+            dimensions::TemperatureValue{1.0}
+        ),
+        eos::EvaluationError
+    );
+}

extension_example/tests/rotating_stellar_model.cpp

@@ -0,0 +1,67 @@
+#include <concepts>
+#include <type_traits>
+
+#include <catch2/catch_test_macros.hpp>
+
+import mean_field;
+import mean_field_extension_example.rotating_stellar_model;
+
+TEST_CASE("The example EOS composes with existing stellar specifications",
+          "[extension-example][model][type]") {
+    using namespace mean_field;
+    namespace example = mean_field::extension_example;
+
+    const auto stellarModel = example::makeRotatingStellarModel({
+        .equationOfState = {
+            .meanMolecularWeight = 0.62,
+            .boltzmannConstant = 1.380649e-16,
+            .atomicMassUnit = 1.66053906660e-24,
+            .radiationConstant = 7.5657e-15
+        },
+        .surfacePressure = dimensions::PressureValue{0.0},
+        .totalMass = dimensions::MassValue{1.75},
+        .totalAngularMomentum = dimensions::AngularMomentumValue{0.3},
+        .rotationAxis = {0.0, 0.0, 4.0},
+        .rotationCenter = {0.1, -0.2, 0.3}
+    });
+    using Model = std::remove_cvref_t<decltype(stellarModel)>;
+
+    STATIC_CHECK(std::same_as<Model, example::RotatingStellarModel>);
+    STATIC_CHECK(model::StellarModelType<Model>);
+    STATIC_CHECK(Model::symbolicallySquare);
+    STATIC_CHECK(Model::specificationCount == 4);
+    STATIC_CHECK(std::same_as<model::EquationOfStateType<Model>, example::IdealGasRadiation>);
+    STATIC_CHECK(Model::template containsSpecification<integral::FixedTotalMass>);
+    STATIC_CHECK(Model::template containsSpecification<integral::FixedAngularMomentum>);
+    STATIC_CHECK(Model::template specificationRoleCount<models::SpecificationRole::constitutive_law> == 1);
+    STATIC_CHECK(Model::template specificationRoleCount<models::SpecificationRole::boundary_condition> == 1);
+    STATIC_CHECK(Model::template specificationRoleCount<models::SpecificationRole::invariant> == 2);
+
+    CHECK(stellarModel.equationOfState().parameters().meanMolecularWeight == 0.62);
+    CHECK(stellarModel.surfaceCondition().targetPressure() == dimensions::PressureValue{0.0});
+    CHECK(stellarModel.specification<integral::FixedTotalMass>().targetMass() == dimensions::MassValue{1.75});
+
+    const auto &angularMomentum = stellarModel.specification<integral::FixedAngularMomentum>();
+    CHECK(angularMomentum.targetAngularMomentum() == dimensions::AngularMomentumValue{0.3});
+    CHECK(angularMomentum.axis()[0] == 0.0);
+    CHECK(angularMomentum.axis()[1] == 0.0);
+    CHECK(angularMomentum.axis()[2] == 1.0);
+    CHECK(angularMomentum.center()[0] == 0.1);
+    CHECK(angularMomentum.center()[1] == -0.2);
+    CHECK(angularMomentum.center()[2] == 0.3);
+    CHECK(stellarModel.runtimeSpecificationDescriptors().size() == 4);
+}
+
+TEST_CASE("The example states the current thermal-runtime boundary explicitly",
+          "[extension-example][model][capability]") {
+    using Model = mean_field::extension_example::RotatingStellarModel;
+
+    /*
+     * This is not a failure of model composition.  It is the intended
+     * compile-time rejection of a thermal EOS by a currently barotropic
+     * numerical core.  See the manual section 'What compiles today'.
+     */
+    STATIC_CHECK(mean_field::model::StellarModelType<Model>);
+    STATIC_CHECK_FALSE(mean_field::extension_example::currentEquilibriumBackendSupportsIdealGasRadiation);
+    STATIC_CHECK_FALSE(mean_field::equilibrium::StellarEquilibriumModel<Model>);
+}

libmeanfield/impl/operators/prepared_angular_momentum.cpp

@@ -0,0 +1,590 @@
+module;
+
+#include <algorithm>
+#include <array>
+#include <cmath>
+#include <utility>
+
+#include <mfem.hpp>
+
+module mean_field;
+
+import :operators.prepared_angular_momentum;
+
+namespace {
+    using DomainSchema = mean_field::utils::domain::CoreEnvelopeVacuumDomainSchema;
+
+    [[nodiscard]] bool is_vacuum_attribute(const int attribute) {
+        return DomainSchema::template attribute_belongs_to<mean_field::utils::domain::Vacuum>(attribute);
+    }
+
+    void validate_finite_vector(const mfem::Vector &vector, const char *message) {
+        for (int index = 0; index < vector.Size(); ++index) {
+            MFEM_VERIFY(std::isfinite(vector(index)), message);
+        }
+    }
+
+    void true_to_local(
+        const mfem::ParFiniteElementSpace &finiteElementSpace,
+        const mfem::Vector &trueVector,
+        mfem::Vector &localVector
+    ) {
+        MFEM_VERIFY(trueVector.Size() == finiteElementSpace.GetTrueVSize(), "True vector has the wrong size.");
+        localVector.SetSize(finiteElementSpace.GetVSize());
+        const mfem::Operator *prolongation = finiteElementSpace.GetProlongationMatrix();
+        if (prolongation != nullptr) {
+            prolongation->Mult(trueVector, localVector);
+        } else {
+            localVector = trueVector;
+        }
+    }
+
+    [[nodiscard]] const mfem::IntegrationRule &get_moment_of_inertia_rule(
+        const mean_field::fem::FEM &f,
+        const mfem::FiniteElement &densityElement,
+        const mfem::ElementTransformation &transformation
+    ) {
+        using DensityField = mean_field::field::Field<mean_field::field::Density>;
+        MFEM_VERIFY(
+            densityElement.GetOrder() == mean_field::field::Density::Scalar::familyOrder,
+            "The angular-momentum element does not match the registered density field."
+        );
+        const mean_field::quadrature::Query query =
+            DensityField::make_query<mean_field::field::Density::Form::Quadrupole>(
+                mean_field::quadrature::QuadratureRole::discretization,
+                transformation.OrderW(),
+                std::array<int, 1>{2},
+                mean_field::utils::DOMAINS::STELLAR,
+                mean_field::quadrature::MappingKind::general
+            );
+        const auto resolution = f.quadratureFactory->get(query, transformation.GetGeometryType());
+        MFEM_VERIFY(
+            resolution.integration_rule != nullptr,
+            "The quadrature policy did not return an angular-momentum integration rule."
+        );
+        return *resolution.integration_rule;
+    }
+
+    void validate_shared_gravity_revisions(
+        const mean_field::operators::context::gravity_field::GravityFieldLinearizationContext &gravityContext,
+        const mean_field::operators::AngularMomentumDependencies &dependencies
+    ) {
+        MFEM_VERIFY(
+            gravityContext.IsPrepared(),
+            "PreparedAngularMomentumOperator requires the shared gravity context to be prepared first."
+        );
+        const auto &revisions = gravityContext.GetRevisions();
+        MFEM_VERIFY(
+            revisions.discretization.value == dependencies.discretization.revision &&
+                revisions.density.value == dependencies.density.revision &&
+                revisions.displacement.value == dependencies.displacement.revision,
+            "PreparedAngularMomentumOperator received revisions that do not match the shared gravity context."
+        );
+    }
+
+    void validate_identity_transition(
+        const mean_field::operators::AngularMomentumDependencyStamp &prepared,
+        const mean_field::operators::AngularMomentumDependencyStamp &requested,
+        const char *message
+    ) {
+        MFEM_VERIFY(prepared.identity == requested.identity || prepared.revision != requested.revision, message);
+    }
+} // namespace
+
+namespace mean_field::operators {
+    PreparedAngularMomentumOperator::PreparedAngularMomentumOperator(
+        const fem::FEM &f,
+        const mapping::DomainMapper &domainMapper,
+        const context::gravity_field::GravityFieldLinearizationContext &gravityContext,
+        models::CompiledFixedAngularMomentum constraint
+    )
+        : m_fem(f),
+          m_domainMapper(domainMapper),
+          m_gravityContext(gravityContext),
+          m_constraint(std::move(constraint)) {
+        MFEM_VERIFY(m_fem.mesh != nullptr, "PreparedAngularMomentumOperator requires a mesh.");
+        MFEM_VERIFY(
+            m_fem.mesh->Dimension() == 3 && m_domainMapper.GetDimension() == 3,
+            "PreparedAngularMomentumOperator currently requires a three-dimensional mapped domain."
+        );
+        MFEM_VERIFY(
+            m_fem.densityFes != nullptr && m_fem.displacementFes != nullptr &&
+                m_fem.compactificationFes != nullptr && m_fem.compactificationCoordinate != nullptr &&
+                m_fem.quadratureFactory != nullptr,
+            "PreparedAngularMomentumOperator requires density, displacement, compactification, and quadrature data."
+        );
+        MFEM_VERIFY(
+            m_gravityContext.GetDensityMap().full_size() == m_fem.densityFes->GetTrueVSize() &&
+                m_gravityContext.GetDisplacementMap().full_size() == m_fem.displacementFes->GetTrueVSize(),
+            "PreparedAngularMomentumOperator received incompatible shared FieldDof maps."
+        );
+        m_densityVariationTrue.SetSize(m_gravityContext.GetDensityMap().full_size());
+        m_displacementVariationTrue.SetSize(m_gravityContext.GetDisplacementMap().full_size());
+    }
+
+    PreparedAngularMomentumReport PreparedAngularMomentumOperator::Prepare(
+        const double angularVelocity,
+        const AngularMomentumDependencies &dependencies
+    ) {
+        MFEM_VERIFY(
+            std::isfinite(angularVelocity),
+            "PreparedAngularMomentumOperator requires a finite angular-velocity coordinate."
+        );
+        validate_shared_gravity_revisions(m_gravityContext, dependencies);
+
+        if (m_isPrepared) {
+            validate_identity_transition(
+                m_preparedDependencies.discretization,
+                dependencies.discretization,
+                "A new angular-momentum discretization identity must change its revision."
+            );
+            validate_identity_transition(
+                m_preparedDependencies.density,
+                dependencies.density,
+                "A new angular-momentum density identity must change its revision."
+            );
+            validate_identity_transition(
+                m_preparedDependencies.displacement,
+                dependencies.displacement,
+                "A new angular-momentum displacement identity must change its revision."
+            );
+            validate_identity_transition(
+                m_preparedDependencies.rotation,
+                dependencies.rotation,
+                "A new angular-momentum rotation identity must change its revision."
+            );
+        }
+
+        const bool rebuildStaticPlan =
+            !m_isPrepared || dependencies.discretization != m_preparedDependencies.discretization;
+        const bool refreshGeometry =
+            rebuildStaticPlan || dependencies.displacement != m_preparedDependencies.displacement;
+        const bool refreshDensity = rebuildStaticPlan || dependencies.density != m_preparedDependencies.density;
+        const bool updateAngularVelocity =
+            !m_isPrepared || dependencies.rotation != m_preparedDependencies.rotation ||
+            angularVelocity != m_angularVelocity;
+
+        m_isPrepared = false;
+        PreparedAngularMomentumReport report;
+        if (rebuildStaticPlan) {
+            BuildStaticPlan();
+            report.rebuiltStaticPlan = true;
+        }
+        if (refreshGeometry) {
+            RefreshGeometry(m_gravityContext.GetGeometryContext().GetDisplacementTrue());
+            report.refreshedGeometry = true;
+        }
+        if (refreshDensity) {
+            RefreshDensity(m_gravityContext.GetDensityTrue());
+            report.refreshedDensity = true;
+        }
+        if (updateAngularVelocity) {
+            m_angularVelocity = angularVelocity;
+            report.updatedAngularVelocity = true;
+        }
+        if (refreshGeometry || refreshDensity || updateAngularVelocity) {
+            AssembleResidual();
+            report.assembledResidual = true;
+        }
+
+        m_preparedDependencies = dependencies;
+        m_isPrepared = true;
+        return report;
+    }
+
+    void PreparedAngularMomentumOperator::BuildStaticPlan() {
+        m_elements.clear();
+        m_elements.reserve(m_fem.mesh->GetNE());
+        int localStellarElementCount = 0;
+        for (int elementId = 0; elementId < m_fem.mesh->GetNE(); ++elementId) {
+            mfem::ElementTransformation *transformation = m_fem.mesh->GetElementTransformation(elementId);
+            MFEM_VERIFY(transformation != nullptr, "Angular-momentum preparation received a null transformation.");
+            if (is_vacuum_attribute(transformation->Attribute)) {
+                continue;
+            }
+            ++localStellarElementCount;
+            m_elements.emplace_back();
+            ElementPAData &data = m_elements.back();
+            data.elementId = elementId;
+            data.densityDofTransformation = m_fem.densityFes->GetElementDofs(elementId, data.densityDofs);
+            data.displacementDofTransformation =
+                m_fem.displacementFes->GetElementVDofs(elementId, data.displacementDofs);
+            data.compactificationDofTransformation =
+                m_fem.compactificationFes->GetElementDofs(elementId, data.compactificationDofs);
+
+            const mfem::FiniteElement &densityElement = *m_fem.densityFes->GetFE(elementId);
+            const mfem::IntegrationRule &integrationRule =
+                get_moment_of_inertia_rule(m_fem, densityElement, *transformation);
+            data.quadraturePoints.resize(integrationRule.GetNPoints());
+            for (int quadraturePoint = 0; quadraturePoint < integrationRule.GetNPoints(); ++quadraturePoint) {
+                QuadraturePointData &point = data.quadraturePoints[quadraturePoint];
+                point.integrationPoint = integrationRule.IntPoint(quadraturePoint);
+                point.densityShape.SetSize(densityElement.GetDof());
+                densityElement.CalcShape(point.integrationPoint, point.densityShape);
+            }
+        }
+        int globalStellarElementCount = 0;
+        MPI_Allreduce(
+            &localStellarElementCount,
+            &globalStellarElementCount,
+            1,
+            MPI_INT,
+            MPI_SUM,
+            m_fem.mesh->GetComm()
+        );
+        MFEM_VERIFY(globalStellarElementCount > 0, "PreparedAngularMomentumOperator found no stellar elements.");
+    }
+
+    void PreparedAngularMomentumOperator::RefreshGeometry(const mfem::Vector &displacement) {
+        MFEM_VERIFY(
+            displacement.Size() == m_fem.displacementFes->GetTrueVSize(),
+            "Angular-momentum geometry has the wrong displacement size."
+        );
+        validate_finite_vector(displacement, "Angular-momentum geometry contains a non-finite displacement.");
+        mfem::Vector displacementLocal;
+        true_to_local(*m_fem.displacementFes, displacement, displacementLocal);
+        mapping::DomainMapper::Workspace workspace(m_fem.mesh->Dimension());
+
+        for (ElementPAData &data : m_elements) {
+            displacementLocal.GetSubVector(data.displacementDofs, data.baseDisplacement);
+            m_fem.compactificationCoordinate->GetSubVector(data.compactificationDofs, data.compactification);
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->InvTransformPrimal(data.baseDisplacement);
+            }
+            if (data.compactificationDofTransformation != nullptr) {
+                data.compactificationDofTransformation->InvTransformPrimal(data.compactification);
+            }
+            const mfem::FiniteElement &displacementElement = *m_fem.displacementFes->GetFE(data.elementId);
+            const mfem::FiniteElement &compactificationElement = *m_fem.compactificationFes->GetFE(data.elementId);
+            const mapping::ElementDisplacementData displacementData =
+                mapping::ElementDisplacementDataFromElementVDofs(displacementElement, data.baseDisplacement);
+            const mapping::ElementCompactificationData compactificationData(
+                compactificationElement,
+                data.compactification
+            );
+            const mapping::ElementMappingData mappingData{
+                .displacement = displacementData,
+                .compactification = compactificationData
+            };
+            mfem::ElementTransformation *transformation = m_fem.mesh->GetElementTransformation(data.elementId);
+            for (QuadraturePointData &point : data.quadraturePoints) {
+                const mapping::MappingStatus status = m_domainMapper.EvaluateVolume(
+                    mappingData,
+                    *transformation,
+                    point.integrationPoint,
+                    workspace,
+                    point.mappingContext
+                );
+                MFEM_VERIFY(
+                    status == mapping::MappingStatus::valid && !point.mappingContext.mapping.compactified,
+                    "Mapped angular-momentum geometry is invalid. Element: " << data.elementId
+                );
+                point.cylindricalRadiusSquared =
+                    CylindricalRadiusSquared(point.mappingContext.mapping.physical_position);
+            }
+        }
+    }
+
+    void PreparedAngularMomentumOperator::RefreshDensity(const mfem::Vector &density) {
+        MFEM_VERIFY(
+            density.Size() == m_fem.densityFes->GetTrueVSize(),
+            "Angular-momentum density has the wrong size."
+        );
+        validate_finite_vector(density, "Angular-momentum density contains a non-finite value.");
+        mfem::Vector densityLocal;
+        true_to_local(*m_fem.densityFes, density, densityLocal);
+        mfem::Vector elementDensity;
+        for (ElementPAData &data : m_elements) {
+            densityLocal.GetSubVector(data.densityDofs, elementDensity);
+            if (data.densityDofTransformation != nullptr) {
+                data.densityDofTransformation->InvTransformPrimal(elementDensity);
+            }
+            for (QuadraturePointData &point : data.quadraturePoints) {
+                point.density = elementDensity * point.densityShape;
+                MFEM_VERIFY(std::isfinite(point.density), "Angular-momentum quadrature density is non-finite.");
+            }
+        }
+    }
+
+    void PreparedAngularMomentumOperator::AssembleResidual() {
+        double localMomentOfInertia = 0.0;
+        for (const ElementPAData &data : m_elements) {
+            for (const QuadraturePointData &point : data.quadraturePoints) {
+                localMomentOfInertia += point.density * point.cylindricalRadiusSquared *
+                                        point.mappingContext.quadrature.weight;
+            }
+        }
+        m_momentOfInertia = GlobalSum(localMomentOfInertia);
+        MFEM_VERIFY(
+            std::isfinite(m_momentOfInertia) && m_momentOfInertia >= 0.0,
+            "PreparedAngularMomentumOperator assembled an invalid moment of inertia."
+        );
+        m_currentAngularMomentum = m_angularVelocity * m_momentOfInertia;
+        m_cachedResidual.SetSize(1);
+        m_cachedResidual(0) = m_currentAngularMomentum - m_constraint.targetAngularMomentum().value();
+        ++m_preparationCount;
+    }
+
+    void PreparedAngularMomentumOperator::BuildResidual(mfem::Vector &residual) const {
+        VerifyPrepared();
+        residual = m_cachedResidual;
+        ++m_residualApplicationCount;
+    }
+
+    double PreparedAngularMomentumOperator::EvaluateDensityMomentActionLocal(
+        const mfem::Vector &densityVariation
+    ) const {
+        MFEM_VERIFY(
+            densityVariation.Size() == m_fem.densityFes->GetTrueVSize(),
+            "Angular-momentum density action has the wrong true-vector size."
+        );
+        true_to_local(*m_fem.densityFes, densityVariation, m_densityVariationLocal);
+        double localAction = 0.0;
+        for (const ElementPAData &data : m_elements) {
+            m_densityVariationLocal.GetSubVector(data.densityDofs, m_elementDensityVariation);
+            if (data.densityDofTransformation != nullptr) {
+                data.densityDofTransformation->InvTransformPrimal(m_elementDensityVariation);
+            }
+            for (const QuadraturePointData &point : data.quadraturePoints) {
+                localAction += (m_elementDensityVariation * point.densityShape) *
+                               point.cylindricalRadiusSquared * point.mappingContext.quadrature.weight;
+            }
+        }
+        return localAction;
+    }
+
+    double PreparedAngularMomentumOperator::EvaluateDisplacementMomentActionLocal(
+        const mfem::Vector &displacementVariation
+    ) const {
+        MFEM_VERIFY(
+            displacementVariation.Size() == m_fem.displacementFes->GetTrueVSize(),
+            "Angular-momentum displacement action has the wrong true-vector size."
+        );
+        true_to_local(*m_fem.displacementFes, displacementVariation, m_displacementVariationLocal);
+        mapping::DomainMapper::Workspace workspace(m_fem.mesh->Dimension());
+        mapping::VolumeMappingVariation variation;
+        double localAction = 0.0;
+        for (const ElementPAData &data : m_elements) {
+            m_displacementVariationLocal.GetSubVector(data.displacementDofs, m_elementDisplacementVariation);
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->InvTransformPrimal(m_elementDisplacementVariation);
+            }
+            const mfem::FiniteElement &displacementElement = *m_fem.displacementFes->GetFE(data.elementId);
+            const mfem::FiniteElement &compactificationElement = *m_fem.compactificationFes->GetFE(data.elementId);
+            const mapping::ElementDisplacementData baseDisplacementData =
+                mapping::ElementDisplacementDataFromElementVDofs(displacementElement, data.baseDisplacement);
+            const mapping::ElementDisplacementData directionData =
+                mapping::ElementDisplacementDataFromElementVDofs(displacementElement, m_elementDisplacementVariation);
+            const mapping::ElementCompactificationData compactificationData(
+                compactificationElement,
+                data.compactification
+            );
+            const mapping::ElementMappingData mappingData{
+                .displacement = baseDisplacementData,
+                .compactification = compactificationData
+            };
+            mfem::ElementTransformation *transformation = m_fem.mesh->GetElementTransformation(data.elementId);
+            for (const QuadraturePointData &point : data.quadraturePoints) {
+                const mapping::MappingStatus status = m_domainMapper.EvaluateVolumeVariation(
+                    mappingData,
+                    directionData,
+                    *transformation,
+                    point.integrationPoint,
+                    point.mappingContext,
+                    workspace,
+                    variation
+                );
+                MFEM_VERIFY(
+                    status == mapping::MappingStatus::valid,
+                    "Mapped angular-momentum variation is invalid. Element: " << data.elementId
+                );
+                const double radiusSquaredVariation = CylindricalRadiusSquaredVariation(
+                    point.mappingContext.mapping.physical_position,
+                    variation.mapping.physical_position_variation
+                );
+                localAction += point.density *
+                               (radiusSquaredVariation * point.mappingContext.quadrature.weight +
+                                point.cylindricalRadiusSquared * variation.weight_variation);
+            }
+        }
+        return localAction;
+    }
+
+    void PreparedAngularMomentumOperator::ApplyDensityJacobianAction(
+        const mfem::Vector &densityVariation,
+        mfem::Vector &action
+    ) const {
+        VerifyPrepared();
+        MFEM_VERIFY(
+            densityVariation.Size() == m_gravityContext.GetDensityMap().reduced_size(),
+            "Angular-momentum density action has the wrong reduced size."
+        );
+        validate_finite_vector(densityVariation, "Angular-momentum density direction is non-finite.");
+        m_gravityContext.GetDensityMap().scatter(densityVariation, m_densityVariationTrue);
+        action.SetSize(1);
+        action(0) = m_angularVelocity * GlobalSum(EvaluateDensityMomentActionLocal(m_densityVariationTrue));
+        ++m_actionStatistics.densityApplications;
+    }
+
+    void PreparedAngularMomentumOperator::ApplyDisplacementJacobianAction(
+        const mfem::Vector &displacementVariation,
+        mfem::Vector &action
+    ) const {
+        VerifyPrepared();
+        MFEM_VERIFY(
+            displacementVariation.Size() == m_gravityContext.GetDisplacementMap().reduced_size(),
+            "Angular-momentum displacement action has the wrong reduced size."
+        );
+        validate_finite_vector(displacementVariation, "Angular-momentum displacement direction is non-finite.");
+        m_gravityContext.GetDisplacementMap().scatter(displacementVariation, m_displacementVariationTrue);
+        action.SetSize(1);
+        action(0) = m_angularVelocity *
+                    GlobalSum(EvaluateDisplacementMomentActionLocal(m_displacementVariationTrue));
+        ++m_actionStatistics.displacementApplications;
+    }
+
+    void PreparedAngularMomentumOperator::ApplyAngularVelocityJacobianAction(
+        const double angularVelocityVariation,
+        mfem::Vector &action
+    ) const {
+        VerifyPrepared();
+        MFEM_VERIFY(std::isfinite(angularVelocityVariation), "Angular-velocity direction is non-finite.");
+        action.SetSize(1);
+        action(0) = m_momentOfInertia * angularVelocityVariation;
+        ++m_actionStatistics.angularVelocityApplications;
+    }
+
+    void PreparedAngularMomentumOperator::ApplyCompleteJacobianAction(
+        const mfem::Vector &densityVariation,
+        const mfem::Vector &displacementVariation,
+        const double angularVelocityVariation,
+        mfem::Vector &action
+    ) const {
+        VerifyPrepared();
+        MFEM_VERIFY(
+            densityVariation.Size() == m_gravityContext.GetDensityMap().reduced_size() &&
+                displacementVariation.Size() == m_gravityContext.GetDisplacementMap().reduced_size(),
+            "Angular-momentum complete action has incompatible reduced coordinates."
+        );
+        validate_finite_vector(densityVariation, "Angular-momentum density direction is non-finite.");
+        validate_finite_vector(displacementVariation, "Angular-momentum displacement direction is non-finite.");
+        MFEM_VERIFY(std::isfinite(angularVelocityVariation), "Angular-velocity direction is non-finite.");
+        m_gravityContext.GetDensityMap().scatter(densityVariation, m_densityVariationTrue);
+        m_gravityContext.GetDisplacementMap().scatter(displacementVariation, m_displacementVariationTrue);
+        const double localMomentAction = EvaluateDensityMomentActionLocal(m_densityVariationTrue) +
+                                         EvaluateDisplacementMomentActionLocal(m_displacementVariationTrue);
+        action.SetSize(1);
+        action(0) = m_angularVelocity * GlobalSum(localMomentAction) +
+                    m_momentOfInertia * angularVelocityVariation;
+        ++m_actionStatistics.completeApplications;
+    }
+
+    double PreparedAngularMomentumOperator::CylindricalRadiusSquared(
+        const mfem::Vector &physicalPosition
+    ) const noexcept {
+        const auto &axis = m_constraint.specification().axis();
+        const auto &center = m_constraint.specification().center();
+        double radiusSquared = 0.0;
+        double axialPosition = 0.0;
+        for (int component = 0; component < 3; ++component) {
+            const double relative = physicalPosition(component) - center[static_cast<std::size_t>(component)];
+            radiusSquared += relative * relative;
+            axialPosition += axis[static_cast<std::size_t>(component)] * relative;
+        }
+        return std::max(0.0, radiusSquared - axialPosition * axialPosition);
+    }
+
+    double PreparedAngularMomentumOperator::CylindricalRadiusSquaredVariation(
+        const mfem::Vector &physicalPosition,
+        const mfem::Vector &physicalPositionVariation
+    ) const noexcept {
+        const auto &axis = m_constraint.specification().axis();
+        const auto &center = m_constraint.specification().center();
+        double relativeDotVariation = 0.0;
+        double axialPosition = 0.0;
+        double axialVariation = 0.0;
+        for (int component = 0; component < 3; ++component) {
+            const double relative = physicalPosition(component) - center[static_cast<std::size_t>(component)];
+            relativeDotVariation += relative * physicalPositionVariation(component);
+            axialPosition += axis[static_cast<std::size_t>(component)] * relative;
+            axialVariation += axis[static_cast<std::size_t>(component)] * physicalPositionVariation(component);
+        }
+        return 2.0 * (relativeDotVariation - axialPosition * axialVariation);
+    }
+
+    double PreparedAngularMomentumOperator::GlobalSum(const double localValue) const {
+        double globalValue = 0.0;
+        MPI_Allreduce(&localValue, &globalValue, 1, MPI_DOUBLE, MPI_SUM, m_fem.mesh->GetComm());
+        return globalValue;
+    }
+
+    bool PreparedAngularMomentumOperator::IsPrepared() const noexcept {
+        if (!m_isPrepared || !m_gravityContext.IsPrepared()) {
+            return false;
+        }
+        const auto &revisions = m_gravityContext.GetRevisions();
+        return revisions.discretization.value == m_preparedDependencies.discretization.revision &&
+               revisions.density.value == m_preparedDependencies.density.revision &&
+               revisions.displacement.value == m_preparedDependencies.displacement.revision;
+    }
+
+    double PreparedAngularMomentumOperator::GetMomentOfInertia() const {
+        VerifyPrepared();
+        return m_momentOfInertia;
+    }
+
+    double PreparedAngularMomentumOperator::GetAngularVelocity() const {
+        VerifyPrepared();
+        return m_angularVelocity;
+    }
+
+    double PreparedAngularMomentumOperator::GetCurrentAngularMomentum() const {
+        VerifyPrepared();
+        return m_currentAngularMomentum;
+    }
+
+    double PreparedAngularMomentumOperator::GetTargetAngularMomentum() const noexcept {
+        return m_constraint.targetAngularMomentum().value();
+    }
+
+    physics::RigidRotation PreparedAngularMomentumOperator::GetRotation() const {
+        VerifyPrepared();
+        return m_constraint.makeRotation(m_angularVelocity);
+    }
+
+    AngularMomentumConstraintReport PreparedAngularMomentumOperator::GetConstraintReport() const {
+        VerifyPrepared();
+        const double target = GetTargetAngularMomentum();
+        const double residual = m_currentAngularMomentum - target;
+        return {
+            .targetAngularMomentum = target,
+            .achievedAngularMomentum = m_currentAngularMomentum,
+            .momentOfInertia = m_momentOfInertia,
+            .angularVelocity = m_angularVelocity,
+            .dimensionalResidual = residual,
+            .scaledResidual = residual / std::max(std::abs(target), 1.0e-300)
+        };
+    }
+
+    std::uint64_t PreparedAngularMomentumOperator::GetPreparationCount() const noexcept {
+        return m_preparationCount;
+    }
+
+    std::uint64_t PreparedAngularMomentumOperator::GetResidualApplicationCount() const noexcept {
+        return m_residualApplicationCount;
+    }
+
+    const PreparedAngularMomentumActionStatistics &
+    PreparedAngularMomentumOperator::GetActionStatistics() const noexcept {
+        return m_actionStatistics;
+    }
+
+    const models::CompiledFixedAngularMomentum &
+    PreparedAngularMomentumOperator::GetCompiledConstraint() const noexcept {
+        return m_constraint;
+    }
+
+    void PreparedAngularMomentumOperator::VerifyPrepared() const {
+        MFEM_VERIFY(IsPrepared(), "The angular-momentum invariant must be prepared before application.");
+    }
+} // namespace mean_field::operators

libmeanfield/impl/operators/prepared_central_density_stellar_equilibrium.cpp

@@ -1,223 +0,0 @@
-module;
-
-#include <array>
-#include <cmath>
-#include <cstdint>
-#include <memory>
-#include <utility>
-
-#include <mfem.hpp>
-
-module mean_field;
-
-import :operators.prepared_central_density_stellar_equilibrium;
-
-namespace {
-    using DomainSchema = mean_field::utils::domain::CoreEnvelopeVacuumDomainSchema;
-    using PhysicalForm = mean_field::utils::blocks::surface_deformed_stellar_equilibrium_form;
-    using BorderedForm = mean_field::operators::CentralDensityStellarEquilibriumForm;
-
-    [[nodiscard]] std::array<
-        int,
-        BorderedForm::value_block_count>
-    make_value_sizes(const mean_field::operators::StellarEquilibriumLayout &physicalLayout) {
-        std::array<int, BorderedForm::value_block_count> sizes{};
-        for (int block = 0; block < PhysicalForm::value_block_count; ++block) {
-            sizes[block] = physicalLayout.value_offsets()[block + 1] - physicalLayout.value_offsets()[block];
-        }
-        sizes[PhysicalForm::value_block_count] = 1;
-        return sizes;
-    }
-
-    [[nodiscard]] std::array<
-        int,
-        BorderedForm::residual_block_count>
-    make_residual_sizes(const mean_field::operators::StellarEquilibriumLayout &physicalLayout) {
-        std::array<int, BorderedForm::residual_block_count> sizes{};
-        for (int block = 0; block < PhysicalForm::residual_block_count; ++block) {
-            sizes[block] = physicalLayout.residual_offsets()[block + 1] - physicalLayout.residual_offsets()[block];
-        }
-        sizes[PhysicalForm::residual_block_count] = 1;
-        return sizes;
-    }
-
-    [[nodiscard]] mean_field::operators::CentralDensityDependencies
-    make_phase_dependencies(const mean_field::operators::StellarEquilibriumDependencies &dependencies) {
-        return {.enthalpy = {.identity = dependencies.enthalpy.identity, .revision = dependencies.enthalpy.revision}};
-    }
-
-    void validate_finite_scalar(
-        const double value,
-        const char *message
-    ) {
-        MFEM_VERIFY(std::isfinite(value), message);
-    }
-} // namespace
-
-namespace mean_field::operators {
-    field::FieldPointDofMap PreparedCentralDensityStellarEquilibriumOperator::MakeCenterDofMap(const fem::FEM &f) {
-        MFEM_VERIFY(
-            f.mesh != nullptr && f.enthalpyFes != nullptr,
-            "The central-density phase requires the mesh and enthalpy finite-element space."
-        );
-        const field::FieldDofMap enthalpyMap = field::make_field_dof_map<field::Enthalpy, DomainSchema>(*f.enthalpyFes);
-        mfem::Vector origin(f.mesh->SpaceDimension());
-        origin = 0.0;
-        return field::make_field_point_dof_map<field::Enthalpy>(*f.enthalpyFes, enthalpyMap, origin, 1.0e-12);
-    }
-
-    PreparedCentralDensityStellarEquilibriumOperator::PreparedCentralDensityStellarEquilibriumOperator(
-        fem::FEM &f,
-        std::unique_ptr<PreparedStellarEquilibriumOperator> physicalOperator,
-        models::CompiledFixedCentralDensity centralDensity,
-        field::FieldPointDofMap centerDof
-    )
-        : mfem::Operator(
-              physicalOperator->Height() + 1,
-              physicalOperator->Width() + 1
-          ),
-          m_physicalOperator(std::move(physicalOperator)),
-          m_centralDensity(std::move(centralDensity)),
-          m_phaseConstraint(
-              std::move(centerDof),
-              f.mesh->GetComm()
-          ),
-          m_rootManifest(
-              make_value_sizes(m_physicalOperator->GetLayout()),
-              make_residual_sizes(m_physicalOperator->GetLayout()),
-              m_physicalOperator->GetTargetMass(),
-              m_physicalOperator->GetSurfaceConstraintOperator().GetPhysicalCondition().targetPressure,
-              m_physicalOperator->GetSurfaceConstraintOperator().GetSurfaceRows().size(),
-              CentralDensityManifestInput{
-                  .targetDensity  = m_centralDensity.targetDensity().value(),
-                  .targetEnthalpy = m_centralDensity.targetEnthalpy().value(),
-                  .centerDofCount = 1
-              }
-          ) {
-        MFEM_VERIFY(
-            Width() == m_rootManifest.layout().value_offsets().Last() &&
-                Height() == m_rootManifest.layout().residual_offsets().Last(),
-            "The central-density bordered root has inconsistent dimensions."
-        );
-    }
-
-    PreparedCentralDensityStellarEquilibriumReport PreparedCentralDensityStellarEquilibriumOperator::Prepare(
-        const mfem::Vector &state,
-        const StellarEquilibriumDependencies &dependencies,
-        const physics::RigidRotation &rotation
-    ) {
-        MFEM_VERIFY(state.Size() == Width(), "The central-density bordered root received a state with the wrong size.");
-        const auto stateView        = m_rootManifest.stateView(state);
-        const mfem::Vector enthalpy = stateView.block(utils::blocks::enthalpy_field.specific_term);
-        const mfem::Vector border   = stateView.block(utils::blocks::fixed_central_density_phase.central_value_term);
-        validate_finite_scalar(border(0), "The central-density bordered root received a non-finite border value.");
-
-        mfem::Vector physicalState(const_cast<mfem::real_t *>(state.GetData()), m_physicalOperator->Width());
-
-        m_isPrepared = false;
-        PreparedCentralDensityStellarEquilibriumReport report;
-        report.physical = m_physicalOperator->Prepare(physicalState, dependencies, rotation);
-        report.phase =
-            m_phaseConstraint.Prepare(m_centralDensity, enthalpy, border(0), make_phase_dependencies(dependencies));
-
-        if (report.physical.assembledResidual || report.phase.DidAnyWork() || m_cachedResidual.Size() != Height()) {
-            AssembleResidual();
-            report.assembledResidual = true;
-        }
-
-        m_isPrepared = true;
-        return report;
-    }
-
-    void PreparedCentralDensityStellarEquilibriumOperator::AssembleResidual() {
-        mfem::Vector physicalResidual;
-        m_physicalOperator->BuildResidual(physicalResidual);
-
-        m_cachedResidual.SetSize(Height());
-        m_cachedResidual = 0.0;
-        mfem::Vector physicalDestination(m_cachedResidual.GetData(), physicalResidual.Size());
-        physicalDestination           = physicalResidual;
-
-        const auto residualView       = m_rootManifest.residualView(m_cachedResidual);
-        mfem::Vector enthalpyResidual = residualView.block(utils::blocks::enthalpy_field.specific_term);
-        mfem::Vector phaseResidual = residualView.block(utils::blocks::fixed_central_density_phase.central_value_term);
-        m_phaseConstraint.AddResidual(enthalpyResidual, phaseResidual);
-    }
-
-    void PreparedCentralDensityStellarEquilibriumOperator::BuildResidual(mfem::Vector &residual) const {
-        VerifyPrepared();
-        residual = m_cachedResidual;
-    }
-
-    void PreparedCentralDensityStellarEquilibriumOperator::Mult(
-        const mfem::Vector &direction,
-        mfem::Vector &action
-    ) const {
-        VerifyPrepared();
-        MFEM_VERIFY(
-            direction.Size() == Width(), "The central-density bordered root received a direction with the wrong size."
-        );
-        const auto directionView             = m_rootManifest.directionView(direction);
-        const mfem::Vector enthalpyDirection = directionView.block(utils::blocks::enthalpy_field.specific_term);
-        const mfem::Vector borderDirection =
-            directionView.block(utils::blocks::fixed_central_density_phase.central_value_term);
-        validate_finite_scalar(
-            borderDirection(0), "The central-density bordered root received a non-finite border direction."
-        );
-
-        mfem::Vector physicalDirection(const_cast<mfem::real_t *>(direction.GetData()), m_physicalOperator->Width());
-        mfem::Vector physicalAction;
-        m_physicalOperator->Mult(physicalDirection, physicalAction);
-
-        action.SetSize(Height());
-        action = 0.0;
-        mfem::Vector physicalDestination(action.GetData(), physicalAction.Size());
-        physicalDestination         = physicalAction;
-
-        const auto actionView       = m_rootManifest.residualView(action);
-        mfem::Vector enthalpyAction = actionView.block(utils::blocks::enthalpy_field.specific_term);
-        mfem::Vector phaseAction    = actionView.block(utils::blocks::fixed_central_density_phase.central_value_term);
-        m_phaseConstraint.ApplyJacobian(
-            {.enthalpyVariation = enthalpyDirection, .borderVariation = borderDirection(0)},
-            {.enthalpyAction = enthalpyAction, .phaseAction = phaseAction}
-        );
-    }
-
-    bool PreparedCentralDensityStellarEquilibriumOperator::IsPrepared() const noexcept {
-        return m_isPrepared && m_physicalOperator->IsPrepared() && m_phaseConstraint.IsPrepared();
-    }
-
-    const CentralDensityStellarEquilibriumLayout &
-    PreparedCentralDensityStellarEquilibriumOperator::GetLayout() const noexcept {
-        return m_rootManifest.layout();
-    }
-
-    const CentralDensityStellarEquilibriumRootManifest &
-    PreparedCentralDensityStellarEquilibriumOperator::GetRootManifest() const noexcept {
-        return m_rootManifest;
-    }
-
-    const PreparedStellarEquilibriumOperator &
-    PreparedCentralDensityStellarEquilibriumOperator::GetPhysicalOperator() const noexcept {
-        return *m_physicalOperator;
-    }
-
-    const PreparedCentralDensityConstraint &
-    PreparedCentralDensityStellarEquilibriumOperator::GetCentralDensityConstraint() const noexcept {
-        return m_phaseConstraint;
-    }
-
-    RootConstraintReport PreparedCentralDensityStellarEquilibriumOperator::GetFixedMassReport() const {
-        VerifyPrepared();
-        return m_physicalOperator->GetFixedMassReport();
-    }
-
-    CentralDensityConstraintReport PreparedCentralDensityStellarEquilibriumOperator::GetCentralDensityReport() const {
-        VerifyPrepared();
-        return m_phaseConstraint.GetConstraintReport();
-    }
-
-    void PreparedCentralDensityStellarEquilibriumOperator::VerifyPrepared() const {
-        MFEM_VERIFY(IsPrepared(), "The central-density bordered root must be prepared before application.");
-    }
-} // namespace mean_field::operators

libmeanfield/impl/operators/prepared_hydrostatic_equilibrium.cpp

@@ -903,6 +903,47 @@ namespace mean_field::operators {
         ++m_algebraicJacobianStatistics.bernoulliConstantApplications;
     }
 
+    void PreparedHydrostaticEquilibriumOperator::ApplyRotationAmplitudeJacobianAction(
+        const double fractionalAngularVelocityVariation,
+        mfem::Vector &action
+    ) const {
+        VerifyPrepared();
+        MFEM_VERIFY(
+            std::isfinite(fractionalAngularVelocityVariation),
+            "Prepared hydrostatic rotation-amplitude Jacobian received a non-finite variation."
+        );
+
+        mfem::Vector localAction(m_fem.enthalpyFes->GetVSize());
+        localAction = 0.0;
+        mfem::Vector weightedVariation;
+        mfem::Vector elementAction;
+
+        for (const ElementPAData &data : m_elements) {
+            const int quadraturePointCount = data.quadratureWeights.Size();
+            MFEM_VERIFY(
+                data.rotationPotential.Size() == quadraturePointCount,
+                "Prepared hydrostatic rotation-amplitude Jacobian has stale rotation data."
+            );
+            weightedVariation.SetSize(quadraturePointCount);
+            for (int quadraturePoint = 0; quadraturePoint < quadraturePointCount; ++quadraturePoint) {
+                weightedVariation(quadraturePoint) =
+                    -2.0 * fractionalAngularVelocityVariation * data.quadratureWeights(quadraturePoint) *
+                    data.rotationPotential(quadraturePoint);
+            }
+            elementAction.SetSize(data.enthalpyDofs.Size());
+            data.enthalpyBasis.MultTranspose(weightedVariation, elementAction);
+            if (data.enthalpyDofTransformation != nullptr) {
+                data.enthalpyDofTransformation->TransformDual(elementAction);
+            }
+            localAction.AddElementVector(data.enthalpyDofs, elementAction);
+        }
+
+        local_to_true(*m_fem.enthalpyFes, localAction, m_fullEnthalpyAction);
+        action.SetSize(m_context.GetEnthalpyMap().reduced_size());
+        m_context.GetEnthalpyMap().gather(m_fullEnthalpyAction, action);
+        ++m_algebraicJacobianStatistics.rotationAmplitudeApplications;
+    }
+
     void PreparedHydrostaticEquilibriumOperator::ApplyAlgebraicJacobianAction(
         const mfem::Vector &enthalpyVariation,
         const mfem::Vector &gravityPotentialVariation,

libmeanfield/impl/operators/prepared_stellar_equilibrium.cpp

@@ -356,8 +356,11 @@ namespace mean_field::operators {
           m_rootManifest(
               constructionData.valueSizes,
               constructionData.residualSizes,
-              fixedMassConstraint.targetMass().value(),
+              StellarEquilibriumSpecificationModel{
-              surfaceConstraint.descriptor().targetPressure,
+                  equationOfState,
+                  surface::Isobaric{
+                      dimensions::PressureValue{surfaceConstraint.descriptor().targetPressure}},
+                  fixedMassConstraint.specification()},
               constructionData.pressureSurfaceRows.size()
           ),
           m_gravityStateOffsets(constructionData.gravityStateOffsets),
@@ -431,6 +434,7 @@ namespace mean_field::operators {
         m_fullMechanicalAction.SetSize(m_domainDeformation.volumeDisplacementSize());
         m_surfaceShapeAction.SetSize(m_domainDeformation.parameterCount());
         m_pullbackDerivativeAction.SetSize(m_domainDeformation.parameterCount());
+        m_densityVolumeIntegralAction.SetSize(1);
 
         m_gravityState                 = 0.0;
         m_gravityDirection             = 0.0;
@@ -441,6 +445,7 @@ namespace mean_field::operators {
         m_fullMechanicalAction         = 0.0;
         m_surfaceShapeAction           = 0.0;
         m_pullbackDerivativeAction     = 0.0;
+        m_densityVolumeIntegralAction  = 0.0;
     }
 
     PreparedStellarEquilibriumReport PreparedStellarEquilibriumOperator::Prepare(
@@ -494,13 +499,13 @@ namespace mean_field::operators {
 
         const auto rootState              = m_rootManifest.stateView(state);
 
-        const mfem::Vector reducedDensity = rootState.block(utils::blocks::density_field.mass_term);
+        const auto reducedDensity = rootState.block(utils::blocks::density_field.mass_term);
-        const mfem::Vector surfaceDeformationParameters =
+        const auto surfaceDeformationParameters =
             rootState.block(utils::blocks::surface_deformation_field.parameters_term);
-        const mfem::Vector gravityGradient  = rootState.block(utils::blocks::gravity_field.gradient_term);
+        const auto gravityGradient  = rootState.block(utils::blocks::gravity_field.gradient_term);
-        const mfem::Vector gravityPotential = rootState.block(utils::blocks::gravity_field.poisson_term);
+        const auto gravityPotential = rootState.block(utils::blocks::gravity_field.poisson_term);
-        const mfem::Vector reducedEnthalpy  = rootState.block(utils::blocks::enthalpy_field.specific_term);
+        const auto reducedEnthalpy  = rootState.block(utils::blocks::enthalpy_field.specific_term);
-        const mfem::Vector bernoulli =
+        const auto bernoulli =
             rootState.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term);
 
         const bool generatedGeometryChanged =
@@ -633,13 +638,13 @@ namespace mean_field::operators {
 
         const auto rootDirection                   = m_rootManifest.directionView(direction);
 
-        const mfem::Vector reducedDensityDirection = rootDirection.block(utils::blocks::density_field.mass_term);
+        const auto reducedDensityDirection = rootDirection.block(utils::blocks::density_field.mass_term);
-        const mfem::Vector surfaceDeformationDirection =
+        const auto surfaceDeformationDirection =
             rootDirection.block(utils::blocks::surface_deformation_field.parameters_term);
-        const mfem::Vector gravityGradientDirection  = rootDirection.block(utils::blocks::gravity_field.gradient_term);
+        const auto gravityGradientDirection  = rootDirection.block(utils::blocks::gravity_field.gradient_term);
-        const mfem::Vector gravityPotentialDirection = rootDirection.block(utils::blocks::gravity_field.poisson_term);
+        const auto gravityPotentialDirection = rootDirection.block(utils::blocks::gravity_field.poisson_term);
-        const mfem::Vector reducedEnthalpyDirection  = rootDirection.block(utils::blocks::enthalpy_field.specific_term);
+        const auto reducedEnthalpyDirection  = rootDirection.block(utils::blocks::enthalpy_field.specific_term);
-        const mfem::Vector bernoulliDirection =
+        const auto bernoulliDirection =
             rootDirection.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term);
 
         m_domainDeformation.applyJacobian(
@@ -794,6 +799,41 @@ namespace mean_field::operators {
         return m_massNormalizationOperator;
     }
 
+    double PreparedStellarEquilibriumOperator::ApplyDensityVolumeIntegralDensityAction(
+        const mfem::Vector &densityDirection
+    ) const {
+        VerifyPrepared();
+        m_massNormalizationOperator.ApplyDensityJacobianAction(
+            densityDirection,
+            m_densityVolumeIntegralAction
+        );
+        MFEM_VERIFY(
+            m_densityVolumeIntegralAction.Size() == 1,
+            "The density-volume integral must produce one global scalar."
+        );
+        return m_densityVolumeIntegralAction(0);
+    }
+
+    double PreparedStellarEquilibriumOperator::ApplyDensityVolumeIntegralSurfaceShapeAction(
+        const mfem::Vector &surfaceShapeDirection
+    ) const {
+        VerifyPrepared();
+        m_domainDeformation.applyJacobian(
+            m_surfaceDeformationParameters,
+            surfaceShapeDirection,
+            m_volumeDisplacementDirection
+        );
+        m_massNormalizationOperator.ApplyDisplacementJacobianAction(
+            m_volumeDisplacementDirection,
+            m_densityVolumeIntegralAction
+        );
+        MFEM_VERIFY(
+            m_densityVolumeIntegralAction.Size() == 1,
+            "The density-volume shape derivative must produce one global scalar."
+        );
+        return m_densityVolumeIntegralAction(0);
+    }
+
     const PreparedPressureSurfaceConstraint &
     PreparedStellarEquilibriumOperator::GetSurfaceConstraintOperator() const noexcept {
         return m_surfaceConstraintOperator;

libmeanfield/impl/seed/stellar_equilibrium_projection.cpp

@@ -3,6 +3,7 @@ module;
 #include <algorithm>
 #include <cmath>
 #include <limits>
+#include <numbers>
 #include <stdexcept>
 
 #include <mfem.hpp>
@@ -133,21 +134,15 @@ namespace {
 
 namespace mean_field::seed::detail {
     ProjectedRadialFields projectRadialFields(
-        const equilibrium::StellarDiscretization &discretization,
+        fem::FEM &finiteElementModel,
         const RadialProfile &profile,
         const dimensions::MassValue targetMass,
-        const dimensions::PressureValue targetSurfacePressure,
         const StellarEquilibriumProjectionOptions &options
     ) {
         validate_profile(profile);
         if (!std::isfinite(options.surfaceRadiusRelativeTolerance) || options.surfaceRadiusRelativeTolerance < 0.0) {
             throw std::invalid_argument("The surface-radius projection tolerance must be finite and nonnegative.");
         }
-        if (targetSurfacePressure.value() != 0.0) {
-            throw std::invalid_argument("A Lane-Emden radial seed requires a zero-pressure isobaric surface.");
-        }
-
-        fem::FEM &finiteElementModel           = discretization.finiteElementModel();
         const SurfaceRadiusRange surfaceRadius = measure_surface_radius(finiteElementModel);
         const double targetRadius              = profile.stellarRadius.value();
         const double comparisonScale           = std::max({targetRadius, surfaceRadius.maximum, 1.0e-300});
@@ -185,6 +180,20 @@ namespace mean_field::seed::detail {
         const physics::GravitySolution gravitySolution =
             physics::solve_gravity_field(finiteElementModel, options.gravity, densityField, displacementField);
 
+        double radialMomentIntegral = 0.0;
+        for (int index = 0; index + 1 < profile.radius.Size(); ++index) {
+            const double leftRadius = profile.radius(index);
+            const double rightRadius = profile.radius(index + 1);
+            const double leftIntegrand = profile.density(index) * std::pow(leftRadius, 4);
+            const double rightIntegrand = profile.density(index + 1) * std::pow(rightRadius, 4);
+            radialMomentIntegral +=
+                0.5 * (rightRadius - leftRadius) * (leftIntegrand + rightIntegrand);
+        }
+        const double sphericalMomentOfInertia = (8.0 * std::numbers::pi / 3.0) * radialMomentIntegral;
+        if (!std::isfinite(sphericalMomentOfInertia) || sphericalMomentOfInertia <= 0.0) {
+            throw std::runtime_error("The radial profile has no finite, positive moment of inertia.");
+        }
+
         const field::FieldDofGridFunctionAdapter densityAdapter =
             field::make_field_dof_grid_function_adapter<field::Density, DomainSchema>(*finiteElementModel.densityFes);
         const field::FieldDofGridFunctionAdapter enthalpyAdapter =
@@ -203,7 +212,8 @@ namespace mean_field::seed::detail {
             .gravityGradient   = gravityFluxAdapter.gather(gravitySolution.gradPhi),
             .gravityPotential  = gravityPotentialAdapter.gather(gravitySolution.phi),
             .specificEnthalpy  = enthalpyAdapter.gather(enthalpyField),
-            .bernoulliConstant = -utils::G * targetMass.value() / targetRadius
+            .bernoulliConstant = -utils::G * targetMass.value() / targetRadius,
+            .sphericalMomentOfInertia = sphericalMomentOfInertia
         };
     }
 } // namespace mean_field::seed::detail

libmeanfield/interface/equilibrium/stellar_discretization.cppm

@@ -1,12 +1,16 @@
 module;
 
+#include <concepts>
 #include <memory>
 #include <stdexcept>
+#include <type_traits>
+#include <utility>
 
 export module mean_field:equilibrium.stellar_discretization;
 
 export import :fem;
 export import :mapping.domain_mapper;
+export import :normalization.physical_riesz;
 
 export namespace mean_field::equilibrium {
     /*
@@ -18,26 +22,78 @@ export namespace mean_field::equilibrium {
      * mutable field workspaces. Separating those workspaces is a prerequisite
      * for shared discretization ownership by solved Structure objects.
      */
-    class StellarDiscretization final {
+    template <normalization::NormalizationPrescription Normalization>
+    class StellarDiscretizationFor final {
     public:
-        explicit StellarDiscretization(fem::FEM &finiteElementModel)
+        using NormalizationPrescriptionType = std::remove_cvref_t<Normalization>;
-            : StellarDiscretization(
+
+        explicit StellarDiscretizationFor(fem::FEM &finiteElementModel)
+            requires std::same_as<NormalizationPrescriptionType, normalization::Unnormalized>
+            : StellarDiscretizationFor(
                   finiteElementModel,
-                  RequireDomainMapper(finiteElementModel)
+                  RequireDomainMapper(finiteElementModel),
+                  normalization::Unnormalized{}
               ) {
         }
 
-        StellarDiscretization(
+        StellarDiscretizationFor(
             fem::FEM &finiteElementModel,
             const mapping::DomainMapper &domainMapper
+        )
+            requires std::same_as<NormalizationPrescriptionType, normalization::Unnormalized>
+            : StellarDiscretizationFor(
+                  finiteElementModel,
+                  domainMapper,
+                  normalization::Unnormalized{}
+              ) {
+        }
+
+        StellarDiscretizationFor(
+            fem::FEM &,
+            mapping::DomainMapper &&
+        ) requires std::same_as<NormalizationPrescriptionType, normalization::Unnormalized> = delete;
+
+        StellarDiscretizationFor(
+            fem::FEM &,
+            const mapping::DomainMapper &&
+        ) requires std::same_as<NormalizationPrescriptionType, normalization::Unnormalized> = delete;
+
+        StellarDiscretizationFor(
+            fem::FEM &finiteElementModel,
+            NormalizationPrescriptionType normalizationPrescription
+        )
+            : StellarDiscretizationFor(
+                  finiteElementModel,
+                  RequireDomainMapper(finiteElementModel),
+                  std::move(normalizationPrescription)
+              ) {
+        }
+
+        StellarDiscretizationFor(
+            fem::FEM &finiteElementModel,
+            const mapping::DomainMapper &domainMapper,
+            NormalizationPrescriptionType normalizationPrescription
         )
             : m_finiteElementModel(std::addressof(finiteElementModel)),
-              m_domainMapper(std::addressof(domainMapper)) {
+              m_domainMapper(std::addressof(domainMapper)),
+              m_normalizationPrescription(std::move(normalizationPrescription)) {
             if (!finiteElementModel.okay()) {
                 throw std::invalid_argument("A stellar discretization requires a complete finite-element model.");
             }
         }
 
+        StellarDiscretizationFor(
+            fem::FEM &,
+            mapping::DomainMapper &&,
+            NormalizationPrescriptionType
+        ) = delete;
+
+        StellarDiscretizationFor(
+            fem::FEM &,
+            const mapping::DomainMapper &&,
+            NormalizationPrescriptionType
+        ) = delete;
+
         [[nodiscard]] fem::FEM &finiteElementModel() const noexcept {
             return *m_finiteElementModel;
         }
@@ -46,6 +102,10 @@ export namespace mean_field::equilibrium {
             return *m_domainMapper;
         }
 
+        [[nodiscard]] const NormalizationPrescriptionType &normalizationPrescription() const noexcept {
+            return m_normalizationPrescription;
+        }
+
         [[nodiscard]] bool isCurrent() const noexcept {
             return m_finiteElementModel != nullptr && m_domainMapper != nullptr && m_finiteElementModel->okay();
         }
@@ -60,5 +120,64 @@ export namespace mean_field::equilibrium {
 
         fem::FEM *m_finiteElementModel;
         const mapping::DomainMapper *m_domainMapper;
+        NormalizationPrescriptionType m_normalizationPrescription;
     };
+
+    template <normalization::NormalizationPrescription Normalization>
+    StellarDiscretizationFor(fem::FEM &, Normalization)
+        -> StellarDiscretizationFor<std::remove_cvref_t<Normalization>>;
+
+    template <normalization::NormalizationPrescription Normalization>
+    StellarDiscretizationFor(fem::FEM &, const mapping::DomainMapper &, Normalization)
+        -> StellarDiscretizationFor<std::remove_cvref_t<Normalization>>;
+
+    using StellarDiscretization = StellarDiscretizationFor<normalization::Unnormalized>;
+
+    template <typename Candidate> struct IsStellarDiscretization : std::false_type { };
+
+    template <normalization::NormalizationPrescription Normalization>
+    struct IsStellarDiscretization<StellarDiscretizationFor<Normalization>> : std::true_type { };
+
+    template <typename Candidate>
+    concept StellarDiscretizationType = IsStellarDiscretization<std::remove_cvref_t<Candidate>>::value;
+
+    template <normalization::NormalizationPrescription Normalization>
+    [[nodiscard]] auto makeStellarDiscretization(
+        fem::FEM &finiteElementModel,
+        Normalization normalizationPrescription
+    ) {
+        return StellarDiscretizationFor<std::remove_cvref_t<Normalization>>{
+            finiteElementModel,
+            std::move(normalizationPrescription)
+        };
+    }
+
+    template <normalization::NormalizationPrescription Normalization>
+    [[nodiscard]] auto makeStellarDiscretization(
+        fem::FEM &finiteElementModel,
+        const mapping::DomainMapper &domainMapper,
+        Normalization normalizationPrescription
+    ) {
+        return StellarDiscretizationFor<std::remove_cvref_t<Normalization>>{
+            finiteElementModel,
+            domainMapper,
+            std::move(normalizationPrescription)
+        };
+    }
+
+    template <normalization::NormalizationPrescription Normalization>
+    StellarDiscretizationFor<std::remove_cvref_t<Normalization>>
+    makeStellarDiscretization(
+        fem::FEM &,
+        mapping::DomainMapper &&,
+        Normalization
+    ) = delete;
+
+    template <normalization::NormalizationPrescription Normalization>
+    StellarDiscretizationFor<std::remove_cvref_t<Normalization>>
+    makeStellarDiscretization(
+        fem::FEM &,
+        const mapping::DomainMapper &&,
+        Normalization
+    ) = delete;
 } // namespace mean_field::equilibrium

libmeanfield/interface/field/field_registry.cppm

@@ -231,6 +231,28 @@ export namespace mean_field::field {
         static_assert(constraintsAreValid);
     };
 
+    // Scalar angular speed generated by FixedAngularMomentum. The axis and
+    // center belong to the compiled invariant, so the nonlinear coordinate
+    // contains only the signed speed along that fixed unit axis.
+    struct AngularVelocity {
+        static constexpr std::string_view name = "angular_velocity";
+
+        using PhysicalQuantity                 = dimensions::quantity::AngularVelocity;
+        using Support                          = NonSpatialSupport;
+
+        struct Scalar final : GlobalScalarQ {
+            static constexpr std::string_view symbol = "Omega";
+        };
+
+        using Quantities                          = TypeList<Scalar>;
+        using Constraints                         = TypeList<>;
+        using FormList                            = TypeList<>;
+
+        static constexpr bool constraintsAreValid = validate_constraints(Constraints{});
+
+        static_assert(constraintsAreValid);
+    };
+
     // Solver border generated by FixedCentralDensity. This is deliberately a
     // non-spatial numerical coordinate rather than a physical stellar field.
     struct CentralDensityBorder {

libmeanfield/interface/mean_field.cppm

@@ -27,6 +27,7 @@ export import :quadrature.mfem;
 export import :solver.fields;
 export import :solver.preconditioning_diagnostics;
 export import :preconditioning;
+export import :normalization;
 export import :utils.blocks;
 export import :operators.gravity_field;
 export import :operators.gravity_field_jacobian;
@@ -60,6 +61,7 @@ export import :model.structure.polytropic;
 export import :model.specifications;
 export import :model.typed_stellar;
 export import :model.compiled_fixed_mass;
+export import :model.compiled_fixed_angular_momentum;
 export import :model.compiled_fixed_central_density;
 export import :eos.quantities;
 export import :eos.relations;
@@ -85,11 +87,13 @@ export import :model.stellar;
 export import :operators.root_manifest;
 export import :operators.prepared_constraint;
 export import :operators.prepared_mass_normalization;
+export import :operators.prepared_angular_momentum;
 export import :operators.prepared_central_density;
 export import :operators.prepared_centering_constraint;
 export import :operators.prepared_surface_constraint;
 export import :operators.prepared_stellar_equilibrium;
-export import :operators.prepared_central_density_stellar_equilibrium;
+export import :operators.stellar_equilibrium_compiler;
+export import :operators.prepared_variadic_stellar_equilibrium;
 export import :equilibrium.stellar_discretization;
 export import :operators.stellar_equilibrium_problem;
 export import :seed.stellar_equilibrium_projection;

libmeanfield/interface/models/compiled_fixed_angular_momentum.cppm

@@ -0,0 +1,74 @@
+module;
+
+#include <concepts>
+#include <cstddef>
+#include <type_traits>
+
+#include <mfem.hpp>
+
+export module mean_field:model.compiled_fixed_angular_momentum;
+
+export import :field.registry;
+export import :model.compiled_fixed_mass;
+export import :physics.rigid_rotation;
+export import :utils.blocks;
+
+export namespace mean_field::models {
+    using FixedAngularMomentumLayoutRequest = ConstraintLayoutRequest<
+        FixedAngularMomentum,
+        PhysicalCoordinateFor<FixedAngularMomentum>,
+        ResidualFor<FixedAngularMomentum>,
+        utils::blocks::fixed_angular_momentum::angular_velocity::value,
+        utils::blocks::fixed_angular_momentum::angular_velocity::residual,
+        utils::blocks::fixed_angular_momentum::angular_velocity,
+        utils::blocks::density::mass::value,
+        utils::blocks::surface_deformation::parameters::value,
+        utils::blocks::fixed_angular_momentum::angular_velocity::value>;
+
+    class CompiledFixedAngularMomentum final {
+    public:
+        using SpecificationType   = FixedAngularMomentum;
+        using LayoutRequest       = FixedAngularMomentumLayoutRequest;
+        using AngularVelocityType  = typename LayoutRequest::GeneratedValueType;
+        using ResidualType         = typename LayoutRequest::GeneratedResidualType;
+        using AngularVelocityField = field::AngularVelocity;
+
+        explicit CompiledFixedAngularMomentum(const FixedAngularMomentum specification) noexcept
+            : m_specification(specification) {
+        }
+
+        [[nodiscard]] const FixedAngularMomentum &specification() const noexcept {
+            return m_specification;
+        }
+
+        [[nodiscard]] dimensions::AngularMomentumValue targetAngularMomentum() const noexcept {
+            return m_specification.targetAngularMomentum();
+        }
+
+        [[nodiscard]] physics::RigidRotation makeRotation(const double angularVelocity) const {
+            mfem::Vector velocity(3);
+            mfem::Vector center(3);
+            for (int component = 0; component < 3; ++component) {
+                velocity(component) = angularVelocity * m_specification.axis()[static_cast<std::size_t>(component)];
+                center(component)   = m_specification.center()[static_cast<std::size_t>(component)];
+            }
+            return {velocity, center};
+        }
+
+        [[nodiscard]] static consteval LayoutRequest layoutRequest() noexcept {
+            return {};
+        }
+
+    private:
+        FixedAngularMomentum m_specification;
+    };
+
+    [[nodiscard]] inline CompiledFixedAngularMomentum compileConstraint(
+        const FixedAngularMomentum specification
+    ) noexcept {
+        return CompiledFixedAngularMomentum{specification};
+    }
+
+    static_assert(ConstraintLayoutRequestType<FixedAngularMomentumLayoutRequest>);
+    static_assert(CompiledConstraint<CompiledFixedAngularMomentum>);
+} // namespace mean_field::models

libmeanfield/interface/models/typed_stellar_model.cppm

@@ -14,21 +14,41 @@ export namespace mean_field::model {
     template <typename SpecificationSet> class StellarModel;
 
     template <models::ModelSpecification... CanonicalSpecifications>
+        requires models::ValidModelSpecificationPack<CanonicalSpecifications...> &&
+                 models::SpecificationOperatorSignature<
+                     models::detail::SpecificationSetStorage<CanonicalSpecifications...>>::symbolicallySquare
     class StellarModel<models::detail::SpecificationSetStorage<CanonicalSpecifications...>> final {
     public:
         using SpecificationTypes = models::detail::SpecificationSetStorage<CanonicalSpecifications...>;
         using OperatorSignature  = models::SpecificationOperatorSignature<SpecificationTypes>;
         using Storage            = models::Model<CanonicalSpecifications...>;
+        using EquationOfStateType =
+            models::SpecificationForRoleT<models::SpecificationRole::constitutive_law, SpecificationTypes>;
 
         static constexpr std::size_t specificationCount = sizeof...(CanonicalSpecifications);
         static constexpr bool symbolicallySquare        = Storage::symbolicallySquare;
-        static constexpr bool hasCompleteEquilibriumCompiler                   = Storage::hasCompleteRootCompiler;
+        static constexpr bool hasCompleteEquilibriumDeclaration =
-        static constexpr models::EquilibriumSystemCompilation compilationClass = Storage::compilationClass;
+            Storage::hasCompleteEquilibriumDeclaration;
+
+        template <models::SpecificationRole Role>
+        using SpecificationsForRole = models::SpecificationsForRoleT<Role, SpecificationTypes>;
+
+        template <models::SpecificationRole Role>
+            requires models::HasUniqueSpecificationForRole<Role, SpecificationTypes>
+        using SpecificationForRole = models::SpecificationForRoleT<Role, SpecificationTypes>;
+
+        template <models::SpecificationRole Role>
+        static constexpr std::size_t specificationRoleCount = models::specificationRoleCount<Role, SpecificationTypes>;
+
+        template <models::SpecificationRole Role>
+        static constexpr bool hasSpecificationsForRole = models::HasSpecificationsForRole<Role, SpecificationTypes>;
+
+        template <models::SpecificationRole Role>
+        static constexpr bool hasUniqueSpecificationForRole =
+            models::HasUniqueSpecificationForRole<Role, SpecificationTypes>;
 
         template <typename... Arguments>
-            requires std::constructible_from<
+            requires std::constructible_from<Storage, Arguments...>
-                Storage,
-                Arguments...>
         explicit StellarModel(Arguments &&...arguments) : m_specifications(std::forward<Arguments>(arguments)...) {
         }
 
@@ -41,6 +61,25 @@ export namespace mean_field::model {
         template <models::ModelSpecification Specification>
         static constexpr bool containsSpecification = Storage::template containsSpecification<Specification>;
 
+        template <models::SpecificationRole Role>
+            requires models::HasUniqueSpecificationForRole<Role, SpecificationTypes>
+        [[nodiscard]] const models::SpecificationForRoleT<Role, SpecificationTypes> &
+        specificationForRole() const noexcept {
+            using Specification = models::SpecificationForRoleT<Role, SpecificationTypes>;
+            return specification<Specification>();
+        }
+
+        [[nodiscard]] const EquationOfStateType &equationOfState() const noexcept {
+            return specificationForRole<models::SpecificationRole::constitutive_law>();
+        }
+
+        template <typename = void>
+            requires models::HasUniqueSpecificationForRole<models::SpecificationRole::boundary_condition,
+                                                           SpecificationTypes>
+        [[nodiscard]] const auto &surfaceCondition() const noexcept {
+            return specificationForRole<models::SpecificationRole::boundary_condition>();
+        }
+
         [[nodiscard]] static constexpr std::span<const models::RuntimeSpecificationDescriptor>
         runtimeSpecificationDescriptors() noexcept {
             return Storage::runtimeSpecificationDescriptors();
@@ -56,11 +95,69 @@ export namespace mean_field::model {
         -> StellarModel<models::SpecificationSet<std::remove_cvref_t<Specifications>...>>;
 
     namespace detail {
-        template <typename Candidate> struct IsStellarModel : std::false_type { };
+        template <typename Candidate, typename = void> struct IsStellarModel : std::false_type {};
 
-        template <typename SpecificationSet> struct IsStellarModel<StellarModel<SpecificationSet>> : std::true_type { };
+        template <typename SpecificationSet>
+        struct IsStellarModel<
+            StellarModel<SpecificationSet>,
+            std::void_t<typename StellarModel<SpecificationSet>::SpecificationTypes,
+                        typename StellarModel<SpecificationSet>::OperatorSignature,
+                        decltype(StellarModel<SpecificationSet>::specificationCount),
+                        decltype(StellarModel<SpecificationSet>::hasCompleteEquilibriumDeclaration)>>
+            : std::true_type {};
+
+        template <models::SpecificationRole Role, typename Candidate, bool = IsStellarModel<Candidate>::value>
+        struct StellarModelRoleSelection {
+            using Types = models::ModelTypeList<>;
+
+            static constexpr std::size_t count = 0;
+        };
+
+        template <models::SpecificationRole Role, typename Candidate>
+        struct StellarModelRoleSelection<Role, Candidate, true> {
+            using Types = models::SpecificationsForRoleT<Role, typename Candidate::SpecificationTypes>;
+
+            static constexpr std::size_t count =
+                models::specificationRoleCount<Role, typename Candidate::SpecificationTypes>;
+        };
     } // namespace detail
 
     template <typename Candidate>
     concept StellarModelType = detail::IsStellarModel<std::remove_cvref_t<Candidate>>::value;
+
+    template <models::SpecificationRole Role, typename Candidate>
+    inline constexpr std::size_t specificationRoleCount =
+        detail::StellarModelRoleSelection<Role, std::remove_cvref_t<Candidate>>::count;
+
+    template <models::SpecificationRole Role, typename Candidate>
+    concept HasSpecificationsForRole = StellarModelType<Candidate> && specificationRoleCount<Role, Candidate> > 0;
+
+    template <models::SpecificationRole Role, typename Candidate>
+    concept HasUniqueSpecificationForRole = StellarModelType<Candidate> && specificationRoleCount<Role, Candidate> == 1;
+
+    template <models::SpecificationRole Role, typename Candidate>
+        requires StellarModelType<Candidate>
+    using SpecificationsForRoleT =
+        typename detail::StellarModelRoleSelection<Role, std::remove_cvref_t<Candidate>>::Types;
+
+    template <models::SpecificationRole Role, typename Candidate>
+        requires HasUniqueSpecificationForRole<Role, Candidate>
+    using SpecificationForRoleT =
+        models::SpecificationForRoleT<Role, typename std::remove_cvref_t<Candidate>::SpecificationTypes>;
+
+    template <typename Candidate>
+    concept HasEquationOfState = HasUniqueSpecificationForRole<models::SpecificationRole::constitutive_law, Candidate>;
+
+    template <typename Candidate>
+    concept HasSurfaceCondition = HasSpecificationsForRole<models::SpecificationRole::boundary_condition, Candidate>;
+
+    template <typename Candidate>
+    concept HasUniqueSurfaceCondition =
+        HasUniqueSpecificationForRole<models::SpecificationRole::boundary_condition, Candidate>;
+
+    template <HasEquationOfState Candidate>
+    using EquationOfStateType = SpecificationForRoleT<models::SpecificationRole::constitutive_law, Candidate>;
+
+    template <HasUniqueSurfaceCondition Candidate>
+    using SurfaceConditionType = SpecificationForRoleT<models::SpecificationRole::boundary_condition, Candidate>;
 } // namespace mean_field::model

libmeanfield/interface/normalization/normalization.cppm

@@ -0,0 +1,6 @@
+export module mean_field:normalization;
+
+export import :normalization.plan;
+export import :normalization.physical_riesz;
+export import :normalization.operators;
+export import :normalization.stellar_equilibrium;

libmeanfield/interface/normalization/operators.cppm

@@ -0,0 +1,621 @@
+module;
+
+#include <array>
+#include <cmath>
+#include <concepts>
+#include <cstdint>
+#include <span>
+#include <stdexcept>
+#include <string>
+#include <type_traits>
+#include <utility>
+#include <vector>
+
+#include <mfem.hpp>
+
+export module mean_field:normalization.operators;
+
+export import :normalization.physical_riesz;
+
+export namespace mean_field::normalization {
+    class DiagonalNormalization final {
+    public:
+        DiagonalNormalization(
+            mfem::Vector stateToNormalized,
+            mfem::Vector residualToNormalized
+        )
+            : m_stateToNormalized(std::move(stateToNormalized)),
+              m_residualToNormalized(std::move(residualToNormalized)) {
+            ValidateFactors(m_stateToNormalized, "state");
+            ValidateFactors(m_residualToNormalized, "residual");
+        }
+
+        [[nodiscard]] static DiagonalNormalization Identity(
+            const int stateSize,
+            const int residualSize
+        ) {
+            if (stateSize < 0 || residualSize < 0) {
+                throw std::invalid_argument("Normalization dimensions cannot be negative.");
+            }
+            mfem::Vector state(stateSize);
+            mfem::Vector residual(residualSize);
+            state = 1.0;
+            residual = 1.0;
+            return {std::move(state), std::move(residual)};
+        }
+
+        [[nodiscard]] int StateSize() const noexcept {
+            return m_stateToNormalized.Size();
+        }
+
+        [[nodiscard]] int ResidualSize() const noexcept {
+            return m_residualToNormalized.Size();
+        }
+
+        [[nodiscard]] const mfem::Vector &StateFactors() const noexcept {
+            return m_stateToNormalized;
+        }
+
+        [[nodiscard]] const mfem::Vector &ResidualFactors() const noexcept {
+            return m_residualToNormalized;
+        }
+
+        void NormalizeState(
+            const mfem::Vector &physical,
+            mfem::Vector &normalized
+        ) const {
+            Apply(m_stateToNormalized, physical, normalized, false, "state");
+        }
+
+        void DenormalizeState(
+            const mfem::Vector &normalized,
+            mfem::Vector &physical
+        ) const {
+            Apply(m_stateToNormalized, normalized, physical, true, "state");
+        }
+
+        void NormalizeResidual(
+            const mfem::Vector &physical,
+            mfem::Vector &normalized
+        ) const {
+            Apply(m_residualToNormalized, physical, normalized, false, "residual");
+        }
+
+        void DenormalizeResidual(
+            const mfem::Vector &normalized,
+            mfem::Vector &physical
+        ) const {
+            Apply(m_residualToNormalized, normalized, physical, true, "residual");
+        }
+
+        [[nodiscard]] double LocalStateNormSquared(const mfem::Vector &physical) const {
+            return LocalNormSquared(m_stateToNormalized, physical, "state");
+        }
+
+        [[nodiscard]] double LocalResidualNormSquared(const mfem::Vector &physical) const {
+            return LocalNormSquared(m_residualToNormalized, physical, "residual");
+        }
+
+    private:
+        static void ValidateFactors(
+            const mfem::Vector &factors,
+            const char *role
+        ) {
+            for (int index = 0; index < factors.Size(); ++index) {
+                if (!std::isfinite(factors(index)) || factors(index) <= 0.0) {
+                    throw std::invalid_argument(
+                        std::string("The ") + role + " normalization factors must be finite and positive."
+                    );
+                }
+            }
+        }
+
+        static void Apply(
+            const mfem::Vector &factors,
+            const mfem::Vector &input,
+            mfem::Vector &output,
+            const bool inverse,
+            const char *role
+        ) {
+            if (input.Size() != factors.Size()) {
+                throw std::invalid_argument(std::string("The ") + role + " vector has the wrong size.");
+            }
+            const bool exactAlias = input.GetData() == output.GetData() && input.Size() == output.Size();
+            if (!exactAlias) {
+                output.SetSize(input.Size());
+            }
+            for (int index = 0; index < input.Size(); ++index) {
+                const double value = input(index);
+                output(index) = inverse ? value / factors(index) : factors(index) * value;
+            }
+        }
+
+        [[nodiscard]] static double LocalNormSquared(
+            const mfem::Vector &factors,
+            const mfem::Vector &physical,
+            const char *role
+        ) {
+            if (physical.Size() != factors.Size()) {
+                throw std::invalid_argument(std::string("The ") + role + " vector has the wrong size.");
+            }
+            double normSquared = 0.0;
+            for (int index = 0; index < physical.Size(); ++index) {
+                const double normalized = factors(index) * physical(index);
+                normSquared += normalized * normalized;
+            }
+            return normSquared;
+        }
+
+        mfem::Vector m_stateToNormalized;
+        mfem::Vector m_residualToNormalized;
+    };
+
+    /* Detection-safe public operation for an ordinary third-party runtime
+     * policy. The exact policy is recovered from the problem type and must own
+     * every method in its compiled plan. Its implementation remains beside
+     * the policy and is found by ADL, so adding a normalization family does
+     * not edit a library registry or switch. */
+    template <typename Problem>
+    concept RuntimePreparedNormalizationOperation =
+        requires(const std::remove_cvref_t<Problem> &problem) {
+            typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType;
+            typename std::remove_cvref_t<Problem>::FormType;
+            requires RuntimePreparedNormalizationFor<
+                typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType,
+                typename std::remove_cvref_t<Problem>::FormType>;
+            {
+                problem.GetNormalizationPrescription()
+            } -> std::same_as<const typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType &>;
+            {
+                prepareStellarNormalization(
+                    problem.GetNormalizationPrescription(),
+                    problem)
+            } -> std::same_as<DiagonalNormalization>;
+        };
+
+    template <typename Form>
+        requires utils::blocks::block_form_is_valid_v<Form>
+    class DiagonalNormalizationBuilder final {
+    public:
+        explicit DiagonalNormalizationBuilder(const utils::blocks::form_layout<Form> &layout)
+            : m_layout(&layout),
+              m_stateFactors(layout.value_offsets().Last()),
+              m_residualFactors(layout.residual_offsets().Last()) {
+        }
+
+        explicit DiagonalNormalizationBuilder(
+            utils::blocks::form_layout<Form> &&
+        ) = delete;
+
+        explicit DiagonalNormalizationBuilder(
+            const utils::blocks::form_layout<Form> &&
+        ) = delete;
+
+        template <typename Block>
+            requires utils::blocks::contains_type_v<Block, typename Form::value_blocks>
+        void SetValueBlock(
+            const double physicalScale,
+            const mfem::Vector &primalGramDiagonal
+        ) {
+            constexpr int block = utils::blocks::type_index_v<Block, typename Form::value_blocks>;
+            RequireUnassigned(m_valueAssigned[block], "value");
+            AssignBlock(
+                m_stateFactors,
+                m_layout->value_offsets()[block],
+                m_layout->value_offsets()[block + 1] - m_layout->value_offsets()[block],
+                physicalScale,
+                primalGramDiagonal,
+                false
+            );
+            m_valueAssigned[block] = true;
+        }
+
+        template <typename Block>
+            requires utils::blocks::contains_type_v<Block, typename Form::residual_blocks>
+        void SetResidualBlock(
+            const double physicalScale,
+            const mfem::Vector &primalGramDiagonal
+        ) {
+            constexpr int block = utils::blocks::type_index_v<Block, typename Form::residual_blocks>;
+            RequireUnassigned(m_residualAssigned[block], "residual");
+            AssignBlock(
+                m_residualFactors,
+                m_layout->residual_offsets()[block],
+                m_layout->residual_offsets()[block + 1] - m_layout->residual_offsets()[block],
+                physicalScale,
+                primalGramDiagonal,
+                true
+            );
+            m_residualAssigned[block] = true;
+        }
+
+        template <typename Block>
+            requires utils::blocks::contains_type_v<Block, typename Form::value_blocks>
+        void SetValueGlobal(const double physicalScale) {
+            constexpr int block = utils::blocks::type_index_v<Block, typename Form::value_blocks>;
+            SetConstantMetricValueBlock<Block>(physicalScale, BlockSize(m_layout->value_offsets(), block));
+        }
+
+        template <typename Block>
+            requires utils::blocks::contains_type_v<Block, typename Form::residual_blocks>
+        void SetResidualGlobal(const double physicalScale) {
+            constexpr int block = utils::blocks::type_index_v<Block, typename Form::residual_blocks>;
+            mfem::Vector metric(BlockSize(m_layout->residual_offsets(), block));
+            metric = 1.0;
+            SetResidualBlock<Block>(physicalScale, metric);
+        }
+
+        template <typename Block>
+            requires utils::blocks::contains_type_v<Block, typename Form::residual_blocks>
+        void SetHybridResidualBlock(
+            const double physicalScale,
+            const mfem::Vector &bulkPrimalGramDiagonal,
+            const std::span<const int> pointRows,
+            const double pointMetric = 1.0
+        ) {
+            constexpr int block = utils::blocks::type_index_v<Block, typename Form::residual_blocks>;
+            const int size = BlockSize(m_layout->residual_offsets(), block);
+            if (bulkPrimalGramDiagonal.Size() != size) {
+                throw std::invalid_argument("The hybrid residual Gram diagonal has the wrong size.");
+            }
+            ValidateMetric(pointMetric);
+
+            std::vector<bool> isPointRow(static_cast<std::size_t>(size), false);
+            for (const int row : pointRows) {
+                if (row < 0 || row >= size) {
+                    throw std::out_of_range("A hybrid point row lies outside its residual block.");
+                }
+                if (isPointRow[static_cast<std::size_t>(row)]) {
+                    throw std::invalid_argument("A hybrid point row was supplied more than once.");
+                }
+                isPointRow[static_cast<std::size_t>(row)] = true;
+            }
+
+            mfem::Vector metric(size);
+            for (int row = 0; row < size; ++row) {
+                metric(row) = isPointRow[static_cast<std::size_t>(row)]
+                                  ? pointMetric
+                                  : bulkPrimalGramDiagonal(row);
+            }
+            SetResidualBlock<Block>(physicalScale, metric);
+        }
+
+        [[nodiscard]] DiagonalNormalization Build() && {
+            for (const bool assigned : m_valueAssigned) {
+                if (!assigned) {
+                    throw std::logic_error("The normalization is missing a value block.");
+                }
+            }
+            for (const bool assigned : m_residualAssigned) {
+                if (!assigned) {
+                    throw std::logic_error("The normalization is missing a residual block.");
+                }
+            }
+            return {std::move(m_stateFactors), std::move(m_residualFactors)};
+        }
+
+    private:
+        template <typename Block>
+        void SetConstantMetricValueBlock(
+            const double physicalScale,
+            const int size
+        ) {
+            mfem::Vector metric(size);
+            metric = 1.0;
+            SetValueBlock<Block>(physicalScale, metric);
+        }
+
+        [[nodiscard]] static int BlockSize(
+            const mfem::Array<int> &offsets,
+            const int block
+        ) noexcept {
+            return offsets[block + 1] - offsets[block];
+        }
+
+        static void RequireUnassigned(
+            const bool assigned,
+            const char *role
+        ) {
+            if (assigned) {
+                throw std::logic_error(std::string("The ") + role + " block normalization was assigned twice.");
+            }
+        }
+
+        static void ValidateMetric(const double metric) {
+            if (!std::isfinite(metric) || metric <= 0.0) {
+                throw std::invalid_argument("Every Riesz Gram diagonal entry must be finite and positive.");
+            }
+        }
+
+        static void AssignBlock(
+            mfem::Vector &factors,
+            const int offset,
+            const int size,
+            const double physicalScale,
+            const mfem::Vector &primalGramDiagonal,
+            const bool dual
+        ) {
+            if (!std::isfinite(physicalScale) || physicalScale <= 0.0) {
+                throw std::invalid_argument("A physical normalization scale must be finite and positive.");
+            }
+            if (primalGramDiagonal.Size() != size) {
+                throw std::invalid_argument("A Riesz Gram diagonal has the wrong block size.");
+            }
+            for (int index = 0; index < size; ++index) {
+                const double metric = primalGramDiagonal(index);
+                ValidateMetric(metric);
+                const double rieszFactor = std::sqrt(metric);
+                const double factor = dual
+                                          ? 1.0 / (physicalScale * rieszFactor)
+                                          : rieszFactor / physicalScale;
+                if (!std::isfinite(factor) || factor <= 0.0) {
+                    throw std::overflow_error("A normalization factor is not finite and positive.");
+                }
+                factors(offset + index) = factor;
+            }
+        }
+
+        const utils::blocks::form_layout<Form> *m_layout;
+        mfem::Vector m_stateFactors;
+        mfem::Vector m_residualFactors;
+        std::array<bool, Form::value_block_count> m_valueAssigned{};
+        std::array<bool, Form::residual_block_count> m_residualAssigned{};
+    };
+
+    class ScaledJacobianOperator final : public mfem::Operator {
+    public:
+        ScaledJacobianOperator(
+            const mfem::Operator &physicalJacobian,
+            const DiagonalNormalization &normalization
+        )
+            : mfem::Operator(normalization.ResidualSize(), normalization.StateSize()),
+              m_physicalJacobian(&physicalJacobian),
+              m_normalization(&normalization),
+              m_physicalDirection(normalization.StateSize()),
+              m_physicalAction(normalization.ResidualSize()) {
+            if (physicalJacobian.Width() != normalization.StateSize() ||
+                physicalJacobian.Height() != normalization.ResidualSize()) {
+                throw std::invalid_argument("The physical Jacobian and normalization dimensions do not agree.");
+            }
+        }
+
+        ScaledJacobianOperator(
+            mfem::Operator &&,
+            const DiagonalNormalization &
+        ) = delete;
+
+        ScaledJacobianOperator(
+            const mfem::Operator &&,
+            const DiagonalNormalization &
+        ) = delete;
+
+        ScaledJacobianOperator(
+            const mfem::Operator &,
+            DiagonalNormalization &&
+        ) = delete;
+
+        ScaledJacobianOperator(
+            const mfem::Operator &,
+            const DiagonalNormalization &&
+        ) = delete;
+
+        void Mult(
+            const mfem::Vector &normalizedDirection,
+            mfem::Vector &normalizedAction
+        ) const override {
+            m_normalization->DenormalizeState(normalizedDirection, m_physicalDirection);
+            m_physicalJacobian->Mult(m_physicalDirection, m_physicalAction);
+            m_normalization->NormalizeResidual(m_physicalAction, normalizedAction);
+        }
+
+    private:
+        const mfem::Operator *m_physicalJacobian;
+        const DiagonalNormalization *m_normalization;
+        mutable mfem::Vector m_physicalDirection;
+        mutable mfem::Vector m_physicalAction;
+    };
+
+    class ScaledInverseOperator final : public mfem::Operator {
+    public:
+        ScaledInverseOperator(
+            const mfem::Operator &physicalInverse,
+            const DiagonalNormalization &normalization
+        )
+            : mfem::Operator(normalization.StateSize(), normalization.ResidualSize()),
+              m_physicalInverse(&physicalInverse),
+              m_normalization(&normalization),
+              m_physicalResidual(normalization.ResidualSize()),
+              m_physicalCorrection(normalization.StateSize()) {
+            if (physicalInverse.Width() != normalization.ResidualSize() ||
+                physicalInverse.Height() != normalization.StateSize()) {
+                throw std::invalid_argument("The physical inverse and normalization dimensions do not agree.");
+            }
+        }
+
+        ScaledInverseOperator(
+            mfem::Operator &&,
+            const DiagonalNormalization &
+        ) = delete;
+
+        ScaledInverseOperator(
+            const mfem::Operator &&,
+            const DiagonalNormalization &
+        ) = delete;
+
+        ScaledInverseOperator(
+            const mfem::Operator &,
+            DiagonalNormalization &&
+        ) = delete;
+
+        ScaledInverseOperator(
+            const mfem::Operator &,
+            const DiagonalNormalization &&
+        ) = delete;
+
+        void Mult(
+            const mfem::Vector &normalizedResidual,
+            mfem::Vector &normalizedCorrection
+        ) const override {
+            m_normalization->DenormalizeResidual(normalizedResidual, m_physicalResidual);
+            m_physicalInverse->Mult(m_physicalResidual, m_physicalCorrection);
+            m_normalization->NormalizeState(m_physicalCorrection, normalizedCorrection);
+        }
+
+    private:
+        const mfem::Operator *m_physicalInverse;
+        const DiagonalNormalization *m_normalization;
+        mutable mfem::Vector m_physicalResidual;
+        mutable mfem::Vector m_physicalCorrection;
+    };
+
+    struct ScaledPreconditionerStatistics final {
+        std::uint64_t operatorBindings{0};
+        std::uint64_t applications{0};
+    };
+
+    /*
+     * Solver-compatible realization of R^{-1} M^{-1} L^{-1}. The wrapped
+     * inverse always sees the dimensional Jacobian, even when an MFEM Krylov
+     * solver binds this object to the normalized Jacobian L J R.
+     */
+    class ScaledPreconditioner final : public mfem::Solver {
+    public:
+        ScaledPreconditioner(
+            mfem::Solver &physicalInverse,
+            const mfem::Operator &physicalJacobian,
+            const mfem::Operator &normalizedJacobian,
+            const DiagonalNormalization &normalization
+        )
+            : mfem::Solver(
+                  normalization.StateSize(),
+                  normalization.ResidualSize(),
+                  physicalInverse.iterative_mode
+              ),
+              m_physicalInverse(&physicalInverse),
+              m_physicalJacobian(&physicalJacobian),
+              m_expectedNormalizedJacobian(&normalizedJacobian),
+              m_normalization(&normalization),
+              m_physicalResidual(normalization.ResidualSize()),
+              m_physicalCorrection(normalization.StateSize()) {
+            if (physicalInverse.Width() != normalization.ResidualSize() ||
+                physicalInverse.Height() != normalization.StateSize() ||
+                physicalJacobian.Width() != normalization.StateSize() ||
+                physicalJacobian.Height() != normalization.ResidualSize()) {
+                throw std::invalid_argument(
+                    "The physical preconditioner, Jacobian, and normalization dimensions do not agree."
+                );
+            }
+            SetOperator(normalizedJacobian);
+        }
+
+        ScaledPreconditioner(
+            mfem::Solver &,
+            mfem::Operator &&,
+            const mfem::Operator &,
+            const DiagonalNormalization &
+        ) = delete;
+
+        ScaledPreconditioner(
+            mfem::Solver &,
+            const mfem::Operator &&,
+            const mfem::Operator &,
+            const DiagonalNormalization &
+        ) = delete;
+
+        ScaledPreconditioner(
+            mfem::Solver &,
+            const mfem::Operator &,
+            mfem::Operator &&,
+            const DiagonalNormalization &
+        ) = delete;
+
+        ScaledPreconditioner(
+            mfem::Solver &,
+            const mfem::Operator &,
+            const mfem::Operator &&,
+            const DiagonalNormalization &
+        ) = delete;
+
+        ScaledPreconditioner(
+            mfem::Solver &,
+            const mfem::Operator &,
+            const mfem::Operator &,
+            DiagonalNormalization &&
+        ) = delete;
+
+        ScaledPreconditioner(
+            mfem::Solver &,
+            const mfem::Operator &,
+            const mfem::Operator &,
+            const DiagonalNormalization &&
+        ) = delete;
+
+        ScaledPreconditioner(const ScaledPreconditioner &)            = delete;
+        ScaledPreconditioner &operator=(const ScaledPreconditioner &) = delete;
+        ScaledPreconditioner(ScaledPreconditioner &&)                 = delete;
+        ScaledPreconditioner &operator=(ScaledPreconditioner &&)      = delete;
+
+        void SetOperator(const mfem::Operator &normalizedJacobian) override {
+            if (normalizedJacobian.Width() != Width() || normalizedJacobian.Height() != Height()) {
+                throw std::invalid_argument(
+                    "The scaled preconditioner received an incompatible normalized Jacobian."
+                );
+            }
+            if (&normalizedJacobian != m_expectedNormalizedJacobian) {
+                throw std::invalid_argument(
+                    "The scaled preconditioner cannot be rebound to a different normalized Jacobian."
+                );
+            }
+            m_physicalInverse->SetOperator(*m_physicalJacobian);
+            m_normalizedJacobian = &normalizedJacobian;
+            ++m_statistics.operatorBindings;
+        }
+
+        void Mult(
+            const mfem::Vector &normalizedResidual,
+            mfem::Vector &normalizedCorrection
+        ) const override {
+            if (m_normalizedJacobian == nullptr) {
+                throw std::logic_error("The scaled preconditioner has not been bound to a normalized Jacobian.");
+            }
+            if (normalizedResidual.Size() != Width() || normalizedCorrection.Size() != Height()) {
+                throw std::invalid_argument(
+                    "The scaled preconditioner requires compatible, preallocated normalized vectors."
+                );
+            }
+            m_normalization->DenormalizeResidual(normalizedResidual, m_physicalResidual);
+            m_physicalInverse->Mult(m_physicalResidual, m_physicalCorrection);
+            m_normalization->NormalizeState(m_physicalCorrection, normalizedCorrection);
+            ++m_statistics.applications;
+        }
+
+        [[nodiscard]] const mfem::Solver &GetPhysicalInverse() const noexcept {
+            return *m_physicalInverse;
+        }
+
+        [[nodiscard]] const mfem::Operator &GetPhysicalJacobian() const noexcept {
+            return *m_physicalJacobian;
+        }
+
+        [[nodiscard]] const mfem::Operator &GetNormalizedJacobian() const {
+            if (m_normalizedJacobian == nullptr) {
+                throw std::logic_error("The scaled preconditioner has not been bound to a normalized Jacobian.");
+            }
+            return *m_normalizedJacobian;
+        }
+
+        [[nodiscard]] const ScaledPreconditionerStatistics &GetStatistics() const noexcept {
+            return m_statistics;
+        }
+
+    private:
+        mfem::Solver *m_physicalInverse;
+        const mfem::Operator *m_physicalJacobian;
+        const mfem::Operator *m_expectedNormalizedJacobian;
+        const mfem::Operator *m_normalizedJacobian{nullptr};
+        const DiagonalNormalization *m_normalization;
+        mutable mfem::Vector m_physicalResidual;
+        mutable mfem::Vector m_physicalCorrection;
+        mutable ScaledPreconditionerStatistics m_statistics;
+    };
+} // namespace mean_field::normalization

libmeanfield/interface/normalization/physical_riesz.cppm

@@ -0,0 +1,728 @@
+module;
+
+#include <cmath>
+#include <concepts>
+#include <stdexcept>
+#include <type_traits>
+
+export module mean_field:normalization.physical_riesz;
+
+export import :dimensions.quantities;
+export import :field.mfem;
+export import :model.specifications;
+export import :normalization.plan;
+
+export namespace mean_field::normalization {
+    struct Unnormalized final : NormalizationPrescriptionTag { };
+
+    struct ReferenceGeometry final { };
+
+    struct FixedMassBranchReference final { };
+
+    template <typename Candidate>
+    concept RieszGeometryPolicy = std::same_as<std::remove_cvref_t<Candidate>, ReferenceGeometry>;
+
+    template <typename Candidate>
+    concept ReferenceScalePolicy = std::same_as<std::remove_cvref_t<Candidate>, FixedMassBranchReference>;
+
+    template <
+        RieszGeometryPolicy GeometryPolicy = ReferenceGeometry,
+        ReferenceScalePolicy ScalePolicy = FixedMassBranchReference>
+    class PhysicalRieszDiagonal final : public NormalizationPrescriptionTag {
+    public:
+        using Geometry = GeometryPolicy;
+        using ScaleSource = ScalePolicy;
+
+        explicit PhysicalRieszDiagonal(
+            const dimensions::LengthValue referenceRadius,
+            const double gravitationalConstant = 1.0
+        )
+            : m_referenceRadius(referenceRadius),
+              m_gravitationalConstant(gravitationalConstant) {
+            if (!std::isfinite(referenceRadius.value()) || referenceRadius.value() <= 0.0) {
+                throw std::invalid_argument("Physical Riesz normalization requires a finite, positive branch radius.");
+            }
+            if (!std::isfinite(gravitationalConstant) || gravitationalConstant <= 0.0) {
+                throw std::invalid_argument(
+                    "Physical Riesz normalization requires a finite, positive gravitational constant."
+                );
+            }
+        }
+
+        [[nodiscard]] dimensions::LengthValue referenceRadius() const noexcept {
+            return m_referenceRadius;
+        }
+
+        [[nodiscard]] double gravitationalConstant() const noexcept {
+            return m_gravitationalConstant;
+        }
+
+    private:
+        dimensions::LengthValue m_referenceRadius;
+        double m_gravitationalConstant;
+    };
+
+    PhysicalRieszDiagonal(dimensions::LengthValue, double = 1.0)
+        -> PhysicalRieszDiagonal<ReferenceGeometry, FixedMassBranchReference>;
+
+    template <typename Candidate> struct IsPhysicalRieszDiagonal : std::false_type { };
+
+    template <RieszGeometryPolicy Geometry, ReferenceScalePolicy ScaleSource>
+    struct IsPhysicalRieszDiagonal<PhysicalRieszDiagonal<Geometry, ScaleSource>> : std::true_type { };
+
+    template <typename Candidate>
+    concept PhysicalRieszDiagonalPrescription =
+        IsPhysicalRieszDiagonal<std::remove_cvref_t<Candidate>>::value;
+
+    struct StellarCharacteristicScales final {
+        dimensions::MassValue mass;
+        dimensions::LengthValue radius;
+        double gravitationalConstant;
+        double density;
+        double acceleration;
+        double inverseTimeSquared;
+        double specificEnergy;
+        double pressure;
+        double angularVelocity;
+        double angularMomentum;
+        double force;
+    };
+
+    [[nodiscard]] inline StellarCharacteristicScales deriveStellarCharacteristicScales(
+        const dimensions::MassValue mass,
+        const dimensions::LengthValue radius,
+        const double gravitationalConstant = 1.0
+    ) {
+        const double massValue = mass.value();
+        const double radiusValue = radius.value();
+        if (!std::isfinite(massValue) || massValue <= 0.0) {
+            throw std::invalid_argument("Characteristic stellar scales require a finite, positive mass.");
+        }
+        if (!std::isfinite(radiusValue) || radiusValue <= 0.0) {
+            throw std::invalid_argument("Characteristic stellar scales require a finite, positive radius.");
+        }
+        if (!std::isfinite(gravitationalConstant) || gravitationalConstant <= 0.0) {
+            throw std::invalid_argument(
+                "Characteristic stellar scales require a finite, positive gravitational constant."
+            );
+        }
+
+        const double radiusSquared = radiusValue * radiusValue;
+        const double radiusCubed = radiusSquared * radiusValue;
+        const double density = massValue / radiusCubed;
+        const double acceleration = gravitationalConstant * massValue / radiusSquared;
+        const double inverseTimeSquared = gravitationalConstant * massValue / radiusCubed;
+        const double specificEnergy = gravitationalConstant * massValue / radiusValue;
+        const double pressure = gravitationalConstant * massValue * massValue /
+                                (radiusSquared * radiusSquared);
+        const double angularVelocity = std::sqrt(inverseTimeSquared);
+        const double angularMomentum = massValue * std::sqrt(gravitationalConstant * massValue * radiusValue);
+        const double force = gravitationalConstant * massValue * massValue / radiusSquared;
+
+        const double derived[] = {
+            density,
+            acceleration,
+            inverseTimeSquared,
+            specificEnergy,
+            pressure,
+            angularVelocity,
+            angularMomentum,
+            force
+        };
+        for (const double value : derived) {
+            if (!std::isfinite(value) || value <= 0.0) {
+                throw std::overflow_error("A derived characteristic stellar scale is not finite and positive.");
+            }
+        }
+
+        return {
+            .mass = mass,
+            .radius = radius,
+            .gravitationalConstant = gravitationalConstant,
+            .density = density,
+            .acceleration = acceleration,
+            .inverseTimeSquared = inverseTimeSquared,
+            .specificEnergy = specificEnergy,
+            .pressure = pressure,
+            .angularVelocity = angularVelocity,
+            .angularMomentum = angularMomentum,
+            .force = force
+        };
+    }
+
+    template <RieszGeometryPolicy Geometry, ReferenceScalePolicy ScaleSource, typename Model>
+        requires requires(const Model &model) {
+            {
+                model.template specification<models::FixedTotalMass>()
+            } -> std::same_as<const models::FixedTotalMass &>;
+            {
+                model.template specification<models::FixedTotalMass>().targetMass()
+            } -> std::same_as<dimensions::MassValue>;
+        }
+    [[nodiscard]] StellarCharacteristicScales deriveStellarCharacteristicScales(
+        const PhysicalRieszDiagonal<Geometry, ScaleSource> &prescription,
+        const Model &model
+    ) {
+        return deriveStellarCharacteristicScales(
+            model.template specification<models::FixedTotalMass>().targetMass(),
+            prescription.referenceRadius(),
+            prescription.gravitationalConstant()
+        );
+    }
+
+    namespace detail {
+        /*
+         * Model definitions live below the numerical normalization layer so
+         * that a physics component can describe its generated coordinates
+         * without importing solver machinery.  These two translations are the
+         * deliberately small boundary between that neutral declaration and the
+         * normalization plan used by the discretization.
+         */
+        template <models::RieszTopology Topology> struct DeclaredRieszTopology {
+            static constexpr bool available = false;
+            static constexpr RieszTopology value = RieszTopology::identity;
+        };
+
+#define MEAN_FIELD_DECLARED_RIESZ_TOPOLOGY(Name)                                                        \
+    template <> struct DeclaredRieszTopology<models::RieszTopology::Name> {                             \
+        static constexpr bool available = true;                                                         \
+        static constexpr RieszTopology value = RieszTopology::Name;                                     \
+    }
+
+        MEAN_FIELD_DECLARED_RIESZ_TOPOLOGY(identity);
+        MEAN_FIELD_DECLARED_RIESZ_TOPOLOGY(scalar_volume_l2);
+        MEAN_FIELD_DECLARED_RIESZ_TOPOLOGY(vector_volume_l2);
+        MEAN_FIELD_DECLARED_RIESZ_TOPOLOGY(scalar_boundary_l2);
+        MEAN_FIELD_DECLARED_RIESZ_TOPOLOGY(hybrid_scalar_volume_point_rows);
+        MEAN_FIELD_DECLARED_RIESZ_TOPOLOGY(global_scalar);
+
+#undef MEAN_FIELD_DECLARED_RIESZ_TOPOLOGY
+
+        template <models::PhysicalScaleLaw Scale> struct DeclaredPhysicalScale {
+            static constexpr bool available = false;
+            static constexpr PhysicalScaleKind value = PhysicalScaleKind::dimensionless;
+        };
+
+#define MEAN_FIELD_DECLARED_PHYSICAL_SCALE(Name)                                                        \
+    template <> struct DeclaredPhysicalScale<models::PhysicalScaleLaw::Name> {                          \
+        static constexpr bool available = true;                                                         \
+        static constexpr PhysicalScaleKind value = PhysicalScaleKind::Name;                             \
+    }
+
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(dimensionless);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(density);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(length);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(acceleration);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(inverse_time_squared);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(specific_energy);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(pressure);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(mass);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(force);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(angular_velocity);
+        MEAN_FIELD_DECLARED_PHYSICAL_SCALE(angular_momentum);
+
+#undef MEAN_FIELD_DECLARED_PHYSICAL_SCALE
+
+        template <typename Declaration, typename = void>
+        struct CompileDeclaredPhysicalRieszCoordinate {
+            using Method = UnsupportedPhysicalRieszCoordinate;
+            static constexpr bool registered = false;
+        };
+
+        template <typename Declaration>
+        struct CompileDeclaredPhysicalRieszCoordinate<
+            Declaration,
+            std::void_t<
+                decltype(std::integral_constant<
+                         models::RieszTopology,
+                         static_cast<models::RieszTopology>(Declaration::topology)>{}),
+                decltype(std::integral_constant<
+                         models::PhysicalScaleLaw,
+                         static_cast<models::PhysicalScaleLaw>(Declaration::scale)>{}),
+                decltype(std::bool_constant<static_cast<bool>(Declaration::available)>{})>> {
+        private:
+            static constexpr models::RieszTopology declaredTopology =
+                static_cast<models::RieszTopology>(Declaration::topology);
+            static constexpr models::PhysicalScaleLaw declaredScale =
+                static_cast<models::PhysicalScaleLaw>(Declaration::scale);
+            using Topology = DeclaredRieszTopology<declaredTopology>;
+            using Scale = DeclaredPhysicalScale<declaredScale>;
+
+        public:
+            static constexpr bool registered = static_cast<bool>(Declaration::available) &&
+                                               Topology::available && Scale::available;
+            using Method = std::conditional_t<
+                registered,
+                PhysicalRieszCoordinate<Topology::value, Scale::value>,
+                UnsupportedPhysicalRieszCoordinate>;
+        };
+
+        template <typename Generated, CoordinateKind Kind, typename = void>
+        struct DeclaredGeneratedPhysicalRieszCoordinate {
+            using Method = UnsupportedPhysicalRieszCoordinate;
+            static constexpr bool registered = false;
+        };
+
+        template <typename Generated>
+        struct DeclaredGeneratedPhysicalRieszCoordinate<
+            Generated,
+            CoordinateKind::value,
+            std::void_t<
+                typename Generated::SpecificationType,
+                typename models::SpecificationContribution<
+                    typename Generated::SpecificationType>::Normalization::Value>>
+            : CompileDeclaredPhysicalRieszCoordinate<
+                  typename models::SpecificationContribution<
+                      typename Generated::SpecificationType>::Normalization::Value> { };
+
+        template <typename Generated>
+        struct DeclaredGeneratedPhysicalRieszCoordinate<
+            Generated,
+            CoordinateKind::residual,
+            std::void_t<
+                typename Generated::SpecificationType,
+                typename models::SpecificationContribution<
+                    typename Generated::SpecificationType>::Normalization::Residual>>
+            : CompileDeclaredPhysicalRieszCoordinate<
+                  typename models::SpecificationContribution<
+                      typename Generated::SpecificationType>::Normalization::Residual> { };
+
+        template <typename GeneratedValues, typename GeneratedResiduals>
+        struct GeneratedPhysicalRieszCoverage {
+            static constexpr bool complete = false;
+        };
+
+        template <typename... GeneratedValues, typename... GeneratedResiduals>
+        struct GeneratedPhysicalRieszCoverage<
+            models::ModelTypeList<GeneratedValues...>,
+            models::ModelTypeList<GeneratedResiduals...>> {
+            static constexpr bool complete =
+                (DeclaredGeneratedPhysicalRieszCoordinate<
+                     GeneratedValues,
+                     CoordinateKind::value>::registered && ...) &&
+                (DeclaredGeneratedPhysicalRieszCoordinate<
+                     GeneratedResiduals,
+                     CoordinateKind::residual>::registered && ...);
+        };
+
+        template <typename Specification, typename = void>
+        struct SpecificationPhysicalRieszCoverage {
+            static constexpr bool complete = false;
+        };
+
+        template <models::ModelSpecification Specification>
+        struct SpecificationPhysicalRieszCoverage<
+            Specification,
+            std::void_t<
+                typename models::SpecificationContribution<Specification>::GeneratedValues,
+                typename models::SpecificationContribution<Specification>::GeneratedResiduals>>
+            : GeneratedPhysicalRieszCoverage<
+                  typename models::SpecificationContribution<Specification>::GeneratedValues,
+                  typename models::SpecificationContribution<Specification>::GeneratedResiduals> { };
+    } // namespace detail
+
+    /*
+     * All generated blocks are normalized from their generating physics
+     * specification.  Adding another constraint therefore does not add a
+     * normalization specialization: its public ModelDefinition is the single
+     * source of both the value and residual Riesz laws.
+     */
+    template <typename Generated>
+    struct PhysicalRieszBlockTraits<utils::blocks::generated_value_block<Generated>>
+        : detail::DeclaredGeneratedPhysicalRieszCoordinate<Generated, CoordinateKind::value> { };
+
+    template <typename Generated>
+    struct PhysicalRieszBlockTraits<utils::blocks::generated_residual_block<Generated>>
+        : detail::DeclaredGeneratedPhysicalRieszCoordinate<Generated, CoordinateKind::residual> { };
+
+    template <typename Generated>
+    concept GeneratedValuePhysicalRieszNormalizable =
+        detail::DeclaredGeneratedPhysicalRieszCoordinate<Generated, CoordinateKind::value>::registered;
+
+    template <typename Generated>
+    concept GeneratedResidualPhysicalRieszNormalizable =
+        detail::DeclaredGeneratedPhysicalRieszCoordinate<Generated, CoordinateKind::residual>::registered;
+
+    template <typename Specification>
+    concept CompleteGeneratedPhysicalRieszNormalizationFor =
+        detail::SpecificationPhysicalRieszCoverage<std::remove_cvref_t<Specification>>::complete;
+
+    /*
+     * Runtime Physical Riesz assembly needs more than a symbolically complete
+     * plan: it must be able to recover the finite-element maps owned by the
+     * selected physical core.  Keep that structural capability in this low
+     * normalization module so both problem formation and the solver-facing
+     * adapter can consult the same authority without importing one another.
+     */
+    template <typename Candidate>
+    concept PhysicalRieszCoreRuntime =
+        requires(const std::remove_cvref_t<Candidate> &core) {
+            {
+                core.GetGravityContext().GetDensityMap()
+            } -> std::same_as<const field::FieldDofMap &>;
+            {
+                core.GetGravityContext().GetGravityGradientMap()
+            } -> std::same_as<const field::FieldDofMap &>;
+            {
+                core.GetGravityContext().GetGravityPotentialMap()
+            } -> std::same_as<const field::FieldDofMap &>;
+            {
+                core.GetHydrostaticOperator().GetEnthalpyMap()
+            } -> std::same_as<const field::FieldDofMap &>;
+            {
+                core.GetDomainDeformation().parameterCount()
+            } -> std::same_as<int>;
+        };
+
+    namespace detail {
+        template <typename Generated, CoordinateKind Kind>
+        using GeneratedPhysicalRieszMethod =
+            typename DeclaredGeneratedPhysicalRieszCoordinate<Generated, Kind>::Method;
+
+        template <typename Generated, CoordinateKind Kind, typename = void>
+        struct GeneratedPhysicalRieszRuntimeCoordinate : std::false_type { };
+
+        template <typename Generated, CoordinateKind Kind>
+        struct GeneratedPhysicalRieszRuntimeCoordinate<
+            Generated,
+            Kind,
+            std::void_t<decltype(GeneratedPhysicalRieszMethod<Generated, Kind>::topology)>>
+            : std::bool_constant<
+                  DeclaredGeneratedPhysicalRieszCoordinate<Generated, Kind>::registered &&
+                  GeneratedPhysicalRieszMethod<Generated, Kind>::topology ==
+                      RieszTopology::global_scalar> { };
+
+        template <typename Specification, typename = void>
+        struct SpecificationPhysicalRieszRuntimeCoverage : std::false_type { };
+
+        template <typename Values, typename Residuals>
+        struct GeneratedPhysicalRieszRuntimeCoverage : std::false_type { };
+
+        template <typename... Values, typename... Residuals>
+        struct GeneratedPhysicalRieszRuntimeCoverage<
+            models::ModelTypeList<Values...>,
+            models::ModelTypeList<Residuals...>>
+            : std::bool_constant<
+                  (GeneratedPhysicalRieszRuntimeCoordinate<Values, CoordinateKind::value>::value && ...) &&
+                  (GeneratedPhysicalRieszRuntimeCoordinate<Residuals, CoordinateKind::residual>::value && ...)> { };
+
+        template <models::ModelSpecification Specification>
+        struct SpecificationPhysicalRieszRuntimeCoverage<
+            Specification,
+            std::void_t<
+                typename models::SpecificationContribution<Specification>::GeneratedValues,
+                typename models::SpecificationContribution<Specification>::GeneratedResiduals>>
+            : GeneratedPhysicalRieszRuntimeCoverage<
+                  typename models::SpecificationContribution<Specification>::GeneratedValues,
+                  typename models::SpecificationContribution<Specification>::GeneratedResiduals> { };
+
+        template <typename SpecificationTypes>
+        struct SpecificationSetPhysicalRieszRuntimeCoverage : std::false_type { };
+
+        template <models::ModelSpecification... Specifications>
+        struct SpecificationSetPhysicalRieszRuntimeCoverage<
+            models::detail::SpecificationSetStorage<Specifications...>>
+            : std::bool_constant<
+                  (SpecificationPhysicalRieszRuntimeCoverage<Specifications>::value && ...)> { };
+    } // namespace detail
+
+    template <typename Specification>
+    concept CompleteGeneratedPhysicalRieszRuntimeNormalizationFor =
+        detail::SpecificationPhysicalRieszRuntimeCoverage<
+            std::remove_cvref_t<Specification>>::value;
+
+#define MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(BlockType, TopologyValue, ScaleValue)                                         \
+    template <> struct PhysicalRieszBlockTraits<BlockType> {                                                         \
+        using Method = PhysicalRieszCoordinate<RieszTopology::TopologyValue, PhysicalScaleKind::ScaleValue>;         \
+        static constexpr bool registered = true;                                                                     \
+    }
+
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::density::mass::value,
+        scalar_volume_l2,
+        density
+    );
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::surface_deformation::parameters::value,
+        scalar_boundary_l2,
+        length
+    );
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::gravity::gradient::value,
+        vector_volume_l2,
+        acceleration
+    );
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::gravity::poisson::value,
+        scalar_volume_l2,
+        specific_energy
+    );
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::enthalpy::specific::value,
+        scalar_volume_l2,
+        specific_energy
+    );
+
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::gravity::gradient::residual,
+        vector_volume_l2,
+        acceleration
+    );
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::gravity::poisson::residual,
+        scalar_volume_l2,
+        inverse_time_squared
+    );
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::density::mass::residual,
+        scalar_volume_l2,
+        density
+    );
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::surface_deformation::shape_equilibrium::residual,
+        scalar_boundary_l2,
+        force
+    );
+    MEAN_FIELD_PHYSICAL_RIESZ_TRAIT(
+        utils::blocks::enthalpy::specific::residual,
+        hybrid_scalar_volume_point_rows,
+        specific_energy
+    );
+#undef MEAN_FIELD_PHYSICAL_RIESZ_TRAIT
+
+    template <typename Block>
+    [[nodiscard]] double physicalScale(
+        const StellarCharacteristicScales &scales
+    ) {
+        static_assert(PhysicalRieszBlockTraits<Block>::registered, "The block has no Physical Riesz normalization.");
+        using Method = typename PhysicalRieszBlockTraits<Block>::Method;
+        constexpr PhysicalScaleKind scale = Method::scale;
+        if constexpr (scale == PhysicalScaleKind::dimensionless) {
+            return 1.0;
+        } else if constexpr (scale == PhysicalScaleKind::density) {
+            return scales.density;
+        } else if constexpr (scale == PhysicalScaleKind::length) {
+            return scales.radius.value();
+        } else if constexpr (scale == PhysicalScaleKind::acceleration) {
+            return scales.acceleration;
+        } else if constexpr (scale == PhysicalScaleKind::inverse_time_squared) {
+            return scales.inverseTimeSquared;
+        } else if constexpr (scale == PhysicalScaleKind::specific_energy) {
+            return scales.specificEnergy;
+        } else if constexpr (scale == PhysicalScaleKind::pressure) {
+            return scales.pressure;
+        } else if constexpr (scale == PhysicalScaleKind::mass) {
+            return scales.mass.value();
+        } else if constexpr (scale == PhysicalScaleKind::force) {
+            return scales.force;
+        } else if constexpr (scale == PhysicalScaleKind::angular_velocity) {
+            return scales.angularVelocity;
+        } else {
+            static_assert(scale == PhysicalScaleKind::angular_momentum);
+            return scales.angularMomentum;
+        }
+    }
+
+    namespace detail {
+        template <typename Values, typename Residuals> struct MakePhysicalRieszPlan;
+
+        template <typename... Values, typename... Residuals>
+        struct MakePhysicalRieszPlan<
+            utils::blocks::type_list<Values...>,
+            utils::blocks::type_list<Residuals...>> {
+            using Type = NormalizationPlan<
+                CoordinateComponent<
+                    CoordinateKind::value,
+                    utils::blocks::type_list<Values>,
+                    typename PhysicalRieszBlockTraits<Values>::Method>...,
+                CoordinateComponent<
+                    CoordinateKind::residual,
+                    utils::blocks::type_list<Residuals>,
+                    typename PhysicalRieszBlockTraits<Residuals>::Method>...>;
+        };
+    } // namespace detail
+
+    template <typename Form>
+        requires utils::blocks::block_form_is_valid_v<Form>
+    using PhysicalRieszNormalizationPlanFor = typename detail::MakePhysicalRieszPlan<
+        typename Form::value_blocks,
+        typename Form::residual_blocks>::Type;
+
+    /*
+     * Public compile-time extension point for a normalization prescription.
+     * A specialization owns both the complete coordinate plan and the
+     * low-level runtime compatibility predicate used before a discretized
+     * problem type is formed.  Keeping those declarations together prevents a
+     * policy from compiling a plan which the selected stellar core cannot
+     * actually prepare.
+     */
+    template <typename Prescription, typename Form> struct NormalizationCompilation {
+        using Plan = NormalizationPlan<>;
+        static constexpr bool registered = false;
+
+        template <typename PhysicalCore, typename SpecificationTypes>
+        static constexpr bool runtimeAvailableFor = false;
+    };
+
+    /* Astronomy/numerics-facing package for a policy which prepares one
+     * runtime diagonal over the complete inferred form and needs no private
+     * facility of a particular stellar core. The generated plan truthfully
+     * labels every coordinate as runtime-prepared by this exact policy. */
+    template <NormalizationPrescription Prescription, typename Form>
+        requires utils::blocks::block_form_is_valid_v<Form>
+    struct RuntimePreparedNormalizationCompilation {
+        using Plan = RuntimePreparedNormalizationPlanFor<Prescription, Form>;
+        static constexpr bool registered = CompleteNormalizationFor<Plan, Form>;
+
+        template <typename PhysicalCore, typename SpecificationTypes>
+        static constexpr bool runtimeAvailableFor = registered;
+    };
+
+    template <typename Form>
+        requires utils::blocks::block_form_is_valid_v<Form>
+    struct NormalizationCompilation<Unnormalized, Form> {
+        using Plan = IdentityNormalizationPlanFor<Form>;
+        static constexpr bool registered = CompleteNormalizationFor<Plan, Form>;
+
+        template <typename PhysicalCore, typename SpecificationTypes>
+        static constexpr bool runtimeAvailableFor = registered;
+    };
+
+    template <RieszGeometryPolicy Geometry, ReferenceScalePolicy ScaleSource, typename Form>
+        requires utils::blocks::block_form_is_valid_v<Form>
+    struct NormalizationCompilation<PhysicalRieszDiagonal<Geometry, ScaleSource>, Form> {
+        using Plan = PhysicalRieszNormalizationPlanFor<Form>;
+        static constexpr bool registered = CompleteNormalizationFor<Plan, Form>;
+
+        template <typename PhysicalCore, typename SpecificationTypes>
+        static constexpr bool runtimeAvailableFor =
+            registered &&
+            PhysicalRieszCoreRuntime<std::remove_cvref_t<PhysicalCore>> &&
+            detail::SpecificationSetPhysicalRieszRuntimeCoverage<
+                std::remove_cvref_t<SpecificationTypes>>::value;
+    };
+
+    namespace detail {
+        template <typename Prescription, typename Form, typename = void>
+        struct NormalizationCompilationAudit {
+            using Plan = NormalizationPlan<>;
+            static constexpr bool registered = false;
+        };
+
+        template <typename Prescription, typename Form>
+            requires NormalizationPrescription<std::remove_cvref_t<Prescription>> &&
+                     utils::blocks::block_form_is_valid_v<std::remove_cvref_t<Form>>
+        struct NormalizationCompilationAudit<
+            Prescription,
+            Form,
+            std::void_t<
+                typename NormalizationCompilation<
+                    std::remove_cvref_t<Prescription>,
+                    std::remove_cvref_t<Form>>::Plan,
+                decltype(std::bool_constant<static_cast<bool>(
+                    NormalizationCompilation<
+                        std::remove_cvref_t<Prescription>,
+                        std::remove_cvref_t<Form>>::registered)>{})>> {
+            using Compilation = NormalizationCompilation<
+                std::remove_cvref_t<Prescription>,
+                std::remove_cvref_t<Form>>;
+            using Plan = typename Compilation::Plan;
+
+            static constexpr bool registered =
+                static_cast<bool>(Compilation::registered) &&
+                CompleteNormalizationFor<Plan, std::remove_cvref_t<Form>>;
+        };
+    } // namespace detail
+
+    template <NormalizationPrescription Prescription, typename Form>
+    using NormalizationPlanFor = typename detail::NormalizationCompilationAudit<
+        std::remove_cvref_t<Prescription>,
+        Form>::Plan;
+
+    template <typename Prescription, typename Form>
+    concept CompilableNormalizationFor =
+        detail::NormalizationCompilationAudit<
+            std::remove_cvref_t<Prescription>,
+            std::remove_cvref_t<Form>>::registered;
+
+    /* The public runtime-preparation adapter is intentionally narrower than
+     * an arbitrary complete plan: every coordinate must name the exact policy
+     * which supplies its runtime factor.  This prevents a custom policy from
+     * advertising IdentityCoordinate (or another policy's method) while
+     * silently installing a different diagonal at runtime. */
+    template <typename Prescription, typename Form>
+    concept RuntimePreparedNormalizationFor =
+        NormalizationPrescription<std::remove_cvref_t<Prescription>> &&
+        utils::blocks::block_form_is_valid_v<std::remove_cvref_t<Form>> &&
+        CompilableNormalizationFor<
+            std::remove_cvref_t<Prescription>,
+            std::remove_cvref_t<Form>> &&
+        std::same_as<
+            NormalizationPlanFor<
+                std::remove_cvref_t<Prescription>,
+                std::remove_cvref_t<Form>>,
+            RuntimePreparedNormalizationPlanFor<
+                std::remove_cvref_t<Prescription>,
+                std::remove_cvref_t<Form>>>;
+
+    namespace detail {
+        template <
+            typename Prescription,
+            typename Form,
+            typename PhysicalCore,
+            typename SpecificationTypes,
+            typename = void>
+        struct StellarNormalizationRuntimeAudit : std::false_type { };
+
+        template <
+            typename Prescription,
+            typename Form,
+            typename PhysicalCore,
+            typename SpecificationTypes>
+        struct StellarNormalizationRuntimeAudit<
+            Prescription,
+            Form,
+            PhysicalCore,
+            SpecificationTypes,
+            std::void_t<
+                std::enable_if_t<NormalizationCompilationAudit<
+                    Prescription,
+                    Form>::registered>,
+                decltype(std::bool_constant<static_cast<bool>(
+                    NormalizationCompilation<
+                        Prescription,
+                        Form>::template runtimeAvailableFor<
+                            PhysicalCore,
+                            SpecificationTypes>)>{})>>
+            : std::bool_constant<
+                  (std::same_as<Prescription, Unnormalized> ||
+                   PhysicalRieszDiagonalPrescription<Prescription> ||
+                   RuntimePreparedNormalizationFor<Prescription, Form>) &&
+                  static_cast<bool>(NormalizationCompilation<
+                      Prescription,
+                      Form>::template runtimeAvailableFor<
+                          PhysicalCore,
+                          SpecificationTypes>)> { };
+    } // namespace detail
+
+    /*
+     * Single detection-safe authority for pairing a compiled stellar form,
+     * its selected physical core, and a runtime normalization prescription.
+     * Each public NormalizationCompilation specialization declares this
+     * compatibility alongside its plan. The identity policy needs only a
+     * complete plan. Physical Riesz also requires every map consumed during
+     * assembly and global-scalar runtime preparation for every generated
+     * coordinate in the specification pack.
+     */
+    template <
+        typename Prescription,
+        typename Form,
+        typename PhysicalCore,
+        typename SpecificationTypes>
+    concept StellarNormalizationRuntimeAvailableFor =
+        detail::StellarNormalizationRuntimeAudit<
+            std::remove_cvref_t<Prescription>,
+            std::remove_cvref_t<Form>,
+            std::remove_cvref_t<PhysicalCore>,
+            std::remove_cvref_t<SpecificationTypes>>::value;
+} // namespace mean_field::normalization

libmeanfield/interface/normalization/plan.cppm

@@ -0,0 +1,376 @@
+module;
+
+#include <concepts>
+#include <type_traits>
+
+export module mean_field:normalization.plan;
+
+export import :utils.blocks;
+
+export namespace mean_field::normalization {
+    struct NormalizationPrescriptionTag { };
+
+    template <typename Candidate>
+    concept NormalizationPrescription =
+        std::derived_from<
+            std::remove_cvref_t<Candidate>,
+            NormalizationPrescriptionTag>;
+
+    enum class CoordinateKind { value, residual };
+
+    enum class RieszTopology {
+        identity,
+        scalar_volume_l2,
+        vector_volume_l2,
+        scalar_boundary_l2,
+        hybrid_scalar_volume_point_rows,
+        global_scalar
+    };
+
+    enum class PhysicalScaleKind {
+        dimensionless,
+        density,
+        length,
+        acceleration,
+        inverse_time_squared,
+        specific_energy,
+        pressure,
+        mass,
+        force,
+        angular_velocity,
+        angular_momentum
+    };
+
+    struct IdentityCoordinate final { };
+
+    /*
+     * Honest compile-time method for a coordinate whose positive diagonal
+     * factor is supplied at runtime by one exact normalization prescription.
+     * Unlike IdentityCoordinate, this category makes no claim about the
+     * numerical value of that factor.  The owner type prevents one policy from
+     * silently presenting another policy's runtime map as its own plan.
+     */
+    template <NormalizationPrescription Prescription>
+    struct RuntimePreparedCoordinate final {
+        using PrescriptionType = std::remove_cvref_t<Prescription>;
+    };
+
+    template <RieszTopology Topology, PhysicalScaleKind Scale> struct PhysicalRieszCoordinate final {
+        static constexpr RieszTopology topology       = Topology;
+        static constexpr PhysicalScaleKind scale      = Scale;
+    };
+
+    struct UnsupportedPhysicalRieszCoordinate final { };
+
+    template <typename Block> struct PhysicalRieszBlockTraits {
+        using Method                         = UnsupportedPhysicalRieszCoordinate;
+        static constexpr bool registered     = false;
+    };
+
+    template <CoordinateKind Kind, typename BlockList, typename MethodType>
+    struct CoordinateComponent final {
+        using Blocks                         = BlockList;
+        using Method                         = MethodType;
+        static constexpr CoordinateKind kind = Kind;
+
+        using ValueBlocks = std::conditional_t<
+            Kind == CoordinateKind::value,
+            BlockList,
+            utils::blocks::type_list<>>;
+        using ResidualBlocks = std::conditional_t<
+            Kind == CoordinateKind::residual,
+            BlockList,
+            utils::blocks::type_list<>>;
+    };
+
+    namespace detail {
+        template <typename Candidate> struct IsTypeList : std::false_type { };
+
+        template <typename... Types>
+        struct IsTypeList<utils::blocks::type_list<Types...>> : std::true_type { };
+
+        template <typename List, typename Base> struct IsUniqueDerivedBlockList : std::false_type { };
+
+        template <typename Base, typename... Blocks>
+        struct IsUniqueDerivedBlockList<utils::blocks::type_list<Blocks...>, Base>
+            : std::bool_constant<
+                  (std::derived_from<Blocks, Base> && ...) &&
+                  utils::blocks::types_are_unique_v<utils::blocks::type_list<Blocks...>>> { };
+
+        template <typename Method> struct IsCoordinateMethod : std::false_type { };
+
+        template <> struct IsCoordinateMethod<IdentityCoordinate> : std::true_type { };
+
+        template <NormalizationPrescription Prescription>
+        struct IsCoordinateMethod<RuntimePreparedCoordinate<Prescription>>
+            : std::true_type { };
+
+        template <RieszTopology Topology, PhysicalScaleKind Scale>
+        struct IsCoordinateMethod<PhysicalRieszCoordinate<Topology, Scale>> : std::true_type { };
+
+        template <typename Method, typename Block> struct MethodSupportsBlock : std::false_type { };
+
+        template <typename Block>
+        struct MethodSupportsBlock<IdentityCoordinate, Block>
+            : std::bool_constant<std::derived_from<Block, utils::blocks::block>> { };
+
+        template <NormalizationPrescription Prescription, typename Block>
+        struct MethodSupportsBlock<RuntimePreparedCoordinate<Prescription>, Block>
+            : std::bool_constant<std::derived_from<Block, utils::blocks::block>> { };
+
+        template <RieszTopology Topology, PhysicalScaleKind Scale, typename Block>
+        struct MethodSupportsBlock<PhysicalRieszCoordinate<Topology, Scale>, Block>
+            : std::bool_constant<
+                  PhysicalRieszBlockTraits<Block>::registered &&
+                  std::same_as<
+                      typename PhysicalRieszBlockTraits<Block>::Method,
+                      PhysicalRieszCoordinate<Topology, Scale>>> { };
+
+        template <typename Method, typename List> struct MethodSupportsEveryBlock : std::false_type { };
+
+        template <typename Method, typename... Blocks>
+        struct MethodSupportsEveryBlock<Method, utils::blocks::type_list<Blocks...>>
+            : std::bool_constant<(MethodSupportsBlock<Method, Blocks>::value && ...)> { };
+
+        template <typename Candidate, typename = void> struct ComponentTraits {
+            static constexpr bool valid = false;
+        };
+
+        template <typename Candidate>
+        struct ComponentTraits<
+            Candidate,
+            std::void_t<
+                typename Candidate::Blocks,
+                typename Candidate::Method,
+                typename Candidate::ValueBlocks,
+                typename Candidate::ResidualBlocks,
+                decltype(Candidate::kind)>> {
+            using Blocks         = typename Candidate::Blocks;
+            using Method         = typename Candidate::Method;
+            using ValueBlocks    = typename Candidate::ValueBlocks;
+            using ResidualBlocks = typename Candidate::ResidualBlocks;
+
+            static constexpr bool hasValidKind =
+                std::same_as<std::remove_cv_t<decltype(Candidate::kind)>, CoordinateKind>;
+
+            static constexpr bool hasValidBlockList = [] {
+                if constexpr (!hasValidKind || !IsTypeList<Blocks>::value) {
+                    return false;
+                } else if constexpr (Candidate::kind == CoordinateKind::value) {
+                    return IsUniqueDerivedBlockList<Blocks, utils::blocks::value_block_base>::value;
+                } else if constexpr (Candidate::kind == CoordinateKind::residual) {
+                    return IsUniqueDerivedBlockList<Blocks, utils::blocks::residual_block_base>::value;
+                } else {
+                    return false;
+                }
+            }();
+
+            static constexpr bool hasCoherentCoordinateLists = [] {
+                if constexpr (!hasValidKind || !IsTypeList<ValueBlocks>::value ||
+                              !IsTypeList<ResidualBlocks>::value) {
+                    return false;
+                } else if constexpr (Candidate::kind == CoordinateKind::value) {
+                    return std::same_as<ValueBlocks, Blocks> &&
+                           std::same_as<ResidualBlocks, utils::blocks::type_list<>>;
+                } else if constexpr (Candidate::kind == CoordinateKind::residual) {
+                    return std::same_as<ValueBlocks, utils::blocks::type_list<>> &&
+                           std::same_as<ResidualBlocks, Blocks>;
+                } else {
+                    return false;
+                }
+            }();
+
+            static constexpr bool valid = hasValidKind && IsTypeList<Blocks>::value &&
+                                          IsCoordinateMethod<Method>::value && hasValidBlockList &&
+                                          hasCoherentCoordinateLists &&
+                                          MethodSupportsEveryBlock<Method, Blocks>::value;
+        };
+
+        template <typename... Lists> struct Concatenate;
+
+        template <> struct Concatenate<> {
+            using Type = utils::blocks::type_list<>;
+        };
+
+        template <typename... Types> struct Concatenate<utils::blocks::type_list<Types...>> {
+            using Type = utils::blocks::type_list<Types...>;
+        };
+
+        template <typename... Left, typename... Right, typename... Remaining>
+        struct Concatenate<utils::blocks::type_list<Left...>, utils::blocks::type_list<Right...>, Remaining...> {
+            using Type = typename Concatenate<utils::blocks::type_list<Left..., Right...>, Remaining...>::Type;
+        };
+
+        template <typename... Lists> using ConcatenateT = typename Concatenate<Lists...>::Type;
+
+        template <typename List, typename Type> struct Append;
+
+        template <typename... Types, typename Appended>
+        struct Append<utils::blocks::type_list<Types...>, Appended> {
+            using Type = utils::blocks::type_list<Types..., Appended>;
+        };
+
+        template <typename List, typename Type> using AppendT = typename Append<List, Type>::Type;
+
+        template <typename List, typename Type>
+        using AppendUniqueT = std::conditional_t<
+            utils::blocks::contains_type_v<Type, List>,
+            List,
+            AppendT<List, Type>>;
+
+        template <typename Source, typename Excluded> struct ListDifference;
+
+        template <typename Excluded>
+        struct ListDifference<utils::blocks::type_list<>, Excluded> {
+            using Type = utils::blocks::type_list<>;
+        };
+
+        template <typename Head, typename... Tail, typename Excluded>
+        struct ListDifference<utils::blocks::type_list<Head, Tail...>, Excluded> {
+        private:
+            using Remaining = typename ListDifference<utils::blocks::type_list<Tail...>, Excluded>::Type;
+
+        public:
+            using Type = std::conditional_t<
+                utils::blocks::contains_type_v<Head, Excluded>,
+                Remaining,
+                ConcatenateT<utils::blocks::type_list<Head>, Remaining>>;
+        };
+
+        template <typename Source, typename Excluded>
+        using ListDifferenceT = typename ListDifference<Source, Excluded>::Type;
+
+        template <typename Remaining, typename Original, typename Repeated> struct CollectRepeatedTypes;
+
+        template <typename Original, typename Repeated>
+        struct CollectRepeatedTypes<utils::blocks::type_list<>, Original, Repeated> {
+            using Type = Repeated;
+        };
+
+        template <typename Head, typename... Tail, typename Original, typename Repeated>
+        struct CollectRepeatedTypes<utils::blocks::type_list<Head, Tail...>, Original, Repeated> {
+        private:
+            using Next = std::conditional_t<
+                (utils::blocks::type_count_v<Head, Original> > 1),
+                AppendUniqueT<Repeated, Head>,
+                Repeated>;
+
+        public:
+            using Type = typename CollectRepeatedTypes<utils::blocks::type_list<Tail...>, Original, Next>::Type;
+        };
+
+        template <typename List>
+        using RepeatedTypesT = typename CollectRepeatedTypes<
+            List,
+            List,
+            utils::blocks::type_list<>>::Type;
+
+        template <typename Candidate, typename = void> struct PlanTraits {
+            static constexpr bool valid = false;
+        };
+    } // namespace detail
+
+    template <typename Candidate>
+    concept NormalizationComponent = detail::ComponentTraits<std::remove_cvref_t<Candidate>>::valid;
+
+    template <typename... Components> struct NormalizationPlan final {
+        using ComponentTypes  = utils::blocks::type_list<Components...>;
+        using ValueBlocks     = detail::ConcatenateT<typename Components::ValueBlocks...>;
+        using ResidualBlocks  = detail::ConcatenateT<typename Components::ResidualBlocks...>;
+    };
+
+    namespace detail {
+        template <typename... Components>
+        struct PlanTraits<NormalizationPlan<Components...>> {
+            static constexpr bool valid = (ComponentTraits<Components>::valid && ...);
+        };
+
+        template <typename Values, typename Residuals> struct MakeIdentityPlan;
+
+        template <typename... Values, typename... Residuals>
+        struct MakeIdentityPlan<
+            utils::blocks::type_list<Values...>,
+            utils::blocks::type_list<Residuals...>> {
+            using Type = NormalizationPlan<
+                CoordinateComponent<CoordinateKind::value, utils::blocks::type_list<Values>, IdentityCoordinate>...,
+                CoordinateComponent<
+                    CoordinateKind::residual,
+                    utils::blocks::type_list<Residuals>,
+                    IdentityCoordinate>...>;
+        };
+
+        template <
+            NormalizationPrescription Prescription,
+            typename Values,
+            typename Residuals>
+        struct MakeRuntimePreparedPlan;
+
+        template <
+            NormalizationPrescription Prescription,
+            typename... Values,
+            typename... Residuals>
+        struct MakeRuntimePreparedPlan<
+            Prescription,
+            utils::blocks::type_list<Values...>,
+            utils::blocks::type_list<Residuals...>> {
+            using Method = RuntimePreparedCoordinate<Prescription>;
+            using Type = NormalizationPlan<
+                CoordinateComponent<
+                    CoordinateKind::value,
+                    utils::blocks::type_list<Values>,
+                    Method>...,
+                CoordinateComponent<
+                    CoordinateKind::residual,
+                    utils::blocks::type_list<Residuals>,
+                    Method>...>;
+        };
+    } // namespace detail
+
+    template <typename Candidate>
+    concept NormalizationPlanType = detail::PlanTraits<std::remove_cvref_t<Candidate>>::valid;
+
+    template <typename Form>
+        requires utils::blocks::block_form_is_valid_v<Form>
+    using IdentityNormalizationPlanFor = typename detail::MakeIdentityPlan<
+        typename Form::value_blocks,
+        typename Form::residual_blocks>::Type;
+
+    template <NormalizationPrescription Prescription, typename Form>
+        requires utils::blocks::block_form_is_valid_v<Form>
+    using RuntimePreparedNormalizationPlanFor =
+        typename detail::MakeRuntimePreparedPlan<
+            std::remove_cvref_t<Prescription>,
+            typename Form::value_blocks,
+            typename Form::residual_blocks>::Type;
+
+    template <typename Form, typename Plan>
+        requires utils::blocks::block_form_is_valid_v<Form>
+    struct NormalizationCoverage final {
+        using DeclaredValueBlocks = typename Plan::ValueBlocks;
+        using DeclaredResidualBlocks = typename Plan::ResidualBlocks;
+
+        using MissingValueBlocks = detail::ListDifferenceT<typename Form::value_blocks, DeclaredValueBlocks>;
+        using UnexpectedValueBlocks = detail::ListDifferenceT<DeclaredValueBlocks, typename Form::value_blocks>;
+        using RepeatedValueBlocks = detail::RepeatedTypesT<DeclaredValueBlocks>;
+
+        using MissingResidualBlocks = detail::ListDifferenceT<typename Form::residual_blocks, DeclaredResidualBlocks>;
+        using UnexpectedResidualBlocks = detail::ListDifferenceT<DeclaredResidualBlocks, typename Form::residual_blocks>;
+        using RepeatedResidualBlocks = detail::RepeatedTypesT<DeclaredResidualBlocks>;
+
+        static constexpr bool hasEveryValueBlock = MissingValueBlocks::size == 0;
+        static constexpr bool hasOnlyValueBlocks = UnexpectedValueBlocks::size == 0;
+        static constexpr bool hasUniqueValueOwners = RepeatedValueBlocks::size == 0;
+        static constexpr bool hasEveryResidualBlock = MissingResidualBlocks::size == 0;
+        static constexpr bool hasOnlyResidualBlocks = UnexpectedResidualBlocks::size == 0;
+        static constexpr bool hasUniqueResidualOwners = RepeatedResidualBlocks::size == 0;
+
+        static constexpr bool complete = hasEveryValueBlock && hasOnlyValueBlocks && hasUniqueValueOwners &&
+                                         hasEveryResidualBlock && hasOnlyResidualBlocks &&
+                                         hasUniqueResidualOwners;
+    };
+
+    template <typename Plan, typename Form>
+    concept CompleteNormalizationFor = utils::blocks::block_form_is_valid_v<Form> &&
+                                       NormalizationPlanType<Plan> &&
+                                       NormalizationCoverage<Form, std::remove_cvref_t<Plan>>::complete;
+} // namespace mean_field::normalization

libmeanfield/interface/normalization/stellar_equilibrium.cppm

@@ -0,0 +1,921 @@
+module;
+
+#include <concepts>
+#include <cstdint>
+#include <memory>
+#include <span>
+#include <stdexcept>
+#include <string>
+#include <type_traits>
+#include <utility>
+
+#include <mfem.hpp>
+
+export module mean_field:normalization.stellar_equilibrium;
+
+export import :normalization.operators;
+export import :operators.stellar_equilibrium_compiler;
+export import :operators.stellar_equilibrium_problem;
+export import :utils.domain;
+
+namespace mean_field::normalization::detail {
+    using DomainSchema = utils::domain::CoreEnvelopeVacuumDomainSchema;
+
+    [[nodiscard]] inline mfem::Vector AssembleScalarMassDiagonal(
+        mfem::ParFiniteElementSpace &space,
+        mfem::Array<int> *domainMarker = nullptr
+    ) {
+        mfem::ParBilinearForm mass(&space);
+        if (domainMarker == nullptr) {
+            mass.AddDomainIntegrator(new mfem::MassIntegrator());
+        } else {
+            mass.AddDomainIntegrator(new mfem::MassIntegrator(), *domainMarker);
+        }
+        mass.Assemble();
+        mass.Finalize();
+        std::unique_ptr<mfem::HypreParMatrix> matrix(mass.ParallelAssemble());
+        if (matrix == nullptr) {
+            throw std::runtime_error("Reference scalar Riesz mass assembly failed.");
+        }
+        mfem::Vector diagonal;
+        matrix->GetDiag(diagonal);
+        return diagonal;
+    }
+
+    [[nodiscard]] inline mfem::Vector AssembleHDivMassDiagonal(mfem::ParFiniteElementSpace &space) {
+        mfem::ParBilinearForm mass(&space);
+        mass.AddDomainIntegrator(new mfem::VectorFEMassIntegrator());
+        mass.Assemble();
+        mass.Finalize();
+        std::unique_ptr<mfem::HypreParMatrix> matrix(mass.ParallelAssemble());
+        if (matrix == nullptr) {
+            throw std::runtime_error("Reference H(div) Riesz mass assembly failed.");
+        }
+        mfem::Vector diagonal;
+        matrix->GetDiag(diagonal);
+        return diagonal;
+    }
+
+    [[nodiscard]] inline mfem::Vector AssembleSurfaceMassDiagonal(
+        const fem::FEM &finiteElements,
+        const field::ScalarBoundaryDofMap &surfaceMap
+    ) {
+        mfem::Array<int> marker(finiteElements.mesh->bdr_attributes.Max());
+        marker = 0;
+        constexpr int attribute = DomainSchema::template boundary_attribute<utils::domain::StellarSurface>();
+        if (attribute <= 0 || attribute > marker.Size()) {
+            throw std::invalid_argument("The reference mesh does not contain the stellar-surface boundary.");
+        }
+        marker[attribute - 1] = 1;
+
+        mfem::ParBilinearForm mass(finiteElements.surfaceDeformationFes.get());
+        mass.AddBoundaryIntegrator(new mfem::MassIntegrator(), marker);
+        mass.Assemble();
+        mass.Finalize();
+        std::unique_ptr<mfem::HypreParMatrix> matrix(mass.ParallelAssemble());
+        if (matrix == nullptr) {
+            throw std::runtime_error("Reference surface Riesz mass assembly failed.");
+        }
+        mfem::Vector ambientDiagonal;
+        matrix->GetDiag(ambientDiagonal);
+        return surfaceMap.gather(ambientDiagonal);
+    }
+
+    template <equilibrium::DiscretizedStellarEquilibriumProblem Problem>
+    [[nodiscard]] const auto &PhysicalOperator(const Problem &problem) {
+        return problem.GetPhysicalOperator();
+    }
+
+    [[nodiscard]] inline mfem::Vector GatherDiagonal(
+        const mfem::Vector &fullDiagonal,
+        const field::FieldDofMap &map,
+        const char *role
+    ) {
+        if (fullDiagonal.Size() != map.full_size()) {
+            throw std::logic_error(std::string("The reference ") + role + " Gram diagonal has an incompatible map.");
+        }
+        return map.gather(fullDiagonal);
+    }
+} // namespace mean_field::normalization::detail
+
+export namespace mean_field::normalization {
+    /*
+     * Runtime preparation paired with the compile-time normalization plan.
+     * The operator compiler is the authority for which blocks a specification
+     * generated, and PhysicalRieszBlockTraits is the authority for their
+     * declared physical laws.  Keeping those responsibilities separate means
+     * this layer never names a concrete integral or phase constraint.
+     */
+    namespace detail {
+        template <typename Block>
+        using PhysicalRieszMethodFor = typename PhysicalRieszBlockTraits<Block>::Method;
+
+        template <typename Block, typename = void>
+        struct IsGlobalGeneratedValueNormalization : std::false_type { };
+
+        template <typename Generated>
+        struct IsGlobalGeneratedValueNormalization<
+            utils::blocks::generated_value_block<Generated>,
+            std::void_t<
+                decltype(PhysicalRieszMethodFor<
+                         utils::blocks::generated_value_block<Generated>>::topology),
+                decltype(PhysicalRieszMethodFor<
+                         utils::blocks::generated_value_block<Generated>>::scale)>>
+            : std::bool_constant<
+                  PhysicalRieszBlockTraits<
+                      utils::blocks::generated_value_block<Generated>>::registered &&
+                  PhysicalRieszMethodFor<
+                      utils::blocks::generated_value_block<Generated>>::topology ==
+                      RieszTopology::global_scalar> { };
+
+        template <typename Blocks, typename Specification>
+        struct GeneratedValueBlocksBelongToSpecification : std::false_type { };
+
+        template <typename Generated, typename Specification, typename = void>
+        struct GeneratedCoordinateBelongsToSpecification : std::false_type { };
+
+        template <typename Generated, typename Specification>
+        struct GeneratedCoordinateBelongsToSpecification<
+            Generated,
+            Specification,
+            std::void_t<typename Generated::SpecificationType>>
+            : std::bool_constant<
+                  std::same_as<typename Generated::SpecificationType, Specification>> { };
+
+        template <typename Specification, typename... Generated>
+        struct GeneratedValueBlocksBelongToSpecification<
+            utils::blocks::type_list<utils::blocks::generated_value_block<Generated>...>,
+            Specification>
+            : std::bool_constant<
+                  (GeneratedCoordinateBelongsToSpecification<Generated, Specification>::value && ...)> { };
+
+        template <typename Block, typename = void>
+        struct IsGlobalGeneratedResidualNormalization : std::false_type { };
+
+        template <typename Generated>
+        struct IsGlobalGeneratedResidualNormalization<
+            utils::blocks::generated_residual_block<Generated>,
+            std::void_t<
+                decltype(PhysicalRieszMethodFor<
+                         utils::blocks::generated_residual_block<Generated>>::topology),
+                decltype(PhysicalRieszMethodFor<
+                         utils::blocks::generated_residual_block<Generated>>::scale)>>
+            : std::bool_constant<
+                  PhysicalRieszBlockTraits<
+                      utils::blocks::generated_residual_block<Generated>>::registered &&
+                  PhysicalRieszMethodFor<
+                      utils::blocks::generated_residual_block<Generated>>::topology ==
+                      RieszTopology::global_scalar> { };
+
+        template <typename Blocks, typename Specification>
+        struct GeneratedResidualBlocksBelongToSpecification : std::false_type { };
+
+        template <typename Specification, typename... Generated>
+        struct GeneratedResidualBlocksBelongToSpecification<
+            utils::blocks::type_list<utils::blocks::generated_residual_block<Generated>...>,
+            Specification>
+            : std::bool_constant<
+                  (GeneratedCoordinateBelongsToSpecification<Generated, Specification>::value && ...)> { };
+
+        template <typename Blocks> struct PrepareGeneratedValueNormalizations {
+            static constexpr bool registered = false;
+
+            template <typename Form>
+            static constexpr bool completeFor = false;
+
+            template <typename Form>
+            static void Apply(
+                DiagonalNormalizationBuilder<Form> &,
+                const StellarCharacteristicScales &
+            ) {
+                static_assert(registered, "Generated value-block normalization metadata is malformed.");
+            }
+        };
+
+        template <typename... Blocks>
+        struct PrepareGeneratedValueNormalizations<utils::blocks::type_list<Blocks...>> {
+            static constexpr bool registered =
+                (IsGlobalGeneratedValueNormalization<Blocks>::value && ...);
+
+            template <typename Form>
+            static constexpr bool completeFor = registered &&
+                utils::blocks::block_form_is_valid_v<Form> &&
+                (utils::blocks::contains_type_v<Blocks, typename Form::value_blocks> && ...);
+
+            template <typename Form>
+            static void Apply(
+                DiagonalNormalizationBuilder<Form> &builder,
+                const StellarCharacteristicScales &scales
+            ) {
+                if constexpr (completeFor<Form>) {
+                    (builder.template SetValueGlobal<Blocks>(physicalScale<Blocks>(scales)), ...);
+                } else {
+                    static_assert(
+                        completeFor<Form>,
+                        "Every generated value block must have a declared global-scalar Physical Riesz law "
+                        "and belong to the compiled equilibrium form."
+                    );
+                }
+            }
+        };
+
+        template <typename Blocks> struct PrepareGeneratedResidualNormalizations {
+            static constexpr bool registered = false;
+
+            template <typename Form>
+            static constexpr bool completeFor = false;
+
+            template <typename Form>
+            static void Apply(
+                DiagonalNormalizationBuilder<Form> &,
+                const StellarCharacteristicScales &
+            ) {
+                static_assert(registered, "Generated residual-block normalization metadata is malformed.");
+            }
+        };
+
+        template <typename... Blocks>
+        struct PrepareGeneratedResidualNormalizations<utils::blocks::type_list<Blocks...>> {
+            static constexpr bool registered =
+                (IsGlobalGeneratedResidualNormalization<Blocks>::value && ...);
+
+            template <typename Form>
+            static constexpr bool completeFor = registered &&
+                utils::blocks::block_form_is_valid_v<Form> &&
+                (utils::blocks::contains_type_v<Blocks, typename Form::residual_blocks> && ...);
+
+            template <typename Form>
+            static void Apply(
+                DiagonalNormalizationBuilder<Form> &builder,
+                const StellarCharacteristicScales &scales
+            ) {
+                if constexpr (completeFor<Form>) {
+                    (builder.template SetResidualGlobal<Blocks>(physicalScale<Blocks>(scales)), ...);
+                } else {
+                    static_assert(
+                        completeFor<Form>,
+                        "Every generated residual block must have a declared global-scalar Physical Riesz law "
+                        "and belong to the compiled equilibrium form."
+                    );
+                }
+            }
+        };
+
+        template <typename Specification, typename = void>
+        struct CompileStellarSpecificationNormalization {
+            using ValuePreparation = PrepareGeneratedValueNormalizations<void>;
+            using ResidualPreparation = PrepareGeneratedResidualNormalizations<void>;
+
+            static constexpr bool registered = false;
+
+            template <typename Form>
+            static constexpr bool completeFor = false;
+
+            template <typename Form>
+            static void Apply(
+                DiagonalNormalizationBuilder<Form> &,
+                const StellarCharacteristicScales &
+            ) {
+                static_assert(
+                    completeFor<Form>,
+                    "The specification has no complete generated-coordinate normalization."
+                );
+            }
+        };
+
+        template <models::ModelSpecification Specification>
+        struct CompileStellarSpecificationNormalization<
+            Specification,
+            std::void_t<
+                typename operators::StellarEquilibriumSpecificationCompilation<
+                    Specification>::GeneratedValueBlocks,
+                typename operators::StellarEquilibriumSpecificationCompilation<
+                    Specification>::GeneratedResidualBlocks>> {
+            using OperatorCompilation =
+                operators::StellarEquilibriumSpecificationCompilation<Specification>;
+            using ValuePreparation = PrepareGeneratedValueNormalizations<
+                typename OperatorCompilation::GeneratedValueBlocks>;
+            using ResidualPreparation = PrepareGeneratedResidualNormalizations<
+                typename OperatorCompilation::GeneratedResidualBlocks>;
+
+            static constexpr bool registered = OperatorCompilation::complete &&
+                                               models::CompleteGeneratedNormalizationFor<
+                                                   Specification> &&
+                                               GeneratedValueBlocksBelongToSpecification<
+                                                   typename OperatorCompilation::GeneratedValueBlocks,
+                                                   Specification>::value &&
+                                               GeneratedResidualBlocksBelongToSpecification<
+                                                   typename OperatorCompilation::GeneratedResidualBlocks,
+                                                   Specification>::value &&
+                                               ValuePreparation::registered &&
+                                               ResidualPreparation::registered;
+
+            template <typename Form>
+            static constexpr bool completeFor = registered &&
+                                                ValuePreparation::template completeFor<Form> &&
+                                                ResidualPreparation::template completeFor<Form>;
+
+            template <typename Form>
+            static void Apply(
+                DiagonalNormalizationBuilder<Form> &builder,
+                const StellarCharacteristicScales &scales
+            ) {
+                if constexpr (completeFor<Form>) {
+                    ValuePreparation::template Apply<Form>(builder, scales);
+                    ResidualPreparation::template Apply<Form>(builder, scales);
+                } else {
+                    static_assert(
+                        completeFor<Form>,
+                        "The specification's generated blocks do not have a complete runtime normalization."
+                    );
+                }
+            }
+        };
+
+        template <typename SpecificationSet> struct PrepareSpecificationNormalizations;
+
+        template <models::ModelSpecification... Specifications>
+        struct PrepareSpecificationNormalizations<models::detail::SpecificationSetStorage<Specifications...>> {
+            static constexpr bool registered =
+                (CompileStellarSpecificationNormalization<Specifications>::registered && ...);
+
+            template <typename Form>
+            static constexpr bool completeFor =
+                (CompileStellarSpecificationNormalization<Specifications>::template completeFor<Form> && ...);
+
+            template <typename Form>
+            static void Apply(
+                DiagonalNormalizationBuilder<Form> &builder,
+                const StellarCharacteristicScales &scales
+            ) {
+                static_assert(
+                    completeFor<Form>,
+                    "Every generated stellar-equilibrium coordinate requires a declared global-scalar "
+                    "Physical Riesz normalization and compiler-owned root block."
+                );
+                (CompileStellarSpecificationNormalization<Specifications>::template Apply<Form>(builder, scales), ...);
+            }
+        };
+
+        template <typename Model, typename Form, typename = void>
+        struct StellarModelNormalizationCoverage : std::false_type { };
+
+        template <typename Model, typename Form>
+            requires model::StellarModelType<Model> && utils::blocks::block_form_is_valid_v<Form>
+        struct StellarModelNormalizationCoverage<
+            Model,
+            Form,
+            std::void_t<typename std::remove_cvref_t<Model>::SpecificationTypes>>
+            : std::bool_constant<
+                  PrepareSpecificationNormalizations<
+                      typename std::remove_cvref_t<Model>::SpecificationTypes>::template completeFor<Form>> { };
+    } // namespace detail
+
+    template <typename Specification>
+    struct StellarSpecificationNormalizationContribution
+        : detail::CompileStellarSpecificationNormalization<std::remove_cvref_t<Specification>> {
+        using Base = detail::CompileStellarSpecificationNormalization<std::remove_cvref_t<Specification>>;
+
+        template <typename Form>
+        static void Apply(
+            DiagonalNormalizationBuilder<Form> &builder,
+            const StellarCharacteristicScales &scales
+        ) {
+            static_assert(
+                Base::template completeFor<Form>,
+                "The specification's generated blocks do not have a complete runtime normalization."
+            );
+            Base::template Apply<Form>(builder, scales);
+        }
+    };
+
+    template <typename Specification>
+    concept RegisteredStellarSpecificationNormalization =
+        StellarSpecificationNormalizationContribution<Specification>::registered;
+
+    template <typename Specification, typename Form>
+    concept CompleteStellarSpecificationNormalizationFor =
+        utils::blocks::block_form_is_valid_v<Form> &&
+        StellarSpecificationNormalizationContribution<Specification>::template completeFor<Form>;
+
+    template <typename Model, typename Form>
+    concept CompleteStellarNormalizationFor =
+        detail::StellarModelNormalizationCoverage<
+            std::remove_cvref_t<Model>,
+            std::remove_cvref_t<Form>>::value;
+
+    /*
+     * Physical Riesz preparation is an optional capability of a physical
+     * core, not part of the protocol needed by the variadic equilibrium root.
+     * Keeping this boundary structural lets a new EOS core opt in by exposing
+     * the same discretization maps without inheriting from, or otherwise
+     * naming, the Polytrope implementation.
+     */
+    template <typename Candidate>
+    concept PhysicalRieszStellarEquilibriumCore =
+        operators::PreparedStellarEquilibriumPhysicalCore<std::remove_cvref_t<Candidate>> &&
+        PhysicalRieszCoreRuntime<std::remove_cvref_t<Candidate>>;
+
+    template <typename Problem>
+    concept PhysicalRieszStellarEquilibriumProblem =
+        equilibrium::DiscretizedStellarEquilibriumProblem<std::remove_cvref_t<Problem>> &&
+        requires {
+            typename std::remove_cvref_t<Problem>::ModelType;
+            typename std::remove_cvref_t<Problem>::FormType;
+            typename std::remove_cvref_t<Problem>::PhysicalCoreType;
+            typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType;
+            requires PhysicalRieszDiagonalPrescription<
+                typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType>;
+            requires CompilableNormalizationFor<
+                typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType,
+                typename std::remove_cvref_t<Problem>::FormType>;
+            requires CompleteStellarNormalizationFor<
+                typename std::remove_cvref_t<Problem>::ModelType,
+                typename std::remove_cvref_t<Problem>::FormType>;
+            requires StellarNormalizationRuntimeAvailableFor<
+                typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType,
+                typename std::remove_cvref_t<Problem>::FormType,
+                typename std::remove_cvref_t<Problem>::PhysicalCoreType,
+                typename std::remove_cvref_t<Problem>::ModelType::SpecificationTypes>;
+        };
+
+    template <equilibrium::DiscretizedStellarEquilibriumProblem Problem>
+        requires std::same_as<
+            typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType,
+            Unnormalized>
+    [[nodiscard]] DiagonalNormalization prepareNormalization(const Problem &problem) {
+        return DiagonalNormalization::Identity(problem.StateSize(), problem.EquationSize());
+    }
+
+    template <PhysicalRieszStellarEquilibriumProblem Problem>
+    [[nodiscard]] DiagonalNormalization prepareNormalization(const Problem &problem) {
+        using ProblemType = std::remove_cvref_t<Problem>;
+        using Form = typename ProblemType::FormType;
+
+        const fem::FEM &finiteElements = problem.GetDiscretization().finiteElementModel();
+        if (!finiteElements.okay()) {
+            throw std::invalid_argument("Physical Riesz preparation requires a current finite-element model.");
+        }
+
+        const auto &physical = detail::PhysicalOperator(problem);
+        const auto &gravityContext = physical.GetGravityContext();
+        const auto &enthalpyMap = physical.GetHydrostaticOperator().GetEnthalpyMap();
+        const auto scales = deriveStellarCharacteristicScales(
+            problem.GetNormalizationPrescription(),
+            problem.GetStellarModel()
+        );
+
+        mfem::Array<int> stellarMarker =
+            utils::domain::make_attribute_marker<utils::domain::Stellar, detail::DomainSchema>(*finiteElements.mesh);
+        const mfem::Vector densityDiagonal = detail::GatherDiagonal(
+            detail::AssembleScalarMassDiagonal(*finiteElements.densityFes, &stellarMarker),
+            gravityContext.GetDensityMap(),
+            "density"
+        );
+        const mfem::Vector enthalpyDiagonal = detail::GatherDiagonal(
+            detail::AssembleScalarMassDiagonal(*finiteElements.enthalpyFes, &stellarMarker),
+            enthalpyMap,
+            "enthalpy"
+        );
+        const mfem::Vector gravityGradientDiagonal = detail::GatherDiagonal(
+            detail::AssembleHDivMassDiagonal(*finiteElements.gravityFluxFes),
+            gravityContext.GetGravityGradientMap(),
+            "gravity-gradient"
+        );
+        const mfem::Vector gravityPotentialDiagonal = detail::GatherDiagonal(
+            detail::AssembleScalarMassDiagonal(*finiteElements.gravityPotentialFes),
+            gravityContext.GetGravityPotentialMap(),
+            "gravity-potential"
+        );
+        const field::ScalarBoundaryDofMap surfaceMap =
+            field::make_stellar_surface_scalar_dof_map<detail::DomainSchema>(*finiteElements.surfaceDeformationFes);
+        const mfem::Vector surfaceDiagonal = detail::AssembleSurfaceMassDiagonal(finiteElements, surfaceMap);
+        if (surfaceDiagonal.Size() != physical.GetDomainDeformation().parameterCount()) {
+            throw std::logic_error("The reference surface Gram diagonal does not match the root surface block.");
+        }
+
+        DiagonalNormalizationBuilder<Form> builder(problem.GetManifest().layout());
+        builder.template SetValueBlock<utils::blocks::density::mass::value>(
+            physicalScale<utils::blocks::density::mass::value>(scales), densityDiagonal
+        );
+        builder.template SetValueBlock<utils::blocks::surface_deformation::parameters::value>(
+            physicalScale<utils::blocks::surface_deformation::parameters::value>(scales), surfaceDiagonal
+        );
+        builder.template SetValueBlock<utils::blocks::gravity::gradient::value>(
+            physicalScale<utils::blocks::gravity::gradient::value>(scales), gravityGradientDiagonal
+        );
+        builder.template SetValueBlock<utils::blocks::gravity::poisson::value>(
+            physicalScale<utils::blocks::gravity::poisson::value>(scales), gravityPotentialDiagonal
+        );
+        builder.template SetValueBlock<utils::blocks::enthalpy::specific::value>(
+            physicalScale<utils::blocks::enthalpy::specific::value>(scales), enthalpyDiagonal
+        );
+        builder.template SetResidualBlock<utils::blocks::gravity::gradient::residual>(
+            physicalScale<utils::blocks::gravity::gradient::residual>(scales), gravityGradientDiagonal
+        );
+        builder.template SetResidualBlock<utils::blocks::gravity::poisson::residual>(
+            physicalScale<utils::blocks::gravity::poisson::residual>(scales), gravityPotentialDiagonal
+        );
+        builder.template SetResidualBlock<utils::blocks::density::mass::residual>(
+            physicalScale<utils::blocks::density::mass::residual>(scales), densityDiagonal
+        );
+        builder.template SetResidualBlock<utils::blocks::surface_deformation::shape_equilibrium::residual>(
+            physicalScale<utils::blocks::surface_deformation::shape_equilibrium::residual>(scales), surfaceDiagonal
+        );
+        const mfem::Array<int> &surfaceRows = problem.GetPressureSurfaceRows().reduced_dofs();
+        builder.template SetHybridResidualBlock<utils::blocks::enthalpy::specific::residual>(
+            physicalScale<utils::blocks::enthalpy::specific::residual>(scales),
+            enthalpyDiagonal,
+            std::span<const int>{surfaceRows.GetData(), static_cast<std::size_t>(surfaceRows.Size())}
+        );
+        detail::PrepareSpecificationNormalizations<typename ProblemType::ModelType::SpecificationTypes>::Apply(
+            builder,
+            scales
+        );
+
+        return std::move(builder).Build();
+    }
+
+    /* Public adapter for a third-party prescription. The implementation stays
+     * beside the policy and has the readable signature
+     *
+     *   prepareStellarNormalization(policy, problem)
+     *
+     * while every solver-facing caller continues to use the uniform
+     * prepareNormalization(problem) operation. */
+    template <equilibrium::DiscretizedStellarEquilibriumProblem Problem>
+        requires(
+            !std::same_as<
+                typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType,
+                Unnormalized> &&
+            !PhysicalRieszDiagonalPrescription<
+                typename std::remove_cvref_t<Problem>::NormalizationPrescriptionType> &&
+            RuntimePreparedNormalizationOperation<Problem>)
+    [[nodiscard]] DiagonalNormalization prepareNormalization(
+        const Problem &problem
+    ) {
+        return prepareStellarNormalization(
+            problem.GetNormalizationPrescription(),
+            problem
+        );
+    }
+
+    /*
+     * Solver-facing normalization exists exactly when runtime preparation for
+     * the problem's compile-time prescription is a valid operation.  This
+     * folds future policy-owned preparation hooks into the same public contract and
+     * turns unsupported core/prescription pairs into ordinary constraint
+     * failure instead of an error in a constructor body.
+     */
+    template <typename Problem>
+    concept NormalizableStellarEquilibriumProblem =
+        equilibrium::DiscretizedStellarEquilibriumProblem<std::remove_cvref_t<Problem>> &&
+        requires(const std::remove_cvref_t<Problem> &problem) {
+            {
+                prepareNormalization(problem)
+            } -> std::same_as<DiagonalNormalization>;
+        };
+
+    struct NormalizedStellarEquilibriumStatistics final {
+        std::uint64_t normalizationPreparations{0};
+        std::uint64_t physicalPreparations{0};
+        std::uint64_t residualRetrievals{0};
+        std::uint64_t jacobianApplications{0};
+    };
+
+    /*
+     * The high-level stellar adapter retains a pointer to a prepared inverse.
+     * Consequently that inverse must identify the exact physical problem and
+     * expose its lifecycle state.  Generic MFEM solvers remain valid inputs to
+     * the lower-level ScaledPreconditioner, where no stellar association is
+     * implied.
+     */
+    template <typename Candidate, typename Problem>
+    concept ProblemBoundStellarInverseFor =
+        NormalizableStellarEquilibriumProblem<std::remove_cvref_t<Problem>> &&
+        std::derived_from<std::remove_cvref_t<Candidate>, mfem::Solver> &&
+        requires(const std::remove_cvref_t<Candidate> &inverse) {
+            {
+                inverse.GetProblem()
+            } -> std::same_as<const std::remove_cvref_t<Problem> &>;
+            {
+                inverse.IsCurrent()
+            } -> std::same_as<bool>;
+        };
+
+    template <NormalizableStellarEquilibriumProblem Problem, typename PhysicalInverse>
+        requires ProblemBoundStellarInverseFor<PhysicalInverse, Problem>
+    class NormalizedStellarPreconditioner;
+
+    /*
+     * Solver-facing coordinates for a dimensional stellar problem. The
+     * physical problem remains the sole source of residual and Jacobian
+     * physics; this adapter performs only the coordinate maps
+     *
+     *     x = R x_hat,   F_hat = L F,   J_hat = L J R.
+     *
+     * Its normalization is immutable during Prepare/BuildResidual/Mult and is
+     * changed only by an explicit RefreshNormalization call.
+     */
+    template <NormalizableStellarEquilibriumProblem Problem>
+    class NormalizedStellarEquilibriumOperator final : public mfem::Operator {
+    private:
+        using ProblemType = std::remove_cvref_t<Problem>;
+
+    public:
+        explicit NormalizedStellarEquilibriumOperator(ProblemType &problem)
+            : mfem::Operator(problem.EquationSize(), problem.StateSize()),
+              m_problem(&problem),
+              m_normalization(prepareNormalization(problem)),
+              m_scaledJacobian(problem.GetLinearizationOperator(), m_normalization),
+              m_physicalState(problem.StateSize()),
+              m_physicalResidual(problem.EquationSize()),
+              m_normalizedResidual(problem.EquationSize()) {
+            if (Width() != Height()) {
+                throw std::invalid_argument("A normalized stellar-equilibrium operator must be square.");
+            }
+            m_statistics.normalizationPreparations = 1;
+        }
+
+        NormalizedStellarEquilibriumOperator(const NormalizedStellarEquilibriumOperator &)            = delete;
+        NormalizedStellarEquilibriumOperator &operator=(const NormalizedStellarEquilibriumOperator &) = delete;
+        NormalizedStellarEquilibriumOperator(NormalizedStellarEquilibriumOperator &&)                 = delete;
+        NormalizedStellarEquilibriumOperator &operator=(NormalizedStellarEquilibriumOperator &&)      = delete;
+
+        [[nodiscard]] auto Prepare(
+            const mfem::Vector &normalizedState,
+            const operators::StellarEquilibriumDependencies &dependencies,
+            const physics::RigidRotation &rotation
+        ) requires(ProblemType::generatedRotationProviderCount == 0) {
+            if (normalizedState.Size() != Width()) {
+                throw std::invalid_argument("The normalized stellar state has the wrong size.");
+            }
+
+            m_isPrepared = false;
+            m_normalization.DenormalizeState(normalizedState, m_physicalState);
+            auto report = m_problem->Prepare(m_physicalState, dependencies, rotation);
+            m_problem->BuildResidual(m_physicalResidual);
+            m_normalization.NormalizeResidual(m_physicalResidual, m_normalizedResidual);
+            m_physicalPreparationGeneration = m_problem->GetPreparationGeneration();
+            m_isPrepared                    = true;
+            ++m_statistics.physicalPreparations;
+            return report;
+        }
+
+        [[nodiscard]] auto Prepare(
+            const mfem::Vector &normalizedState,
+            const operators::StellarEquilibriumDependencies &dependencies
+        ) requires(ProblemType::generatedRotationProviderCount == 1) {
+            if (normalizedState.Size() != Width()) {
+                throw std::invalid_argument("The normalized stellar state has the wrong size.");
+            }
+
+            m_isPrepared = false;
+            m_normalization.DenormalizeState(normalizedState, m_physicalState);
+            auto report = m_problem->Prepare(m_physicalState, dependencies);
+            m_problem->BuildResidual(m_physicalResidual);
+            m_normalization.NormalizeResidual(m_physicalResidual, m_normalizedResidual);
+            m_physicalPreparationGeneration = m_problem->GetPreparationGeneration();
+            m_isPrepared                    = true;
+            ++m_statistics.physicalPreparations;
+            return report;
+        }
+
+        void BuildResidual(mfem::Vector &normalizedResidual) const {
+            VerifyPrepared();
+            normalizedResidual = m_normalizedResidual;
+            ++m_statistics.residualRetrievals;
+        }
+
+        void Mult(
+            const mfem::Vector &normalizedDirection,
+            mfem::Vector &normalizedAction
+        ) const override {
+            VerifyPrepared();
+            if (normalizedDirection.Size() != Width()) {
+                throw std::invalid_argument("The normalized stellar direction has the wrong size.");
+            }
+            m_scaledJacobian.Mult(normalizedDirection, normalizedAction);
+            ++m_statistics.jacobianApplications;
+        }
+
+        void RefreshNormalization() {
+            DiagonalNormalization refreshed = prepareNormalization(*m_problem);
+            m_normalization                  = std::move(refreshed);
+            m_isPrepared                     = false;
+            ++m_statistics.normalizationPreparations;
+        }
+
+        void NormalizeState(
+            const mfem::Vector &physicalState,
+            mfem::Vector &normalizedState
+        ) const {
+            m_normalization.NormalizeState(physicalState, normalizedState);
+        }
+
+        void DenormalizeState(
+            const mfem::Vector &normalizedState,
+            mfem::Vector &physicalState
+        ) const {
+            m_normalization.DenormalizeState(normalizedState, physicalState);
+        }
+
+        void NormalizeResidual(
+            const mfem::Vector &physicalResidual,
+            mfem::Vector &normalizedResidual
+        ) const {
+            m_normalization.NormalizeResidual(physicalResidual, normalizedResidual);
+        }
+
+        void DenormalizeResidual(
+            const mfem::Vector &normalizedResidual,
+            mfem::Vector &physicalResidual
+        ) const {
+            m_normalization.DenormalizeResidual(normalizedResidual, physicalResidual);
+        }
+
+        template <typename PhysicalInverse>
+            requires ProblemBoundStellarInverseFor<PhysicalInverse, Problem>
+        [[nodiscard]] NormalizedStellarPreconditioner<Problem, std::remove_cvref_t<PhysicalInverse>>
+        MakeScaledPreconditioner(PhysicalInverse &physicalInverse) const;
+
+        [[nodiscard]] bool IsPrepared() const noexcept {
+            return m_isPrepared && m_problem->IsPrepared() &&
+                   m_physicalPreparationGeneration == m_problem->GetPreparationGeneration();
+        }
+
+        [[nodiscard]] ProblemType &GetPhysicalProblem() noexcept {
+            return *m_problem;
+        }
+
+        [[nodiscard]] const ProblemType &GetPhysicalProblem() const noexcept {
+            return *m_problem;
+        }
+
+        [[nodiscard]] const mfem::Operator &GetPhysicalJacobian() const noexcept {
+            return m_problem->GetLinearizationOperator();
+        }
+
+        [[nodiscard]] const ProblemType &GetProblem() const noexcept {
+            return *m_problem;
+        }
+
+        [[nodiscard]] const DiagonalNormalization &GetNormalization() const noexcept {
+            return m_normalization;
+        }
+
+        [[nodiscard]] const mfem::Vector &GetPhysicalState() const {
+            VerifyPrepared();
+            return m_physicalState;
+        }
+
+        [[nodiscard]] const mfem::Vector &GetPhysicalResidual() const {
+            VerifyPrepared();
+            return m_physicalResidual;
+        }
+
+        [[nodiscard]] const NormalizedStellarEquilibriumStatistics &GetStatistics() const noexcept {
+            return m_statistics;
+        }
+
+    private:
+        void VerifyPrepared() const {
+            if (!IsPrepared()) {
+                throw std::logic_error(
+                    "The normalized stellar-equilibrium operator must be prepared and current before application."
+                );
+            }
+        }
+
+        ProblemType *m_problem;
+        DiagonalNormalization m_normalization;
+        ScaledJacobianOperator m_scaledJacobian;
+        mfem::Vector m_physicalState;
+        mfem::Vector m_physicalResidual;
+        mfem::Vector m_normalizedResidual;
+        std::uint64_t m_physicalPreparationGeneration{0};
+        mutable NormalizedStellarEquilibriumStatistics m_statistics;
+        bool m_isPrepared{false};
+    };
+
+    template <NormalizableStellarEquilibriumProblem Problem, typename PhysicalInverse>
+        requires ProblemBoundStellarInverseFor<PhysicalInverse, Problem>
+    class NormalizedStellarPreconditioner final : public mfem::Solver {
+    private:
+        using ProblemType = std::remove_cvref_t<Problem>;
+        using NormalizedOperator = NormalizedStellarEquilibriumOperator<ProblemType>;
+        using PhysicalInverseType = std::remove_cvref_t<PhysicalInverse>;
+
+        [[nodiscard]] static PhysicalInverseType &RequireAssociatedPhysicalInverse(
+            const NormalizedOperator &normalizedOperator,
+            PhysicalInverseType &physicalInverse
+        ) {
+            if (std::addressof(physicalInverse.GetProblem()) !=
+                std::addressof(normalizedOperator.GetProblem())) {
+                throw std::invalid_argument(
+                    "A normalized stellar preconditioner and its physical inverse must belong to the same problem."
+                );
+            }
+            return physicalInverse;
+        }
+
+    public:
+        NormalizedStellarPreconditioner(
+            const NormalizedOperator &normalizedOperator,
+            PhysicalInverseType &physicalInverse
+        )
+            : mfem::Solver(
+                  normalizedOperator.Width(),
+                  normalizedOperator.Height(),
+                  physicalInverse.iterative_mode
+              ),
+              m_normalizedOperator(&normalizedOperator),
+              m_physicalInverse(&physicalInverse),
+              m_scaled(
+                  RequireAssociatedPhysicalInverse(normalizedOperator, physicalInverse),
+                  normalizedOperator.GetPhysicalJacobian(),
+                  normalizedOperator,
+                  normalizedOperator.GetNormalization()
+              ) {
+        }
+
+        NormalizedStellarPreconditioner(const NormalizedStellarPreconditioner &)            = delete;
+        NormalizedStellarPreconditioner &operator=(const NormalizedStellarPreconditioner &) = delete;
+        NormalizedStellarPreconditioner(NormalizedStellarPreconditioner &&)                 = delete;
+        NormalizedStellarPreconditioner &operator=(NormalizedStellarPreconditioner &&)      = delete;
+
+        void SetOperator(const mfem::Operator &normalizedJacobian) override {
+            VerifyCurrent();
+            if (&normalizedJacobian != m_normalizedOperator) {
+                throw std::invalid_argument(
+                    "The normalized stellar preconditioner cannot be rebound to a different Jacobian."
+                );
+            }
+            m_scaled.SetOperator(normalizedJacobian);
+        }
+
+        void Mult(
+            const mfem::Vector &normalizedResidual,
+            mfem::Vector &normalizedCorrection
+        ) const override {
+            VerifyCurrent();
+            m_scaled.Mult(normalizedResidual, normalizedCorrection);
+        }
+
+        [[nodiscard]] bool IsCurrent() const {
+            return m_normalizedOperator->IsPrepared() &&
+                   m_physicalInverse->IsCurrent();
+        }
+
+        [[nodiscard]] PhysicalInverseType &GetPhysicalInverse() noexcept {
+            return *m_physicalInverse;
+        }
+
+        [[nodiscard]] const PhysicalInverseType &GetPhysicalInverse() const noexcept {
+            return *m_physicalInverse;
+        }
+
+        [[nodiscard]] const mfem::Operator &GetPhysicalJacobian() const noexcept {
+            return m_scaled.GetPhysicalJacobian();
+        }
+
+        [[nodiscard]] const mfem::Operator &GetNormalizedJacobian() const {
+            return m_scaled.GetNormalizedJacobian();
+        }
+
+        [[nodiscard]] const ScaledPreconditionerStatistics &GetStatistics() const noexcept {
+            return m_scaled.GetStatistics();
+        }
+
+    private:
+        void VerifyCurrent() const {
+            if (!IsCurrent()) {
+                throw std::logic_error(
+                    "The normalized stellar preconditioner cannot be used while its normalized operator or physical "
+                    "inverse is stale."
+                );
+            }
+        }
+
+        const NormalizedOperator *m_normalizedOperator;
+        PhysicalInverseType *m_physicalInverse;
+        ScaledPreconditioner m_scaled;
+    };
+
+    template <NormalizableStellarEquilibriumProblem Problem>
+    template <typename PhysicalInverse>
+        requires ProblemBoundStellarInverseFor<PhysicalInverse, Problem>
+    NormalizedStellarPreconditioner<Problem, std::remove_cvref_t<PhysicalInverse>>
+    NormalizedStellarEquilibriumOperator<Problem>::MakeScaledPreconditioner(PhysicalInverse &physicalInverse) const {
+        VerifyPrepared();
+        return NormalizedStellarPreconditioner<Problem, std::remove_cvref_t<PhysicalInverse>>{
+            *this,
+            physicalInverse
+        };
+    }
+
+    template <NormalizableStellarEquilibriumProblem Problem>
+    [[nodiscard]] auto makeNormalizedStellarEquilibriumOperator(Problem &problem) {
+        return NormalizedStellarEquilibriumOperator<Problem>{problem};
+    }
+} // namespace mean_field::normalization

libmeanfield/interface/operators/prepared_angular_momentum.cppm

@@ -0,0 +1,196 @@
+module;
+
+#include <compare>
+#include <cstdint>
+#include <vector>
+
+#include <mfem.hpp>
+
+export module mean_field:operators.prepared_angular_momentum;
+
+export import :fem;
+export import :mapping.domain_mapper;
+export import :model.compiled_fixed_angular_momentum;
+export import :operators.context.gravity_field;
+
+export namespace mean_field::operators {
+    struct AngularMomentumDependencyStamp final {
+        std::uint64_t identity{0};
+        std::uint64_t revision{0};
+
+        constexpr auto operator<=>(const AngularMomentumDependencyStamp &) const = default;
+    };
+
+    struct AngularMomentumDependencies final {
+        AngularMomentumDependencyStamp discretization;
+        AngularMomentumDependencyStamp density;
+        AngularMomentumDependencyStamp displacement;
+        AngularMomentumDependencyStamp rotation;
+
+        constexpr auto operator<=>(const AngularMomentumDependencies &) const = default;
+    };
+
+    struct PreparedAngularMomentumReport final {
+        bool rebuiltStaticPlan{false};
+        bool refreshedGeometry{false};
+        bool refreshedDensity{false};
+        bool updatedAngularVelocity{false};
+        bool assembledResidual{false};
+
+        [[nodiscard]] bool DidAnyWork() const noexcept {
+            return rebuiltStaticPlan || refreshedGeometry || refreshedDensity || updatedAngularVelocity ||
+                   assembledResidual;
+        }
+
+        constexpr auto operator<=>(const PreparedAngularMomentumReport &) const = default;
+    };
+
+    struct AngularMomentumConstraintReport final {
+        double targetAngularMomentum;
+        double achievedAngularMomentum;
+        double momentOfInertia;
+        double angularVelocity;
+        double dimensionalResidual;
+        double scaledResidual;
+    };
+
+    struct PreparedAngularMomentumActionStatistics final {
+        std::uint64_t densityApplications{0};
+        std::uint64_t displacementApplications{0};
+        std::uint64_t angularVelocityApplications{0};
+        std::uint64_t completeApplications{0};
+
+        constexpr auto operator<=>(const PreparedAngularMomentumActionStatistics &) const = default;
+    };
+
+    /*
+     * Prepared scalar invariant
+     *
+     *   R_J(rho, d, Omega) = Omega I_axis(rho, d) - J_target,
+     *   I_axis             = integral rho |(x-x_0)_perp|^2 dV.
+     *
+     * The axis is normalized by CompiledFixedAngularMomentum. Density and
+     * geometry are borrowed from the shared gravity context, so this row is
+     * linearized at exactly the same mapped state as every physical equation.
+     */
+    class PreparedAngularMomentumOperator final {
+    public:
+        using SpecificationType      = models::FixedAngularMomentum;
+        using CompiledConstraintType = models::CompiledFixedAngularMomentum;
+        using Dependencies           = AngularMomentumDependencies;
+        using Report                 = PreparedAngularMomentumReport;
+
+        PreparedAngularMomentumOperator(
+            const fem::FEM &f,
+            const mapping::DomainMapper &domainMapper,
+            const context::gravity_field::GravityFieldLinearizationContext &gravityContext,
+            models::CompiledFixedAngularMomentum constraint
+        );
+
+        PreparedAngularMomentumOperator(const PreparedAngularMomentumOperator &)            = delete;
+        PreparedAngularMomentumOperator &operator=(const PreparedAngularMomentumOperator &) = delete;
+        PreparedAngularMomentumOperator(PreparedAngularMomentumOperator &&)                 = delete;
+        PreparedAngularMomentumOperator &operator=(PreparedAngularMomentumOperator &&)      = delete;
+
+        PreparedAngularMomentumReport Prepare(
+            double angularVelocity,
+            const AngularMomentumDependencies &dependencies
+        );
+
+        void BuildResidual(mfem::Vector &residual) const;
+
+        void ApplyDensityJacobianAction(
+            const mfem::Vector &densityVariation,
+            mfem::Vector &action
+        ) const;
+
+        void ApplyDisplacementJacobianAction(
+            const mfem::Vector &displacementVariation,
+            mfem::Vector &action
+        ) const;
+
+        void ApplyAngularVelocityJacobianAction(
+            double angularVelocityVariation,
+            mfem::Vector &action
+        ) const;
+
+        void ApplyCompleteJacobianAction(
+            const mfem::Vector &densityVariation,
+            const mfem::Vector &displacementVariation,
+            double angularVelocityVariation,
+            mfem::Vector &action
+        ) const;
+
+        [[nodiscard]] bool IsPrepared() const noexcept;
+        [[nodiscard]] double GetMomentOfInertia() const;
+        [[nodiscard]] double GetAngularVelocity() const;
+        [[nodiscard]] double GetCurrentAngularMomentum() const;
+        [[nodiscard]] double GetTargetAngularMomentum() const noexcept;
+        [[nodiscard]] physics::RigidRotation GetRotation() const;
+        [[nodiscard]] AngularMomentumConstraintReport GetConstraintReport() const;
+        [[nodiscard]] std::uint64_t GetPreparationCount() const noexcept;
+        [[nodiscard]] std::uint64_t GetResidualApplicationCount() const noexcept;
+        [[nodiscard]] const PreparedAngularMomentumActionStatistics &GetActionStatistics() const noexcept;
+        [[nodiscard]] const models::CompiledFixedAngularMomentum &GetCompiledConstraint() const noexcept;
+
+    private:
+        struct QuadraturePointData final {
+            mfem::IntegrationPoint integrationPoint;
+            mfem::Vector densityShape;
+            mapping::VolumeMappingContext mappingContext;
+            double density{0.0};
+            double cylindricalRadiusSquared{0.0};
+        };
+
+        struct ElementPAData final {
+            int elementId{-1};
+            mfem::Array<int> densityDofs;
+            mfem::Array<int> displacementDofs;
+            mfem::Array<int> compactificationDofs;
+            mfem::DofTransformation *densityDofTransformation{nullptr};
+            mfem::DofTransformation *displacementDofTransformation{nullptr};
+            mfem::DofTransformation *compactificationDofTransformation{nullptr};
+            mfem::Vector baseDisplacement;
+            mfem::Vector compactification;
+            std::vector<QuadraturePointData> quadraturePoints;
+        };
+
+        void BuildStaticPlan();
+        void RefreshGeometry(const mfem::Vector &displacement);
+        void RefreshDensity(const mfem::Vector &density);
+        void AssembleResidual();
+        void VerifyPrepared() const;
+
+        [[nodiscard]] double EvaluateDensityMomentActionLocal(const mfem::Vector &densityVariation) const;
+        [[nodiscard]] double EvaluateDisplacementMomentActionLocal(const mfem::Vector &displacementVariation) const;
+        [[nodiscard]] double CylindricalRadiusSquared(const mfem::Vector &physicalPosition) const noexcept;
+        [[nodiscard]] double CylindricalRadiusSquaredVariation(
+            const mfem::Vector &physicalPosition,
+            const mfem::Vector &physicalPositionVariation
+        ) const noexcept;
+        [[nodiscard]] double GlobalSum(double localValue) const;
+
+        const fem::FEM &m_fem;
+        const mapping::DomainMapper &m_domainMapper;
+        const context::gravity_field::GravityFieldLinearizationContext &m_gravityContext;
+        models::CompiledFixedAngularMomentum m_constraint;
+
+        std::vector<ElementPAData> m_elements;
+        AngularMomentumDependencies m_preparedDependencies;
+        mfem::Vector m_cachedResidual;
+        mutable mfem::Vector m_densityVariationTrue;
+        mutable mfem::Vector m_displacementVariationTrue;
+        mutable mfem::Vector m_densityVariationLocal;
+        mutable mfem::Vector m_displacementVariationLocal;
+        mutable mfem::Vector m_elementDensityVariation;
+        mutable mfem::Vector m_elementDisplacementVariation;
+
+        double m_momentOfInertia{0.0};
+        double m_angularVelocity{0.0};
+        double m_currentAngularMomentum{0.0};
+        std::uint64_t m_preparationCount{0};
+        mutable std::uint64_t m_residualApplicationCount{0};
+        mutable PreparedAngularMomentumActionStatistics m_actionStatistics;
+        bool m_isPrepared{false};
+    };
+} // namespace mean_field::operators

libmeanfield/interface/operators/prepared_central_density_stellar_equilibrium.cppm

@@ -1,117 +0,0 @@
-module;
-
-#include <concepts>
-#include <memory>
-#include <type_traits>
-#include <utility>
-
-#include <mfem.hpp>
-
-export module mean_field:operators.prepared_central_density_stellar_equilibrium;
-
-export import :model.compiled_fixed_central_density;
-export import :operators.prepared_central_density;
-export import :operators.prepared_stellar_equilibrium;
-
-export namespace mean_field::operators {
-    using CentralDensityStellarEquilibriumSpecificationModel = model::StellarModel<
-        models::
-            SpecificationSet<eos::Polytrope, models::FixedTotalMass, surface::Isobaric, models::FixedCentralDensity>>;
-
-    using CentralDensityStellarEquilibriumForm = utils::blocks::central_density_bordered_stellar_equilibrium_form;
-    using CentralDensityStellarEquilibriumJacobianForm =
-        utils::blocks::central_density_bordered_stellar_equilibrium_jacobian_form;
-    using CentralDensityStellarEquilibriumLayout = utils::blocks::form_layout<CentralDensityStellarEquilibriumForm>;
-    using CentralDensityStellarEquilibriumSystemManifest = EquilibriumSystemManifest<
-        CentralDensityStellarEquilibriumSpecificationModel,
-        CentralDensityStellarEquilibriumForm,
-        CentralDensityStellarEquilibriumJacobianForm>;
-
-    using CentralDensityStellarEquilibriumRootManifest = CentralDensityStellarEquilibriumSystemManifest;
-
-    struct PreparedCentralDensityStellarEquilibriumReport final {
-        PreparedStellarEquilibriumReport physical;
-        PreparedCentralDensityReport phase;
-        bool assembledResidual{false};
-
-        [[nodiscard]] bool DidAnyWork() const noexcept {
-            return physical.DidAnyWork() || phase.DidAnyWork() || assembledResidual;
-        }
-    };
-
-    class PreparedCentralDensityStellarEquilibriumOperator final : public mfem::Operator {
-    public:
-        PreparedCentralDensityStellarEquilibriumOperator(
-            fem::FEM &f,
-            const mapping::DomainMapper &domainMapper,
-            const eos::Polytrope &equationOfState,
-            models::CompiledFixedMass fixedMassConstraint,
-            PressureSurfaceConstraintView surfaceConstraint,
-            deformation::PreparedDomainDeformationRuntime domainDeformation,
-            models::CompiledFixedCentralDensity centralDensity
-        )
-            : PreparedCentralDensityStellarEquilibriumOperator(
-                  f,
-                  std::make_unique<PreparedStellarEquilibriumOperator>(
-                      f,
-                      domainMapper,
-                      equationOfState,
-                      std::move(fixedMassConstraint),
-                      surfaceConstraint,
-                      std::move(domainDeformation)
-                  ),
-                  std::move(centralDensity),
-                  MakeCenterDofMap(f)
-              ) {
-        }
-
-        PreparedCentralDensityStellarEquilibriumOperator(const PreparedCentralDensityStellarEquilibriumOperator &) =
-            delete;
-        PreparedCentralDensityStellarEquilibriumOperator &
-        operator=(const PreparedCentralDensityStellarEquilibriumOperator &)                                   = delete;
-        PreparedCentralDensityStellarEquilibriumOperator(PreparedCentralDensityStellarEquilibriumOperator &&) = delete;
-        PreparedCentralDensityStellarEquilibriumOperator &
-        operator=(PreparedCentralDensityStellarEquilibriumOperator &&) = delete;
-
-        PreparedCentralDensityStellarEquilibriumReport Prepare(
-            const mfem::Vector &state,
-            const StellarEquilibriumDependencies &dependencies,
-            const physics::RigidRotation &rotation
-        );
-
-        void BuildResidual(mfem::Vector &residual) const;
-
-        void Mult(
-            const mfem::Vector &direction,
-            mfem::Vector &action
-        ) const override;
-
-        [[nodiscard]] bool IsPrepared() const noexcept;
-        [[nodiscard]] const CentralDensityStellarEquilibriumLayout &GetLayout() const noexcept;
-        [[nodiscard]] const CentralDensityStellarEquilibriumRootManifest &GetRootManifest() const noexcept;
-        [[nodiscard]] const PreparedStellarEquilibriumOperator &GetPhysicalOperator() const noexcept;
-        [[nodiscard]] const PreparedCentralDensityConstraint &GetCentralDensityConstraint() const noexcept;
-        [[nodiscard]] RootConstraintReport GetFixedMassReport() const;
-        [[nodiscard]] CentralDensityConstraintReport GetCentralDensityReport() const;
-
-    private:
-        static field::FieldPointDofMap MakeCenterDofMap(const fem::FEM &f);
-
-        PreparedCentralDensityStellarEquilibriumOperator(
-            fem::FEM &f,
-            std::unique_ptr<PreparedStellarEquilibriumOperator> physicalOperator,
-            models::CompiledFixedCentralDensity centralDensity,
-            field::FieldPointDofMap centerDof
-        );
-
-        void AssembleResidual();
-        void VerifyPrepared() const;
-
-        std::unique_ptr<PreparedStellarEquilibriumOperator> m_physicalOperator;
-        models::CompiledFixedCentralDensity m_centralDensity;
-        PreparedCentralDensityConstraint m_phaseConstraint;
-        CentralDensityStellarEquilibriumRootManifest m_rootManifest;
-        mfem::Vector m_cachedResidual;
-        bool m_isPrepared{false};
-    };
-} // namespace mean_field::operators

libmeanfield/interface/operators/prepared_hydrostatic_equilibrium_operator.cppm

@@ -35,6 +35,7 @@ export namespace mean_field::operators {
         std::uint64_t enthalpyApplications{0};
         std::uint64_t gravityPotentialApplications{0};
         std::uint64_t bernoulliConstantApplications{0};
+        std::uint64_t rotationAmplitudeApplications{0};
         std::uint64_t combinedApplications{0};
 
         constexpr auto operator<=>(const PreparedHydrostaticAlgebraicJacobianStatistics &) const = default;
@@ -118,6 +119,14 @@ export namespace mean_field::operators {
             mfem::Vector &action
         ) const;
 
+        // Differentiates a multiplicative change Omega -> (1 + alpha) Omega
+        // at the frozen rigid rotation. Since Psi_rotation is quadratic in
+        // Omega, this contributes -2 alpha Psi_rotation to the hydrostatic row.
+        void ApplyRotationAmplitudeJacobianAction(
+            double fractionalAngularVelocityVariation,
+            mfem::Vector &action
+        ) const;
+
         void ApplyAlgebraicJacobianAction(
             const mfem::Vector &enthalpyVariation,
             const mfem::Vector &gravityPotentialVariation,

libmeanfield/interface/operators/prepared_stellar_equilibrium.cppm

@@ -96,6 +96,20 @@ export namespace mean_field::operators {
 
     class PreparedStellarEquilibriumOperator final : public mfem::Operator {
     public:
+        /*
+         * Privileged aggregate runtimes can inspect this complete numerical
+         * core. Keep their allow-list on the concrete core itself: a custom
+         * EOS may reuse this class, but it cannot extend the class's backend
+         * privileges. Ordinary specifications use restricted nested physics
+         * and never interact with this list.
+         */
+        using BackendSpecifications = models::ModelTypeList<
+            eos::Polytrope,
+            surface::Isobaric,
+            models::FixedTotalMass,
+            models::FixedAngularMomentum,
+            models::FixedCentralDensity>;
+
         template <models::StellarModelType Model>
             requires std::same_as<
                          typename std::remove_cvref_t<Model>::EquationOfStateType,
@@ -174,6 +188,12 @@ export namespace mean_field::operators {
         [[nodiscard]] const PreparedHydrostaticEquilibriumOperator &GetHydrostaticOperator() const noexcept;
         [[nodiscard]] const PreparedDisplacementResidualOperator &GetDisplacementOperator() const noexcept;
         [[nodiscard]] const PreparedMassNormalizationOperator &GetMassNormalizationOperator() const noexcept;
+        [[nodiscard]] double ApplyDensityVolumeIntegralDensityAction(
+            const mfem::Vector &densityDirection
+        ) const;
+        [[nodiscard]] double ApplyDensityVolumeIntegralSurfaceShapeAction(
+            const mfem::Vector &surfaceShapeDirection
+        ) const;
         [[nodiscard]] const PreparedPressureSurfaceConstraint &GetSurfaceConstraintOperator() const noexcept;
         [[nodiscard]] const deformation::PreparedDomainDeformationRuntime &GetDomainDeformation() const noexcept;
         [[nodiscard]] const mfem::Vector &GetSurfaceDeformationParameters() const;
@@ -234,5 +254,6 @@ export namespace mean_field::operators {
         mutable mfem::Vector m_fullMechanicalAction;
         mutable mfem::Vector m_surfaceShapeAction;
         mutable mfem::Vector m_pullbackDerivativeAction;
+        mutable mfem::Vector m_densityVolumeIntegralAction;
     };
 } // namespace mean_field::operators

libmeanfield/interface/operators/stellar_equilibrium_compiler.cppm

@@ -0,0 +1,815 @@
+module;
+
+#include <concepts>
+#include <type_traits>
+
+export module mean_field:operators.stellar_equilibrium_compiler;
+
+export import :model.compiled_fixed_angular_momentum;
+export import :model.compiled_fixed_central_density;
+export import :model.typed_stellar;
+export import :utils.blocks;
+
+export namespace mean_field::operators {
+/*
+ * A coupling is the symbolic statement that one Jacobian block may be
+ * nonzero. Specifications contribute these statements independently of
+ * the final row and column layout.
+ */
+template <typename ResidualBlock, typename ValueBlock>
+struct StellarEquilibriumJacobianCoupling final {
+  using Residual = ResidualBlock;
+  using Value = ValueBlock;
+
+  using ResidualBlockType = ResidualBlock;
+  using ValueBlockType = ValueBlock;
+};
+
+template <typename ResidualBlock, typename ValueBlock>
+using EquilibriumJacobianCoupling =
+    StellarEquilibriumJacobianCoupling<ResidualBlock, ValueBlock>;
+
+namespace detail {
+template <typename... Lists> struct ConcatenateBlockLists;
+
+template <> struct ConcatenateBlockLists<> {
+  using Type = utils::blocks::type_list<>;
+};
+
+template <typename... Types>
+struct ConcatenateBlockLists<utils::blocks::type_list<Types...>> {
+  using Type = utils::blocks::type_list<Types...>;
+};
+
+template <typename... First, typename... Second, typename... Remaining>
+struct ConcatenateBlockLists<utils::blocks::type_list<First...>,
+                             utils::blocks::type_list<Second...>,
+                             Remaining...> {
+  using Type = typename ConcatenateBlockLists<
+      utils::blocks::type_list<First..., Second...>, Remaining...>::Type;
+};
+
+template <typename... Lists>
+using ConcatenateBlockListsT = typename ConcatenateBlockLists<Lists...>::Type;
+
+template <typename List, typename Type> struct AppendUniqueBlockType;
+
+template <typename... Types, typename Type>
+struct AppendUniqueBlockType<utils::blocks::type_list<Types...>, Type> {
+  using TypeValue = std::conditional_t<
+      utils::blocks::contains_type_v<Type, utils::blocks::type_list<Types...>>,
+      utils::blocks::type_list<Types...>,
+      utils::blocks::type_list<Types..., Type>>;
+};
+
+template <typename Accumulated, typename Remaining> struct UniqueBlockListImpl;
+
+template <typename Accumulated>
+struct UniqueBlockListImpl<Accumulated, utils::blocks::type_list<>> {
+  using Type = Accumulated;
+};
+
+template <typename Accumulated, typename Head, typename... Tail>
+struct UniqueBlockListImpl<Accumulated,
+                           utils::blocks::type_list<Head, Tail...>> {
+  using Type = typename UniqueBlockListImpl<
+      typename AppendUniqueBlockType<Accumulated, Head>::TypeValue,
+      utils::blocks::type_list<Tail...>>::Type;
+};
+
+template <typename List>
+using UniqueBlockListT =
+    typename UniqueBlockListImpl<utils::blocks::type_list<>, List>::Type;
+
+template <typename... Lists>
+using UniqueConcatenateBlockListsT =
+    UniqueBlockListT<ConcatenateBlockListsT<Lists...>>;
+
+template <typename Candidate> struct IsValueBlockList : std::false_type {};
+
+template <typename... Blocks>
+struct IsValueBlockList<utils::blocks::type_list<Blocks...>>
+    : std::bool_constant<
+          (std::derived_from<Blocks, utils::blocks::value_block_base> && ...) &&
+          utils::blocks::types_are_unique_v<
+              utils::blocks::type_list<Blocks...>>> {};
+
+template <typename Candidate> struct IsResidualBlockList : std::false_type {};
+
+template <typename... Blocks>
+struct IsResidualBlockList<utils::blocks::type_list<Blocks...>>
+    : std::bool_constant<
+          (std::derived_from<Blocks, utils::blocks::residual_block_base> &&
+           ...) &&
+          utils::blocks::types_are_unique_v<
+              utils::blocks::type_list<Blocks...>>> {};
+
+template <typename GeneratedValues> struct GeneratedValueBlocksFor;
+
+template <typename... GeneratedValues>
+struct GeneratedValueBlocksFor<models::ModelTypeList<GeneratedValues...>> {
+  using Type = utils::blocks::type_list<
+      utils::blocks::generated_value_block<GeneratedValues>...>;
+};
+
+template <typename GeneratedResiduals> struct GeneratedResidualBlocksFor;
+
+template <typename... GeneratedResiduals>
+struct GeneratedResidualBlocksFor<
+    models::ModelTypeList<GeneratedResiduals...>> {
+  using Type = utils::blocks::type_list<
+      utils::blocks::generated_residual_block<GeneratedResiduals>...>;
+};
+
+/*
+ * One translation boundary turns physics-facing stellar names into backend
+ * blocks.  Existing backend block types pass through unchanged, which keeps
+ * the advanced extension API open without making built-in physics declarations
+ * depend on utils.blocks.
+ */
+template <typename DeclaredDependency>
+struct UnmappedStellarDependency final {};
+
+template <typename Blocks, typename DeclaredDependency>
+struct SingleGeneratedBlock {
+  using Type = UnmappedStellarDependency<DeclaredDependency>;
+  static constexpr bool available = false;
+};
+
+template <typename Block, typename DeclaredDependency>
+struct SingleGeneratedBlock<utils::blocks::type_list<Block>,
+                            DeclaredDependency> {
+  using Type = Block;
+  static constexpr bool available = true;
+};
+
+template <models::ModelSpecification Specification, typename Dependency>
+struct StellarDependencyBlock {
+  using Type = UnmappedStellarDependency<Dependency>;
+  static constexpr bool mapped = false;
+};
+
+template <models::ModelSpecification Specification, typename Block>
+  requires(std::derived_from<Block, utils::blocks::value_block_base> ||
+           std::derived_from<Block, utils::blocks::residual_block_base>)
+struct StellarDependencyBlock<Specification, Block> {
+  using Type = Block;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<Specification, models::stellar::state::Density> {
+  using Type = utils::blocks::density::mass::value;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<Specification,
+                              models::stellar::state::SurfaceShape> {
+  using Type = utils::blocks::surface_deformation::parameters::value;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<Specification,
+                              models::stellar::state::GravityGradient> {
+  using Type = utils::blocks::gravity::gradient::value;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<
+    Specification, models::stellar::state::GravitationalPotential> {
+  using Type = utils::blocks::gravity::poisson::value;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<Specification,
+                              models::stellar::state::SpecificEnthalpy> {
+  using Type = utils::blocks::enthalpy::specific::value;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<
+    Specification, models::stellar::state::OwnGeneratedCoordinate> {
+private:
+  using GeneratedBlocks = typename GeneratedValueBlocksFor<
+      typename models::SpecificationContribution<
+          Specification>::GeneratedValues>::Type;
+  using Selection = SingleGeneratedBlock<
+      GeneratedBlocks, models::stellar::state::OwnGeneratedCoordinate>;
+
+public:
+  using Type = typename Selection::Type;
+  static constexpr bool mapped = Selection::available;
+};
+
+template <models::ModelSpecification Specification,
+          models::ModelSpecification Owner>
+struct StellarDependencyBlock<
+    Specification, models::stellar::state::GeneratedCoordinateOf<Owner>> {
+private:
+  using GeneratedBlocks = typename GeneratedValueBlocksFor<
+      typename models::SpecificationContribution<Owner>::GeneratedValues>::Type;
+  using Dependency = models::stellar::state::GeneratedCoordinateOf<Owner>;
+  using Selection = SingleGeneratedBlock<GeneratedBlocks, Dependency>;
+
+public:
+  using Type = typename Selection::Type;
+  static constexpr bool mapped = Selection::available;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<
+    Specification, models::stellar::equation::GravityGradientDefinition> {
+  using Type = utils::blocks::gravity::gradient::residual;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<Specification,
+                              models::stellar::equation::PoissonEquation> {
+  using Type = utils::blocks::gravity::poisson::residual;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<Specification,
+                              models::stellar::equation::DensityClosure> {
+  using Type = utils::blocks::density::mass::residual;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<
+    Specification, models::stellar::equation::SurfaceShapeBalance> {
+  using Type =
+      utils::blocks::surface_deformation::shape_equilibrium::residual;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<
+    Specification, models::stellar::equation::HydrostaticBalance> {
+  using Type = utils::blocks::enthalpy::specific::residual;
+  static constexpr bool mapped = true;
+};
+
+template <models::ModelSpecification Specification>
+struct StellarDependencyBlock<Specification,
+                              models::stellar::equation::OwnConstraint> {
+private:
+  using GeneratedBlocks = typename GeneratedResidualBlocksFor<
+      typename models::SpecificationContribution<
+          Specification>::GeneratedResiduals>::Type;
+  using Selection = SingleGeneratedBlock<
+      GeneratedBlocks, models::stellar::equation::OwnConstraint>;
+
+public:
+  using Type = typename Selection::Type;
+  static constexpr bool mapped = Selection::available;
+};
+
+template <models::ModelSpecification Specification,
+          models::ModelSpecification Owner>
+struct StellarDependencyBlock<
+    Specification, models::stellar::equation::ConstraintOf<Owner>> {
+private:
+  using GeneratedBlocks = typename GeneratedResidualBlocksFor<
+      typename models::SpecificationContribution<Owner>::GeneratedResiduals>::Type;
+  using Dependency = models::stellar::equation::ConstraintOf<Owner>;
+  using Selection = SingleGeneratedBlock<GeneratedBlocks, Dependency>;
+
+public:
+  using Type = typename Selection::Type;
+  static constexpr bool mapped = Selection::available;
+};
+
+template <models::ModelSpecification Specification, typename Dependencies>
+struct CompileStellarDependencies {
+  using Type = utils::blocks::type_list<
+      UnmappedStellarDependency<Dependencies>>;
+  static constexpr bool complete = false;
+};
+
+template <models::ModelSpecification Specification, typename... Dependencies>
+struct CompileStellarDependencies<Specification,
+                                  models::ModelTypeList<Dependencies...>> {
+  using Type = utils::blocks::type_list<
+      typename StellarDependencyBlock<Specification, Dependencies>::Type...>;
+  static constexpr bool complete =
+      (StellarDependencyBlock<Specification, Dependencies>::mapped && ...);
+};
+
+template <typename Residual, typename Values> struct CoupleResidualToValues;
+
+template <typename Residual, typename... Values>
+struct CoupleResidualToValues<Residual, utils::blocks::type_list<Values...>> {
+  using Type = utils::blocks::type_list<
+      StellarEquilibriumJacobianCoupling<Residual, Values>...>;
+};
+
+template <typename Residuals, typename Values>
+struct CartesianJacobianCouplings;
+
+template <typename... Residuals, typename Values>
+struct CartesianJacobianCouplings<utils::blocks::type_list<Residuals...>,
+                                  Values> {
+  using Type = ConcatenateBlockListsT<
+      typename CoupleResidualToValues<Residuals, Values>::Type...>;
+};
+
+template <typename Candidate> struct IsJacobianCoupling : std::false_type {};
+
+template <typename Residual, typename Value>
+struct IsJacobianCoupling<StellarEquilibriumJacobianCoupling<Residual, Value>>
+    : std::bool_constant<
+          std::derived_from<Residual, utils::blocks::residual_block_base> &&
+          std::derived_from<Value, utils::blocks::value_block_base>> {};
+
+template <typename Candidate>
+struct IsJacobianCouplingList : std::false_type {};
+
+template <typename... Couplings>
+struct IsJacobianCouplingList<utils::blocks::type_list<Couplings...>>
+    : std::bool_constant<(IsJacobianCoupling<Couplings>::value && ...) &&
+                         utils::blocks::types_are_unique_v<
+                             utils::blocks::type_list<Couplings...>>> {};
+
+template <typename Candidate> struct IsGeneratedValueBlock : std::false_type {};
+
+template <typename Owner>
+struct IsGeneratedValueBlock<utils::blocks::generated_value_block<Owner>>
+    : std::true_type {};
+
+template <typename Candidate>
+struct IsGeneratedResidualBlock : std::false_type {};
+
+template <typename Owner>
+struct IsGeneratedResidualBlock<utils::blocks::generated_residual_block<Owner>>
+    : std::true_type {};
+
+template <typename Coupling>
+inline constexpr bool isGeneratedBorderIncidentCoupling =
+    IsGeneratedValueBlock<typename Coupling::Value>::value ||
+    IsGeneratedResidualBlock<typename Coupling::Residual>::value;
+
+template <typename Couplings> struct GeneratedBorderIncidentCouplings;
+
+template <>
+struct GeneratedBorderIncidentCouplings<utils::blocks::type_list<>> {
+  using Type = utils::blocks::type_list<>;
+};
+
+template <typename Head, typename... Tail>
+struct GeneratedBorderIncidentCouplings<
+    utils::blocks::type_list<Head, Tail...>> {
+private:
+  using Remaining = typename GeneratedBorderIncidentCouplings<
+      utils::blocks::type_list<Tail...>>::Type;
+
+public:
+  using Type = std::conditional_t<
+      isGeneratedBorderIncidentCoupling<Head>,
+      ConcatenateBlockListsT<utils::blocks::type_list<Head>, Remaining>,
+      Remaining>;
+};
+
+template <bool Registered, typename GeneratedValues,
+          typename GeneratedResiduals, typename DependsOn, typename Affects>
+struct DeclarativeStellarEquilibriumSpecificationCompilation {
+  using GeneratedValueBlocks = GeneratedValues;
+  using GeneratedResidualBlocks = GeneratedResiduals;
+  using DependsOnValueBlocks = DependsOn;
+  using AffectedResidualBlocks = Affects;
+
+  /*
+   * Preserve the two physical meanings in the declaration instead of
+   * flattening their endpoints into independent unions:
+   *
+   *   constraint equation <- everything named in Reads
+   *   changed equations    <- Reads plus the generated coordinate
+   *
+   * The second group deliberately includes Affects x Reads.  Nonlinear
+   * constraints and multiplier forces generally contribute Hessian-like
+   * state derivatives there.  Linear contributions simply assemble zero on
+   * those structurally permitted edges.
+   */
+  using ConstraintInputValueBlocks = DependsOnValueBlocks;
+  using ConstraintOutputResidualBlocks = GeneratedResidualBlocks;
+  using ChangedEquationInputValueBlocks = UniqueConcatenateBlockListsT<
+      DependsOnValueBlocks, GeneratedValueBlocks>;
+  using ChangedEquationOutputResidualBlocks = AffectedResidualBlocks;
+
+  using ConstraintJacobianCouplings =
+      typename CartesianJacobianCouplings<GeneratedResidualBlocks,
+                                          DependsOnValueBlocks>::Type;
+  using ChangedEquationJacobianCouplings =
+      typename CartesianJacobianCouplings<AffectedResidualBlocks,
+                                          ChangedEquationInputValueBlocks>::Type;
+
+  // Compatibility names retained for backend code that distinguishes the
+  // generated row from the generated-coordinate column.
+  using GeneratedRowJacobianCouplings = ConstraintJacobianCouplings;
+  using AffectedRowJacobianCouplings =
+      typename CartesianJacobianCouplings<AffectedResidualBlocks,
+                                          GeneratedValueBlocks>::Type;
+  using AffectedStateJacobianCouplings =
+      typename CartesianJacobianCouplings<AffectedResidualBlocks,
+                                          DependsOnValueBlocks>::Type;
+
+  using JacobianCouplings = UniqueConcatenateBlockListsT<
+      ConstraintJacobianCouplings, ChangedEquationJacobianCouplings>;
+  using IncidentJacobianCouplings =
+      typename GeneratedBorderIncidentCouplings<JacobianCouplings>::Type;
+
+  // Correction is the Newton-facing name for a value coordinate.
+  using GeneratedCorrectionBlocks = GeneratedValueBlocks;
+
+  static constexpr bool registered = Registered;
+  static constexpr bool complete =
+      registered && IsValueBlockList<GeneratedValueBlocks>::value &&
+      IsResidualBlockList<GeneratedResidualBlocks>::value &&
+      IsValueBlockList<DependsOnValueBlocks>::value &&
+      IsResidualBlockList<AffectedResidualBlocks>::value &&
+      IsJacobianCouplingList<JacobianCouplings>::value &&
+      (GeneratedValueBlocks::size == GeneratedResidualBlocks::size) &&
+      ((GeneratedValueBlocks::size == 0 && DependsOnValueBlocks::size == 0 &&
+        AffectedResidualBlocks::size == 0) ||
+       (GeneratedValueBlocks::size > 0 && DependsOnValueBlocks::size > 0 &&
+        AffectedResidualBlocks::size > 0));
+};
+
+using EmptySpecificationCompilation =
+    DeclarativeStellarEquilibriumSpecificationCompilation<
+        false, utils::blocks::type_list<>, utils::blocks::type_list<>,
+        utils::blocks::type_list<>, utils::blocks::type_list<>>;
+
+template <models::ModelSpecification Specification>
+struct SelfDescribingSpecificationCompilationInputs {
+  using Contribution = models::SpecificationContribution<Specification>;
+  using DependsOn =
+      CompileStellarDependencies<Specification, typename Contribution::DependsOn>;
+  using Affects =
+      CompileStellarDependencies<Specification, typename Contribution::Affects>;
+
+  using GeneratedValueBlocks = typename GeneratedValueBlocksFor<
+      typename Contribution::GeneratedValues>::Type;
+  using GeneratedResidualBlocks = typename GeneratedResidualBlocksFor<
+      typename Contribution::GeneratedResiduals>::Type;
+  using DependsOnValueBlocks = typename DependsOn::Type;
+  using AffectedResidualBlocks = typename Affects::Type;
+
+  static constexpr bool registered =
+      Contribution::hasDeclarativeDefinition && DependsOn::complete &&
+      Affects::complete;
+};
+
+template <models::ModelSpecification Specification>
+struct SelfDescribingSpecificationCompilation
+    : DeclarativeStellarEquilibriumSpecificationCompilation<
+          SelfDescribingSpecificationCompilationInputs<Specification>::registered,
+          typename SelfDescribingSpecificationCompilationInputs<
+              Specification>::GeneratedValueBlocks,
+          typename SelfDescribingSpecificationCompilationInputs<
+              Specification>::GeneratedResidualBlocks,
+          typename SelfDescribingSpecificationCompilationInputs<
+              Specification>::DependsOnValueBlocks,
+          typename SelfDescribingSpecificationCompilationInputs<
+              Specification>::AffectedResidualBlocks> {};
+} // namespace detail
+
+/*
+ * Public, inspectable per-specification compilation metadata. The primary
+ * is deliberately well formed and incomplete, so testing an arbitrary type
+ * in a requires-expression never triggers a diagnostic.
+ */
+template <typename Specification>
+struct StellarEquilibriumSpecificationCompilation
+    : detail::EmptySpecificationCompilation {};
+
+template <models::ModelSpecification Specification>
+struct StellarEquilibriumSpecificationCompilation<Specification>
+    : detail::SelfDescribingSpecificationCompilation<Specification> {};
+
+namespace detail {
+template <typename Specification, typename = void>
+struct SpecificationCompilationIsComplete : std::false_type {};
+
+template <typename Specification>
+struct SpecificationCompilationIsComplete<
+    Specification,
+    std::void_t<typename StellarEquilibriumSpecificationCompilation<
+                    Specification>::GeneratedValueBlocks,
+                typename StellarEquilibriumSpecificationCompilation<
+                    Specification>::GeneratedResidualBlocks,
+                typename StellarEquilibriumSpecificationCompilation<
+                    Specification>::DependsOnValueBlocks,
+                typename StellarEquilibriumSpecificationCompilation<
+                    Specification>::AffectedResidualBlocks,
+                typename StellarEquilibriumSpecificationCompilation<
+                    Specification>::JacobianCouplings,
+                std::bool_constant<StellarEquilibriumSpecificationCompilation<
+                    Specification>::registered>,
+                std::bool_constant<StellarEquilibriumSpecificationCompilation<
+                    Specification>::complete>>>
+    : std::bool_constant<
+          StellarEquilibriumSpecificationCompilation<
+              Specification>::registered &&
+          StellarEquilibriumSpecificationCompilation<Specification>::complete &&
+          IsValueBlockList<typename StellarEquilibriumSpecificationCompilation<
+              Specification>::GeneratedValueBlocks>::value &&
+          IsResidualBlockList<
+              typename StellarEquilibriumSpecificationCompilation<
+                  Specification>::GeneratedResidualBlocks>::value &&
+          IsValueBlockList<typename StellarEquilibriumSpecificationCompilation<
+              Specification>::DependsOnValueBlocks>::value &&
+          IsResidualBlockList<
+              typename StellarEquilibriumSpecificationCompilation<
+                  Specification>::AffectedResidualBlocks>::value &&
+          IsJacobianCouplingList<
+              typename StellarEquilibriumSpecificationCompilation<
+                  Specification>::JacobianCouplings>::value> {};
+} // namespace detail
+
+template <typename Candidate>
+inline constexpr bool stellarEquilibriumSpecificationCompilationComplete =
+    detail::SpecificationCompilationIsComplete<
+        std::remove_cvref_t<Candidate>>::value;
+
+template <typename Candidate>
+concept StellarEquilibriumSpecificationCompilable =
+    stellarEquilibriumSpecificationCompilationComplete<Candidate>;
+
+namespace detail {
+/*
+ * This five-by-five physical core is independent of global constraints.
+ * Even FixedTotalMass is compiled as a contribution, keeping C and R_M
+ * visible in that specification's metadata.
+ */
+using StellarPhysicsValueBlocks = utils::blocks::type_list<
+    utils::blocks::density::mass::value,
+    utils::blocks::surface_deformation::parameters::value,
+    utils::blocks::gravity::gradient::value,
+    utils::blocks::gravity::poisson::value,
+    utils::blocks::enthalpy::specific::value>;
+
+using StellarPhysicsResidualBlocks = utils::blocks::type_list<
+    utils::blocks::gravity::gradient::residual,
+    utils::blocks::gravity::poisson::residual,
+    utils::blocks::density::mass::residual,
+    utils::blocks::surface_deformation::shape_equilibrium::residual,
+    utils::blocks::enthalpy::specific::residual>;
+
+using StellarPhysicsJacobianRows = utils::blocks::type_list<
+    utils::blocks::block_row<
+        utils::blocks::gravity::gradient::residual,
+        utils::blocks::gravity::gradient::value,
+        utils::blocks::gravity::poisson::value,
+        utils::blocks::surface_deformation::parameters::value>,
+    utils::blocks::block_row<
+        utils::blocks::gravity::poisson::residual,
+        utils::blocks::gravity::gradient::value,
+        utils::blocks::density::mass::value,
+        utils::blocks::surface_deformation::parameters::value>,
+    utils::blocks::block_row<
+        utils::blocks::density::mass::residual,
+        utils::blocks::density::mass::value,
+        utils::blocks::enthalpy::specific::value,
+        utils::blocks::surface_deformation::parameters::value>,
+    utils::blocks::block_row<
+        utils::blocks::surface_deformation::shape_equilibrium::residual,
+        utils::blocks::density::mass::value,
+        utils::blocks::surface_deformation::parameters::value,
+        utils::blocks::gravity::gradient::value,
+        utils::blocks::enthalpy::specific::value>,
+    utils::blocks::block_row<
+        utils::blocks::enthalpy::specific::residual,
+        utils::blocks::enthalpy::specific::value,
+        utils::blocks::gravity::poisson::value,
+        utils::blocks::surface_deformation::parameters::value>>;
+
+template <typename Row> struct JacobianRowCouplings;
+
+template <typename Residual, typename... Values>
+struct JacobianRowCouplings<utils::blocks::block_row<Residual, Values...>> {
+  using Type = utils::blocks::type_list<
+      StellarEquilibriumJacobianCoupling<Residual, Values>...>;
+};
+
+template <typename Rows> struct FlattenJacobianRows;
+
+template <typename... Rows>
+struct FlattenJacobianRows<utils::blocks::type_list<Rows...>> {
+  using Type =
+      ConcatenateBlockListsT<typename JacobianRowCouplings<Rows>::Type...>;
+};
+
+using StellarPhysicsJacobianCouplings =
+    typename FlattenJacobianRows<StellarPhysicsJacobianRows>::Type;
+
+template <typename SpecificationSet>
+struct SpecificationSetCompilationsAreComplete;
+
+template <models::ModelSpecification... Specifications>
+struct SpecificationSetCompilationsAreComplete<
+    models::detail::SpecificationSetStorage<Specifications...>>
+    : std::bool_constant<(
+          stellarEquilibriumSpecificationCompilationComplete<Specifications> &&
+          ...)> {};
+
+template <typename SpecificationSet, bool Complete>
+struct CollectStellarEquilibriumContributionsImpl {
+  using GeneratedValueBlocks = utils::blocks::type_list<>;
+  using GeneratedResidualBlocks = utils::blocks::type_list<>;
+  using ContributionJacobianCouplings = utils::blocks::type_list<>;
+  using IncidentJacobianCouplings = ContributionJacobianCouplings;
+
+  static constexpr bool complete = false;
+};
+
+template <models::ModelSpecification... Specifications>
+struct CollectStellarEquilibriumContributionsImpl<
+    models::detail::SpecificationSetStorage<Specifications...>, true> {
+  using GeneratedValueBlocks = ConcatenateBlockListsT<
+      typename StellarEquilibriumSpecificationCompilation<
+          Specifications>::GeneratedValueBlocks...>;
+  using GeneratedResidualBlocks = ConcatenateBlockListsT<
+      typename StellarEquilibriumSpecificationCompilation<
+          Specifications>::GeneratedResidualBlocks...>;
+  using ContributionJacobianCouplings = UniqueConcatenateBlockListsT<
+      typename StellarEquilibriumSpecificationCompilation<
+          Specifications>::JacobianCouplings...>;
+  using IncidentJacobianCouplings = UniqueConcatenateBlockListsT<
+      typename StellarEquilibriumSpecificationCompilation<
+          Specifications>::IncidentJacobianCouplings...>;
+
+  static constexpr bool complete = true;
+};
+
+template <typename SpecificationSet>
+using CollectStellarEquilibriumContributions =
+    CollectStellarEquilibriumContributionsImpl<
+        SpecificationSet,
+        SpecificationSetCompilationsAreComplete<SpecificationSet>::value>;
+
+template <typename Residual, typename Couplings> struct ValuesCoupledToResidual;
+
+template <typename Residual>
+struct ValuesCoupledToResidual<Residual, utils::blocks::type_list<>> {
+  using Type = utils::blocks::type_list<>;
+};
+
+template <typename Residual, typename HeadResidual, typename HeadValue,
+          typename... Tail>
+struct ValuesCoupledToResidual<
+    Residual,
+    utils::blocks::type_list<
+        StellarEquilibriumJacobianCoupling<HeadResidual, HeadValue>, Tail...>> {
+private:
+  using Remaining =
+      typename ValuesCoupledToResidual<Residual,
+                                       utils::blocks::type_list<Tail...>>::Type;
+
+public:
+  using Type = std::conditional_t<
+      std::same_as<Residual, HeadResidual>,
+      ConcatenateBlockListsT<utils::blocks::type_list<HeadValue>, Remaining>,
+      Remaining>;
+};
+
+template <typename Residual, typename Values> struct MakeJacobianRow;
+
+template <typename Residual, typename... Values>
+struct MakeJacobianRow<Residual, utils::blocks::type_list<Values...>> {
+  using Type = utils::blocks::block_row<Residual, Values...>;
+};
+
+template <typename Residuals, typename Couplings> struct SynthesizeJacobianRows;
+
+template <typename... Residuals, typename Couplings>
+struct SynthesizeJacobianRows<utils::blocks::type_list<Residuals...>,
+                              Couplings> {
+  using Type = utils::blocks::type_list<typename MakeJacobianRow<
+      Residuals,
+      typename ValuesCoupledToResidual<Residuals, Couplings>::Type>::Type...>;
+};
+
+template <typename Couplings, typename ValueBlocks, typename ResidualBlocks>
+struct CouplingEndpointsBelongToForm : std::false_type {};
+
+template <typename ValueBlocks, typename ResidualBlocks, typename... Couplings>
+struct CouplingEndpointsBelongToForm<utils::blocks::type_list<Couplings...>,
+                                     ValueBlocks, ResidualBlocks>
+    : std::bool_constant<((utils::blocks::contains_type_v<
+                               typename Couplings::Value, ValueBlocks> &&
+                           utils::blocks::contains_type_v<
+                               typename Couplings::Residual, ResidualBlocks>) &&
+                          ...)> {};
+
+template <typename Candidate> struct CompileStellarEquilibriumSystem {
+  using GeneratedValueBlocks = utils::blocks::type_list<>;
+  using GeneratedCorrectionBlocks = GeneratedValueBlocks;
+  using GeneratedResidualBlocks = utils::blocks::type_list<>;
+  using BaseJacobianCouplings = utils::blocks::type_list<>;
+  using ContributionJacobianCouplings = utils::blocks::type_list<>;
+  using IncidentJacobianCouplings = ContributionJacobianCouplings;
+  using JacobianCouplings = utils::blocks::type_list<>;
+
+  static constexpr bool compilable = false;
+};
+
+template <model::StellarModelType Model>
+struct CompileStellarEquilibriumSystem<Model> {
+  using ModelType = std::remove_cvref_t<Model>;
+  using Contributions = CollectStellarEquilibriumContributions<
+      typename ModelType::SpecificationTypes>;
+
+  using GeneratedValueBlocks = typename Contributions::GeneratedValueBlocks;
+  using GeneratedCorrectionBlocks = GeneratedValueBlocks;
+  using GeneratedResidualBlocks =
+      typename Contributions::GeneratedResidualBlocks;
+
+  using ValueBlocks =
+      ConcatenateBlockListsT<StellarPhysicsValueBlocks, GeneratedValueBlocks>;
+  using ResidualBlocks = ConcatenateBlockListsT<StellarPhysicsResidualBlocks,
+                                                GeneratedResidualBlocks>;
+  using FormType = utils::blocks::block_form<ValueBlocks, ResidualBlocks>;
+
+  using BaseJacobianCouplings = StellarPhysicsJacobianCouplings;
+  using ContributionJacobianCouplings =
+      typename Contributions::ContributionJacobianCouplings;
+  using IncidentJacobianCouplings = ContributionJacobianCouplings;
+  using JacobianCouplings =
+      UniqueConcatenateBlockListsT<BaseJacobianCouplings,
+                                   ContributionJacobianCouplings>;
+
+  // Pass two: materialize rows only after all contributed blocks are
+  // present in the final form.
+  using JacobianType =
+      typename SynthesizeJacobianRows<ResidualBlocks, JacobianCouplings>::Type;
+
+  static constexpr bool compilable =
+      Contributions::complete &&
+      utils::blocks::block_form_is_valid_v<FormType> &&
+      IsJacobianCouplingList<JacobianCouplings>::value &&
+      CouplingEndpointsBelongToForm<JacobianCouplings, ValueBlocks,
+                                    ResidualBlocks>::value &&
+      utils::blocks::jacobian_form_is_valid_v<FormType, JacobianType>;
+};
+} // namespace detail
+
+template <typename Candidate>
+inline constexpr bool stellarEquilibriumSystemIsCompilable =
+    detail::CompileStellarEquilibriumSystem<
+        std::remove_cvref_t<Candidate>>::compilable;
+
+/*
+ * This compiler proves the symbolic block topology only.  Keep the explicit
+ * name available to extension authors and tests so that success here is not
+ * mistaken for an assembled numerical runtime.  The established spelling is
+ * retained below as a compatibility alias.
+ */
+template <typename Candidate>
+inline constexpr bool stellarEquilibriumIsSymbolicallyCompilable =
+    stellarEquilibriumSystemIsCompilable<Candidate>;
+
+template <typename Candidate>
+concept StellarEquilibriumSymbolicallyCompilable =
+    stellarEquilibriumIsSymbolicallyCompilable<Candidate>;
+
+template <typename Candidate>
+concept StellarEquilibriumSystemCompilable =
+    StellarEquilibriumSymbolicallyCompilable<Candidate>;
+
+template <model::StellarModelType Model>
+  requires StellarEquilibriumSystemCompilable<Model>
+struct CompiledStellarEquilibriumSystem final
+    : detail::CompileStellarEquilibriumSystem<std::remove_cvref_t<Model>> {
+  using Base =
+      detail::CompileStellarEquilibriumSystem<std::remove_cvref_t<Model>>;
+
+  using FormType = typename Base::FormType;
+  using JacobianType = typename Base::JacobianType;
+
+  // This classification is exposed only after the complete compiler concept
+  // has succeeded; model declarations intentionally do not predict it.
+  static constexpr models::EquilibriumSystemCompilation compilationClass =
+      models::EquilibriumSystemCompilation::complete_equilibrium_system;
+
+  static_assert(utils::blocks::block_form_is_valid_v<FormType>);
+  static_assert(utils::blocks::valid_jacobian_form<FormType, JacobianType>);
+};
+
+template <model::StellarModelType Model>
+  requires StellarEquilibriumSystemCompilable<Model>
+using CompiledStellarEquilibriumForm =
+    typename CompiledStellarEquilibriumSystem<Model>::FormType;
+
+template <model::StellarModelType Model>
+  requires StellarEquilibriumSystemCompilable<Model>
+using CompiledStellarEquilibriumJacobianForm =
+    typename CompiledStellarEquilibriumSystem<Model>::JacobianType;
+} // namespace mean_field::operators

libmeanfield/interface/operators/stellar_equilibrium_problem.cppm

@@ -2,6 +2,8 @@ module;
 
 #include <concepts>
 #include <cstddef>
+#include <cstdint>
+#include <memory>
 #include <type_traits>
 #include <utility>
 
@@ -13,46 +15,126 @@ export import :deformation.domain_deformation;
 export import :equilibrium.stellar_discretization;
 export import :material.thermodynamic_equations;
 export import :model.typed_stellar;
-export import :operators.prepared_central_density_stellar_equilibrium;
+export import :normalization.operators;
+export import :operators.prepared_variadic_stellar_equilibrium;
 export import :surface.compiler;
 
 export namespace mean_field::equilibrium {
-    template <typename Candidate>
+    namespace detail {
-    concept StellarEquilibriumModel = model::StellarModelType<Candidate> && requires {
+        template <
-        requires std::remove_cvref_t<Candidate>::template containsSpecification<eos::Polytrope>;
+            model::StellarModelType Model,
-        requires std::remove_cvref_t<Candidate>::template containsSpecification<surface::Isobaric>;
+            bool SymbolicallyCompilable = operators::StellarEquilibriumSystemCompilable<Model>>
-        requires std::remove_cvref_t<Candidate>::template containsSpecification<models::FixedTotalMass>;
+        struct StellarSurfaceCompilationAudit {
-        requires std::remove_cvref_t<Candidate>::specificationCount ==
+            static constexpr bool complete = false;
-                     3 + static_cast<std::size_t>(
-                             std::remove_cvref_t<Candidate>::template containsSpecification<models::FixedCentralDensity>
-                         );
         };
 
-    template <StellarEquilibriumModel Model> class StellarEquilibriumProblem final {
+        template <model::StellarModelType Model>
+        struct StellarSurfaceCompilationAudit<Model, true> {
+        private:
+            using ModelType = std::remove_cvref_t<Model>;
+            using EquationOfState = typename ModelType::EquationOfStateType;
+            using Form = operators::CompiledStellarEquilibriumForm<ModelType>;
+            using AvailableEquations = material::StellarEquilibriumThermodynamicEquations;
+
+            static constexpr bool thermodynamicsCompilable =
+                material::ThermodynamicEquationsCompilable<EquationOfState, Form, AvailableEquations>;
+
+        public:
+            static constexpr bool complete = [] {
+                if constexpr (!thermodynamicsCompilable) {
+                    return false;
+                } else {
+                    using ThermodynamicEquations =
+                        material::CompiledThermodynamicEquationsT<EquationOfState, Form, AvailableEquations>;
+                    using Formulation = typename ThermodynamicEquations::PressureSurfaceFormulation;
+                    using CompiledSurface =
+                        surface::CompiledPressureSurfaceConstraintT<Formulation, EquationOfState>;
+                    return requires(const ModelType &model) {
+                        {
+                            surface::compilePressureSurfaceConstraint<Formulation>(
+                                model.surfaceCondition(),
+                                model.equationOfState()
+                            )
+                        } -> std::same_as<CompiledSurface>;
+                    };
+                }
+            }();
+        };
+    } // namespace detail
+
+    template <model::StellarModelType Model>
+    inline constexpr bool hasStellarEquilibriumSurfaceCompilation =
+        detail::StellarSurfaceCompilationAudit<std::remove_cvref_t<Model>>::complete;
+
+    template <typename Candidate>
+    concept StellarEquilibriumModel = model::StellarModelType<Candidate> && requires {
+        typename std::remove_cvref_t<Candidate>::EquationOfStateType;
+        requires(
+            std::remove_cvref_t<Candidate>::template specificationRoleCount<
+                models::SpecificationRole::boundary_condition> == 1
+        );
+        requires std::remove_cvref_t<Candidate>::template containsSpecification<models::FixedTotalMass>;
+        requires operators::StellarEquilibriumSystemCompilable<std::remove_cvref_t<Candidate>>;
+        requires hasStellarEquilibriumSurfaceCompilation<std::remove_cvref_t<Candidate>>;
+        requires operators::CompilableRootManifestFor<
+            std::remove_cvref_t<Candidate>,
+            operators::CompiledStellarEquilibriumForm<std::remove_cvref_t<Candidate>>>;
+        requires operators::hasStellarEquilibriumCoreRuntime<std::remove_cvref_t<Candidate>>;
+        requires operators::hasCompleteStellarEquilibriumRuntime<std::remove_cvref_t<Candidate>>;
+        requires operators::stellarEquilibriumRotationProviderCount<std::remove_cvref_t<Candidate>> <= 1;
+    };
+
+    namespace detail {
+        template <typename Model, typename Discretization, typename = void>
+        struct StellarEquilibriumModelDiscretizationStructureAudit : std::false_type { };
+
+        template <typename Model, typename Discretization>
+            requires StellarEquilibriumModel<std::remove_cvref_t<Model>> &&
+                     StellarDiscretizationType<std::remove_cvref_t<Discretization>>
+        struct StellarEquilibriumModelDiscretizationStructureAudit<
+            Model,
+            Discretization,
+            std::void_t<
+                typename std::remove_cvref_t<Discretization>::NormalizationPrescriptionType,
+                typename std::remove_cvref_t<Model>::SpecificationTypes,
+                operators::CompiledStellarEquilibriumForm<std::remove_cvref_t<Model>>,
+                operators::StellarEquilibriumPhysicalCoreType<std::remove_cvref_t<Model>>>>
+            : std::bool_constant<normalization::StellarNormalizationRuntimeAvailableFor<
+                  typename std::remove_cvref_t<Discretization>::NormalizationPrescriptionType,
+                  operators::CompiledStellarEquilibriumForm<std::remove_cvref_t<Model>>,
+                  operators::StellarEquilibriumPhysicalCoreType<std::remove_cvref_t<Model>>,
+                  typename std::remove_cvref_t<Model>::SpecificationTypes>> { };
+
+        struct StellarEquilibriumProblemFactory;
+    } // namespace detail
+
+    template <
+        StellarEquilibriumModel Model,
+        StellarDiscretizationType Discretization = StellarDiscretization>
+        requires detail::StellarEquilibriumModelDiscretizationStructureAudit<
+            std::remove_cvref_t<Model>,
+            std::remove_cvref_t<Discretization>>::value
+    class StellarEquilibriumProblem final {
     public:
         using ModelType = std::remove_cvref_t<Model>;
+        using DiscretizationType = std::remove_cvref_t<Discretization>;
+        using NormalizationPrescriptionType = typename DiscretizationType::NormalizationPrescriptionType;
 
         static constexpr bool hasFixedCentralDensity =
             ModelType::template containsSpecification<models::FixedCentralDensity>;
+        static constexpr bool hasFixedAngularMomentum =
+            ModelType::template containsSpecification<models::FixedAngularMomentum>;
+        static constexpr std::size_t generatedRotationProviderCount =
+            operators::stellarEquilibriumRotationProviderCount<ModelType>;
         static constexpr bool symbolicallySquare = ModelType::symbolicallySquare;
 
-        using PreparedOperatorType               = std::conditional_t<
+        using PreparedOperatorType = operators::PreparedVariadicStellarEquilibriumOperator<ModelType>;
-            hasFixedCentralDensity,
+        using PhysicalCoreType = typename PreparedOperatorType::PhysicalCoreType;
-            operators::PreparedCentralDensityStellarEquilibriumOperator,
+        using FormType = operators::CompiledStellarEquilibriumForm<ModelType>;
-            operators::PreparedStellarEquilibriumOperator>;
+        using JacobianFormType = operators::CompiledStellarEquilibriumJacobianForm<ModelType>;
-        using FormType = std::conditional_t<
+        using ManifestType = operators::EquilibriumSystemManifest<ModelType, FormType, JacobianFormType>;
-            hasFixedCentralDensity,
+        using EquationOfStateType             = model::EquationOfStateType<ModelType>;
-            operators::CentralDensityStellarEquilibriumForm,
+        using SurfaceConditionType            = model::SurfaceConditionType<ModelType>;
-            utils::blocks::surface_deformed_stellar_equilibrium_form>;
-        using JacobianFormType = std::conditional_t<
-            hasFixedCentralDensity,
-            operators::CentralDensityStellarEquilibriumJacobianForm,
-            utils::blocks::surface_deformed_stellar_equilibrium_jacobian_form>;
-        using ManifestType = std::conditional_t<
-            hasFixedCentralDensity,
-            operators::CentralDensityStellarEquilibriumSystemManifest,
-            operators::StellarEquilibriumSystemManifest>;
-        using EquationOfStateType             = eos::Polytrope;
         using AvailableThermodynamicEquations = material::StellarEquilibriumThermodynamicEquations;
         using ThermodynamicEquationsType =
             material::CompiledThermodynamicEquationsT<EquationOfStateType, FormType, AvailableThermodynamicEquations>;
@@ -60,44 +142,22 @@ export namespace mean_field::equilibrium {
             typename ThermodynamicEquationsType::PressureSurfaceFormulation,
             EquationOfStateType>;
 
-        StellarEquilibriumProblem(
+    private:
-            ModelType stellarModel,
+        friend struct detail::StellarEquilibriumProblemFactory;
-            const StellarDiscretization discretization
-        )
-            requires(!hasFixedCentralDensity)
-            : m_stellarModel(std::move(stellarModel)),
-              m_discretization(discretization),
-              m_compiledSurfaceConstraint(CompileSurfaceConstraint(m_stellarModel)),
-              m_preparedOperator(
-                  m_discretization.finiteElementModel(),
-                  m_discretization.domainMapper(),
-                  m_stellarModel.template specification<eos::Polytrope>(),
-                  models::compileConstraint(m_stellarModel.template specification<models::FixedTotalMass>()),
-                  operators::PressureSurfaceConstraintView{m_compiledSurfaceConstraint},
-                  CompileDefaultDomainDeformation(m_discretization.finiteElementModel())
-              ) {
-            VerifyProblem();
-        }
 
         StellarEquilibriumProblem(
             ModelType stellarModel,
-            const StellarDiscretization discretization
+            DiscretizationType discretization
         )
-            requires hasFixedCentralDensity
+            : m_stellarModel(std::make_shared<ModelType>(std::move(stellarModel))),
-            : m_stellarModel(std::move(stellarModel)),
+              m_discretization(std::move(discretization)),
-              m_discretization(discretization),
+              m_compiledSurfaceConstraint(CompileSurfaceConstraint(*m_stellarModel)),
-              m_compiledSurfaceConstraint(CompileSurfaceConstraint(m_stellarModel)),
               m_preparedOperator(
                   m_discretization.finiteElementModel(),
                   m_discretization.domainMapper(),
-                  m_stellarModel.template specification<eos::Polytrope>(),
+                  m_stellarModel,
-                  models::compileConstraint(m_stellarModel.template specification<models::FixedTotalMass>()),
                   operators::PressureSurfaceConstraintView{m_compiledSurfaceConstraint},
-                  CompileDefaultDomainDeformation(m_discretization.finiteElementModel()),
+                  CompileDefaultDomainDeformation(m_discretization.finiteElementModel())
-                  models::compileConstraint(
-                      m_stellarModel.template specification<models::FixedCentralDensity>(),
-                      m_stellarModel.template specification<eos::Polytrope>()
-                  )
               ) {
             VerifyProblem();
         }
@@ -107,14 +167,19 @@ export namespace mean_field::equilibrium {
         StellarEquilibriumProblem(StellarEquilibriumProblem &&)                 = delete;
         StellarEquilibriumProblem &operator=(StellarEquilibriumProblem &&)      = delete;
 
+    public:
         [[nodiscard]] const ModelType &GetStellarModel() const noexcept {
-            return m_stellarModel;
+            return *m_stellarModel;
         }
 
-        [[nodiscard]] const StellarDiscretization &GetDiscretization() const noexcept {
+        [[nodiscard]] const DiscretizationType &GetDiscretization() const noexcept {
             return m_discretization;
         }
 
+        [[nodiscard]] const NormalizationPrescriptionType &GetNormalizationPrescription() const noexcept {
+            return m_discretization.normalizationPrescription();
+        }
+
         [[nodiscard]] const CompiledSurfaceConstraintType &GetCompiledSurfaceConstraint() const noexcept {
             return m_compiledSurfaceConstraint;
         }
@@ -127,6 +192,10 @@ export namespace mean_field::equilibrium {
             return m_preparedOperator;
         }
 
+        [[nodiscard]] const PhysicalCoreType &GetPhysicalOperator() const noexcept {
+            return m_preparedOperator.GetPhysicalOperator();
+        }
+
         [[nodiscard]] const auto &GetManifest() const noexcept {
             return m_preparedOperator.GetRootManifest();
         }
@@ -135,28 +204,20 @@ export namespace mean_field::equilibrium {
             return m_preparedOperator.IsPrepared();
         }
 
-        [[nodiscard]] const operators::StellarEquilibriumDependencies &GetLinearizationDependencies() const {
+        [[nodiscard]] std::uint64_t GetPreparationGeneration() const noexcept {
-            if constexpr (hasFixedCentralDensity) {
+            return m_preparationGeneration;
-                return m_preparedOperator.GetPhysicalOperator().GetDependencies();
-            } else {
-                return m_preparedOperator.GetDependencies();
         }
+
+        [[nodiscard]] const operators::StellarEquilibriumDependencies &GetLinearizationDependencies() const {
+            return GetPhysicalOperator().GetDependencies();
         }
 
         [[nodiscard]] const operators::StellarEquilibriumDependencyStamp &GetGeometryDependency() const {
-            if constexpr (hasFixedCentralDensity) {
+            return GetPhysicalOperator().GetGeneratedDisplacementDependency();
-                return m_preparedOperator.GetPhysicalOperator().GetGeneratedDisplacementDependency();
-            } else {
-                return m_preparedOperator.GetGeneratedDisplacementDependency();
-            }
         }
 
         [[nodiscard]] const field::FieldBoundaryDofMap &GetPressureSurfaceRows() const noexcept {
-            if constexpr (hasFixedCentralDensity) {
+            return GetPhysicalOperator().GetSurfaceConstraintOperator().GetSurfaceRows();
-                return m_preparedOperator.GetPhysicalOperator().GetSurfaceConstraintOperator().GetSurfaceRows();
-            } else {
-                return m_preparedOperator.GetSurfaceConstraintOperator().GetSurfaceRows();
-            }
         }
 
         [[nodiscard]] int StateSize() const noexcept {
@@ -175,8 +236,19 @@ export namespace mean_field::equilibrium {
             const mfem::Vector &state,
             const operators::StellarEquilibriumDependencies &dependencies,
             const physics::RigidRotation &rotation
-        ) {
+        ) requires(generatedRotationProviderCount == 0) {
-            return m_preparedOperator.Prepare(state, dependencies, rotation);
+            auto report = m_preparedOperator.Prepare(state, dependencies, rotation);
+            ++m_preparationGeneration;
+            return report;
+        }
+
+        [[nodiscard]] auto Prepare(
+            const mfem::Vector &state,
+            const operators::StellarEquilibriumDependencies &dependencies
+        ) requires(generatedRotationProviderCount == 1) {
+            auto report = m_preparedOperator.Prepare(state, dependencies);
+            ++m_preparationGeneration;
+            return report;
         }
 
         void BuildResidual(mfem::Vector &residual) const {
@@ -194,8 +266,8 @@ export namespace mean_field::equilibrium {
         [[nodiscard]] static CompiledSurfaceConstraintType CompileSurfaceConstraint(const ModelType &stellarModel) {
             return surface::compilePressureSurfaceConstraint<
                 typename ThermodynamicEquationsType::PressureSurfaceFormulation>(
-                stellarModel.template specification<surface::Isobaric>(),
+                stellarModel.surfaceCondition(),
-                stellarModel.template specification<EquationOfStateType>()
+                stellarModel.equationOfState()
             );
         }
 
@@ -224,34 +296,112 @@ export namespace mean_field::equilibrium {
             MFEM_VERIFY(m_discretization.isCurrent(), "The stellar equilibrium problem has a stale discretization.");
         }
 
-        ModelType m_stellarModel;
+        std::shared_ptr<const ModelType> m_stellarModel;
-        StellarDiscretization m_discretization;
+        DiscretizationType m_discretization;
         CompiledSurfaceConstraintType m_compiledSurfaceConstraint;
         PreparedOperatorType m_preparedOperator;
+        std::uint64_t m_preparationGeneration{0};
     };
 
-    template <StellarEquilibriumModel Model>
+    template <typename Candidate> struct IsStellarEquilibriumProblem : std::false_type { };
-    [[nodiscard]] auto discretize(
+
+    template <StellarEquilibriumModel Model, StellarDiscretizationType Discretization>
+        requires detail::StellarEquilibriumModelDiscretizationStructureAudit<
+            std::remove_cvref_t<Model>,
+            std::remove_cvref_t<Discretization>>::value
+    struct IsStellarEquilibriumProblem<StellarEquilibriumProblem<Model, Discretization>> : std::true_type { };
+
+    template <typename Candidate>
+    concept DiscretizedStellarEquilibriumProblem = IsStellarEquilibriumProblem<std::remove_cvref_t<Candidate>>::value;
+
+    namespace detail {
+        template <
+            typename Model,
+            typename Discretization,
+            bool StructurallyCompatible =
+                StellarEquilibriumModelDiscretizationStructureAudit<
+                    std::remove_cvref_t<Model>,
+                    std::remove_cvref_t<Discretization>>::value>
+        struct StellarEquilibriumModelDiscretizationOperationAudit : std::false_type { };
+
+        template <typename Model, typename Discretization>
+        struct StellarEquilibriumModelDiscretizationOperationAudit<
+            Model,
+            Discretization,
+            true> {
+        private:
+            using ModelType = std::remove_cvref_t<Model>;
+            using DiscretizationType = std::remove_cvref_t<Discretization>;
+            using Problem = StellarEquilibriumProblem<ModelType, DiscretizationType>;
+            using Prescription = typename DiscretizationType::NormalizationPrescriptionType;
+
+        public:
+            static constexpr bool value = [] {
+                if constexpr (
+                    std::same_as<Prescription, normalization::Unnormalized> ||
+                    normalization::PhysicalRieszDiagonalPrescription<Prescription>) {
+                    return true;
+                } else {
+                    return normalization::RuntimePreparedNormalizationOperation<Problem>;
+                }
+            }();
+        };
+    } // namespace detail
+
+    /*
+     * A model and a discretization are separate compile-time choices. Their
+     * pairing is valid only when the normalization plan covers the inferred
+     * form, the selected physical runtime supports it, and a third-party
+     * runtime policy provides its exact preparation operation. Keeping this
+     * as a detection-safe public factory boundary rejects incomplete policies
+     * at discretize(), before a solver-facing problem can be constructed.
+     */
+    template <typename Model, typename Discretization>
+    concept StellarEquilibriumModelDiscretizationCompatible =
+        detail::StellarEquilibriumModelDiscretizationOperationAudit<
+            std::remove_cvref_t<Model>,
+            std::remove_cvref_t<Discretization>>::value;
+
+    namespace detail {
+        /* The structurally formed problem type is needed to probe the ADL
+         * operation without a recursive concept. Its constructor remains
+         * private, and this factory is the single construction authority after
+         * the complete public compatibility contract has succeeded. */
+        struct StellarEquilibriumProblemFactory final {
+            template <StellarEquilibriumModel Model, StellarDiscretizationType Discretization>
+                requires StellarEquilibriumModelDiscretizationCompatible<Model, Discretization>
+            [[nodiscard]] static auto Create(
                 Model &&stellarModel,
-        const StellarDiscretization discretization
+                Discretization discretization
             ) {
                 using ModelType = std::remove_cvref_t<Model>;
-        return StellarEquilibriumProblem<ModelType>{std::forward<Model>(stellarModel), discretization};
+                using DiscretizationType = std::remove_cvref_t<Discretization>;
+                return StellarEquilibriumProblem<ModelType, DiscretizationType>{
+                    std::forward<Model>(stellarModel),
+                    std::move(discretization)
+                };
+            }
+        };
+    } // namespace detail
+
+    template <StellarEquilibriumModel Model, StellarDiscretizationType Discretization>
+        requires StellarEquilibriumModelDiscretizationCompatible<Model, Discretization>
+    [[nodiscard]] auto discretize(
+        Model &&stellarModel,
+        Discretization discretization
+    ) {
+        return detail::StellarEquilibriumProblemFactory::Create(
+            std::forward<Model>(stellarModel),
+            std::move(discretization)
+        );
     }
 
     template <StellarEquilibriumModel Model>
+        requires StellarEquilibriumModelDiscretizationCompatible<Model, StellarDiscretization>
     [[nodiscard]] auto discretize(
         Model &&stellarModel,
         fem::FEM &finiteElementModel
     ) {
         return discretize(std::forward<Model>(stellarModel), StellarDiscretization{finiteElementModel});
     }
-
-    template <typename Candidate> struct IsStellarEquilibriumProblem : std::false_type { };
-
-    template <StellarEquilibriumModel Model>
-    struct IsStellarEquilibriumProblem<StellarEquilibriumProblem<Model>> : std::true_type { };
-
-    template <typename Candidate>
-    concept DiscretizedStellarEquilibriumProblem = IsStellarEquilibriumProblem<std::remove_cvref_t<Candidate>>::value;
 } // namespace mean_field::equilibrium

libmeanfield/interface/preconditioning/equilibrium_coordinates.cppm

@@ -292,7 +292,8 @@ export namespace mean_field::preconditioning {
     };
 
     template <equilibrium::DiscretizedStellarEquilibriumProblem Problem, SpecificationBorderBlockType Block>
-        requires EquilibriumCoordinateComponentFor<Block, typename std::remove_cvref_t<Problem>::FormType>
+        requires EquilibriumCoordinateComponentFor<Block, typename std::remove_cvref_t<Problem>::FormType> &&
+                 SpecificationBorderPreparableFor<Problem, Block>
     class PreparedStellarPreconditioner final : public mfem::Solver {
     private:
         using ProblemType = std::remove_cvref_t<Problem>;
@@ -324,6 +325,9 @@ export namespace mean_field::preconditioning {
             }
         }
 
+        PreparedStellarPreconditioner(ProblemType &&, BlockType) = delete;
+        PreparedStellarPreconditioner(const ProblemType &&, BlockType) = delete;
+
         PreparedStellarPreconditioner(const PreparedStellarPreconditioner &)            = delete;
         PreparedStellarPreconditioner &operator=(const PreparedStellarPreconditioner &) = delete;
         PreparedStellarPreconditioner(PreparedStellarPreconditioner &&)                 = delete;
@@ -367,6 +371,10 @@ export namespace mean_field::preconditioning {
             return m_grouped.GetBlock();
         }
 
+        [[nodiscard]] const ProblemType &GetProblem() const noexcept {
+            return m_grouped.GetProblem();
+        }
+
         [[nodiscard]] const GroupedPreconditioner &GetGroupedPreconditioner() const noexcept {
             return m_grouped;
         }
@@ -392,11 +400,26 @@ export namespace mean_field::preconditioning {
         SpecificationBorderBlockType Block>
         requires EquilibriumCoordinateComponentFor<
             Block,
-            typename std::remove_cvref_t<Problem>::FormType>
+            typename std::remove_cvref_t<Problem>::FormType> &&
+                 SpecificationBorderPreparableFor<Problem, Block>
     [[nodiscard]] auto prepare(
         const Problem &problem,
         Block block
     ) {
         return PreparedStellarPreconditioner<Problem, Block>{problem, std::move(block)};
     }
+
+    template <typename Problem, SpecificationBorderBlockType Block>
+        requires (!std::is_lvalue_reference_v<Problem>) &&
+                 equilibrium::DiscretizedStellarEquilibriumProblem<std::remove_cvref_t<Problem>> &&
+                 EquilibriumCoordinateComponentFor<
+                     Block,
+                     typename std::remove_cvref_t<Problem>::FormType> &&
+                 SpecificationBorderPreparableFor<std::remove_cvref_t<Problem>, Block>
+    [[nodiscard]] auto prepare(
+        Problem &&,
+        Block
+    ) -> PreparedStellarPreconditioner<
+        std::remove_cvref_t<Problem>,
+        std::remove_cvref_t<Block>> = delete;
 } // namespace mean_field::preconditioning

libmeanfield/interface/preconditioning/material_surface.cppm

@@ -231,10 +231,90 @@ export namespace mean_field::preconditioning {
     template <typename Candidate>
     concept MaterialSurfaceDescriptor = detail::IsMaterialSurfaceDescriptor<std::remove_cvref_t<Candidate>>::value;
 
+    /*
+     * Capability boundary for EOS-specific material/surface surrogate
+     * assembly. The current kernels remain polytropic, but selection no
+     * longer embeds that closed-world type test in the descriptor concept.
+     */
+    template <typename EquationOfState>
+    struct MaterialSurfaceEquationOfStateBackend {
+        static constexpr bool registered = false;
+    };
+
+    template <>
+    struct MaterialSurfaceEquationOfStateBackend<eos::Polytrope> {
+        static constexpr bool registered = true;
+        using CoreType = operators::PreparedStellarEquilibriumOperator;
+    };
+
+    template <typename EquationOfState>
+    concept ImplementedMaterialSurfaceEquationOfState = requires {
+        {
+            MaterialSurfaceEquationOfStateBackend<std::remove_cvref_t<EquationOfState>>::registered
+        } -> std::convertible_to<bool>;
+        requires MaterialSurfaceEquationOfStateBackend<
+            std::remove_cvref_t<EquationOfState>>::registered;
+        typename MaterialSurfaceEquationOfStateBackend<std::remove_cvref_t<EquationOfState>>::CoreType;
+    };
+
+    /*
+     * Registering an EOS-to-core association is intentionally not enough to
+     * claim that the material/surface preconditioner can execute it.  Every
+     * implementation listed here must have matching prepared operators and
+     * prepare(...) overloads below.  A future backend should add its pair only
+     * after those executable pieces exist; this keeps capability queries
+     * truthful while the current kernels still consume the legacy physical
+     * core directly.
+     */
+    template <typename EquationOfState, typename PhysicalCore>
+    struct MaterialSurfaceExecutableRuntime {
+        static constexpr bool available = false;
+    };
+
+    template <>
+    struct MaterialSurfaceExecutableRuntime<eos::Polytrope, operators::PreparedStellarEquilibriumOperator> {
+        static constexpr bool available = true;
+    };
+
+    template <typename EquationOfState, typename PhysicalCore>
+    concept ExecutableMaterialSurfaceRuntimeFor = requires {
+        {
+            MaterialSurfaceExecutableRuntime<
+                std::remove_cvref_t<EquationOfState>,
+                std::remove_cvref_t<PhysicalCore>>::available
+        } -> std::convertible_to<bool>;
+        requires MaterialSurfaceExecutableRuntime<
+            std::remove_cvref_t<EquationOfState>,
+            std::remove_cvref_t<PhysicalCore>>::available;
+    };
+
     template <typename Descriptor>
     concept ImplementedMaterialSurfaceDescriptor =
         MaterialSurfaceDescriptor<Descriptor> &&
-        std::same_as<typename Descriptor::ThermodynamicEquations::EquationOfStateType, eos::Polytrope>;
+        ImplementedMaterialSurfaceEquationOfState<
+            typename Descriptor::ThermodynamicEquations::EquationOfStateType>;
+
+    template <typename Descriptor, typename PhysicalCore>
+    concept MaterialSurfaceRuntimeFor =
+        ImplementedMaterialSurfaceDescriptor<Descriptor> && requires {
+            typename MaterialSurfaceEquationOfStateBackend<
+                typename std::remove_cvref_t<Descriptor>::ThermodynamicEquations::EquationOfStateType>::CoreType;
+            requires std::same_as<
+                std::remove_cvref_t<PhysicalCore>,
+                typename MaterialSurfaceEquationOfStateBackend<
+                    typename std::remove_cvref_t<Descriptor>::ThermodynamicEquations::EquationOfStateType>::CoreType>;
+            requires ExecutableMaterialSurfaceRuntimeFor<
+                typename std::remove_cvref_t<Descriptor>::ThermodynamicEquations::EquationOfStateType,
+                PhysicalCore>;
+        };
+
+    template <typename Candidate>
+    concept MaterialSurfacePreconditionerProblem =
+        equilibrium::DiscretizedStellarEquilibriumProblem<Candidate> && requires {
+            requires MaterialSurfaceRuntimeFor<
+                MaterialSurfaceDescriptorFor<std::remove_cvref_t<Candidate>>,
+                typename std::remove_cvref_t<Candidate>::PhysicalCoreType>;
+        };
 
     using DensityMassDiagonalCharacteristics = OperatorCharacteristics<
         OperatorCategory::mass_like,
@@ -605,7 +685,7 @@ export namespace mean_field::preconditioning {
     };
 
     template <
-        equilibrium::DiscretizedStellarEquilibriumProblem Problem,
+        MaterialSurfacePreconditionerProblem Problem,
         backend::Registered MaterialBackend       = backend::Diagonal,
         backend::Registered SurfaceBackend        = backend::Diagonal,
         MaterialSurfaceFactorizationPolicy Policy = SurfaceThenMaterialTriangular>
@@ -623,7 +703,7 @@ export namespace mean_field::preconditioning {
     }
 
     template <
-        equilibrium::DiscretizedStellarEquilibriumProblem Problem,
+        MaterialSurfacePreconditionerProblem Problem,
         backend::Registered MaterialBackend,
         backend::Registered SurfaceBackend,
         MaterialSurfaceFactorizationPolicy Policy,
@@ -688,7 +768,7 @@ export namespace mean_field::preconditioning {
             // direct coupling actions and does not pay for a full Jacobian
             // application.
             m_fullDirection                   = 0.0;
-            const auto fullDirectionView      = m_operation->GetRootManifest().directionView(m_fullDirection);
+            const auto fullDirectionView      = m_operation->GetRootManifest().stateView(m_fullDirection);
             mfem::Vector fullDensityDirection = fullDirectionView.block(utils::blocks::density_field.mass_term);
             mfem::Vector fullSurfaceDirection =
                 fullDirectionView.block(utils::blocks::surface_deformation_field.parameters_term);
@@ -699,10 +779,10 @@ export namespace mean_field::preconditioning {
 
             m_operation->Mult(m_fullDirection, m_fullAction);
             const auto fullActionView            = m_operation->GetRootManifest().residualView(m_fullAction);
-            const mfem::Vector fullDensityAction = fullActionView.block(utils::blocks::density_field.mass_term);
+            const auto fullDensityAction = fullActionView.block(utils::blocks::density_field.mass_term);
-            const mfem::Vector fullSurfaceAction =
+            const auto fullSurfaceAction =
                 fullActionView.block(utils::blocks::surface_deformation_field.shape_equilibrium_term);
-            const mfem::Vector fullEnthalpyAction = fullActionView.block(utils::blocks::enthalpy_field.specific_term);
+            const auto fullEnthalpyAction = fullActionView.block(utils::blocks::enthalpy_field.specific_term);
             densityAction                         = fullDensityAction;
             surfaceAction                         = fullSurfaceAction;
             enthalpyAction                        = fullEnthalpyAction;
@@ -1108,7 +1188,8 @@ export namespace mean_field::preconditioning {
         std::uint64_t surfaceH1Assemblies{0};
     };
 
-    template <ImplementedMaterialSurfaceDescriptor Descriptor, MaterialSurfaceFactorizationPolicy Policy>
+    template <MaterialSurfaceDescriptor Descriptor, MaterialSurfaceFactorizationPolicy Policy>
+        requires MaterialSurfaceRuntimeFor<Descriptor, operators::PreparedStellarEquilibriumOperator>
     class PreparedMaterialSurfaceBlock final : public mfem::Solver {
     public:
         using Block = MaterialSurfaceBlock<Descriptor, backend::Diagonal, backend::Diagonal, Policy>;
@@ -1562,9 +1643,10 @@ export namespace mean_field::preconditioning {
     };
 
     template <
-        ImplementedMaterialSurfaceDescriptor Descriptor,
+        MaterialSurfaceDescriptor Descriptor,
         MaterialSurfaceFactorizationPolicy Policy,
         backend::ApplicationMode Mode>
+        requires MaterialSurfaceRuntimeFor<Descriptor, operators::PreparedStellarEquilibriumOperator>
     class PreparedH1MaterialSurfaceBlock final : public mfem::Solver {
     public:
         using SurfaceBackend = backend::HypreBoomerAMG<Mode>;
@@ -2027,8 +2109,9 @@ export namespace mean_field::preconditioning {
     };
 
     template <
-        ImplementedMaterialSurfaceDescriptor Descriptor,
+        MaterialSurfaceDescriptor Descriptor,
         MaterialSurfaceFactorizationPolicy Policy>
+        requires MaterialSurfaceRuntimeFor<Descriptor, operators::PreparedStellarEquilibriumOperator>
     [[nodiscard]] auto prepare(
         const operators::PreparedStellarEquilibriumOperator &operation,
         MaterialSurfaceBlock<
@@ -2042,26 +2125,26 @@ export namespace mean_field::preconditioning {
 
     template <
         equilibrium::StellarEquilibriumModel Model,
+        equilibrium::StellarDiscretizationType Discretization,
         MaterialSurfaceFactorizationPolicy Policy>
+        requires MaterialSurfacePreconditionerProblem<
+            equilibrium::StellarEquilibriumProblem<Model, Discretization>>
     [[nodiscard]] auto prepare(
-        const equilibrium::StellarEquilibriumProblem<Model> &problem,
+        const equilibrium::StellarEquilibriumProblem<Model, Discretization> &problem,
         MaterialSurfaceBlock<
-            MaterialSurfaceDescriptorFor<equilibrium::StellarEquilibriumProblem<Model>>,
+            MaterialSurfaceDescriptorFor<equilibrium::StellarEquilibriumProblem<Model, Discretization>>,
             backend::Diagonal,
             backend::Diagonal,
             Policy> block
     ) {
-        if constexpr (equilibrium::StellarEquilibriumProblem<Model>::hasFixedCentralDensity) {
+        return prepare(problem.GetPhysicalOperator(), std::move(block));
-            return prepare(problem.GetPreparedOperator().GetPhysicalOperator(), std::move(block));
-        } else {
-            return prepare(problem.GetPreparedOperator(), std::move(block));
-        }
     }
 
     template <
-        ImplementedMaterialSurfaceDescriptor Descriptor,
+        MaterialSurfaceDescriptor Descriptor,
         MaterialSurfaceFactorizationPolicy Policy,
         backend::ApplicationMode Mode>
+        requires MaterialSurfaceRuntimeFor<Descriptor, operators::PreparedStellarEquilibriumOperator>
     [[nodiscard]] auto prepare(
         const operators::PreparedStellarEquilibriumOperator &operation,
         MaterialSurfaceBlock<
@@ -2076,21 +2159,20 @@ export namespace mean_field::preconditioning {
 
     template <
         equilibrium::StellarEquilibriumModel Model,
+        equilibrium::StellarDiscretizationType Discretization,
         MaterialSurfaceFactorizationPolicy Policy,
         backend::ApplicationMode Mode>
+        requires MaterialSurfacePreconditionerProblem<
+            equilibrium::StellarEquilibriumProblem<Model, Discretization>>
     [[nodiscard]] auto prepare(
-        const equilibrium::StellarEquilibriumProblem<Model> &problem,
+        const equilibrium::StellarEquilibriumProblem<Model, Discretization> &problem,
         MaterialSurfaceBlock<
-            MaterialSurfaceDescriptorFor<equilibrium::StellarEquilibriumProblem<Model>>,
+            MaterialSurfaceDescriptorFor<equilibrium::StellarEquilibriumProblem<Model, Discretization>>,
             backend::Diagonal,
             backend::HypreBoomerAMG<Mode>,
             Policy,
             SurfaceH1MassStiffness> block
     ) {
-        if constexpr (equilibrium::StellarEquilibriumProblem<Model>::hasFixedCentralDensity) {
+        return prepare(problem.GetPhysicalOperator(), std::move(block));
-            return prepare(problem.GetPreparedOperator().GetPhysicalOperator(), std::move(block));
-        } else {
-            return prepare(problem.GetPreparedOperator(), std::move(block));
-        }
     }
 } // namespace mean_field::preconditioning

libmeanfield/interface/preconditioning/stellar_equilibrium.cppm

@@ -24,6 +24,7 @@ export namespace mean_field::preconditioning {
         operators::StellarEquilibriumDependencyStamp geometry;
         const void *equationOfStateIdentity{nullptr};
         operators::StellarEquilibriumDependencies linearization;
+        std::uint64_t preparedOperatorGeneration{0};
 
         constexpr bool operator==(const StellarPreconditionerLifecycleSnapshot &) const = default;
     };
@@ -61,7 +62,8 @@ export namespace mean_field::preconditioning {
             .discretization  = prepared.discretization != current.discretization,
             .geometry        = prepared.geometry != current.geometry,
             .equationOfState = prepared.equationOfStateIdentity != current.equationOfStateIdentity,
-            .linearization   = prepared.linearization != current.linearization
+            .linearization   = prepared.linearization != current.linearization ||
+                             prepared.preparedOperatorGeneration != current.preparedOperatorGeneration
         };
     }
 
@@ -95,9 +97,11 @@ export namespace mean_field::preconditioning {
         static constexpr bool registered = false;
     };
 
-    template <equilibrium::StellarEquilibriumModel Model>
+    template <
-    struct StellarEquilibriumProblemTraits<equilibrium::StellarEquilibriumProblem<Model>> {
+        equilibrium::StellarEquilibriumModel Model,
-        using Problem                    = equilibrium::StellarEquilibriumProblem<Model>;
+        equilibrium::StellarDiscretizationType Discretization>
+    struct StellarEquilibriumProblemTraits<equilibrium::StellarEquilibriumProblem<Model, Discretization>> {
+        using Problem                    = equilibrium::StellarEquilibriumProblem<Model, Discretization>;
         using Form                       = typename Problem::FormType;
         using JacobianForm               = typename Problem::JacobianFormType;
         using Manifest                   = typename Problem::ManifestType;
@@ -130,8 +134,9 @@ export namespace mean_field::preconditioning {
                 .discretization = dependencies.discretization,
                 .geometry       = problem.GetGeometryDependency(),
                 .equationOfStateIdentity =
-                    std::addressof(problem.GetStellarModel().template specification<eos::Polytrope>()),
+                    std::addressof(problem.GetStellarModel().equationOfState()),
-                .linearization = dependencies
+                .linearization              = dependencies,
+                .preparedOperatorGeneration = problem.GetPreparationGeneration()
             };
         }
     };
@@ -139,6 +144,243 @@ export namespace mean_field::preconditioning {
     template <typename Candidate>
     concept StellarPreconditionerProblem = StellarEquilibriumProblemTraits<std::remove_cvref_t<Candidate>>::registered;
 
+    namespace detail {
+        template <typename Block>
+        struct IsGeneratedStellarValueBlock : std::false_type { };
+
+        template <typename Generated>
+        struct IsGeneratedStellarValueBlock<utils::blocks::generated_value_block<Generated>>
+            : std::true_type { };
+
+        template <typename Block>
+        struct IsGeneratedStellarResidualBlock : std::false_type { };
+
+        template <typename Generated>
+        struct IsGeneratedStellarResidualBlock<utils::blocks::generated_residual_block<Generated>>
+            : std::true_type { };
+
+        template <typename Coupling>
+        inline constexpr bool isPurePhysicalStellarCoupling =
+            !IsGeneratedStellarValueBlock<
+                std::remove_cvref_t<typename Coupling::Value>>::value &&
+            !IsGeneratedStellarResidualBlock<
+                std::remove_cvref_t<typename Coupling::Residual>>::value;
+
+        /* Pure structure contributions owned by a trusted backend are exact
+         * (core, specification, coupling) capabilities.  Future cores and new
+         * edges start with no privilege: changing a built-in declaration must
+         * be accompanied by an explicit preconditioner decision. Generated-
+         * border terms remain the responsibility of specification-border
+         * machinery. */
+        template <typename PhysicalCore, typename Specification>
+        struct StellarStructureBackendHandledCouplings {
+            using Type = utils::blocks::type_list<>;
+        };
+
+        template <>
+        struct StellarStructureBackendHandledCouplings<
+            operators::PreparedStellarEquilibriumOperator,
+            models::FixedTotalMass> {
+            using Type = utils::blocks::type_list<
+                operators::StellarEquilibriumJacobianCoupling<
+                    utils::blocks::enthalpy::specific::residual,
+                    utils::blocks::density::mass::value>,
+                operators::StellarEquilibriumJacobianCoupling<
+                    utils::blocks::enthalpy::specific::residual,
+                    utils::blocks::surface_deformation::parameters::value>>;
+        };
+
+        template <>
+        struct StellarStructureBackendHandledCouplings<
+            operators::PreparedStellarEquilibriumOperator,
+            models::FixedAngularMomentum> {
+            using Type = utils::blocks::type_list<
+                operators::StellarEquilibriumJacobianCoupling<
+                    utils::blocks::surface_deformation::shape_equilibrium::residual,
+                    utils::blocks::density::mass::value>,
+                operators::StellarEquilibriumJacobianCoupling<
+                    utils::blocks::surface_deformation::shape_equilibrium::residual,
+                    utils::blocks::surface_deformation::parameters::value>,
+                operators::StellarEquilibriumJacobianCoupling<
+                    utils::blocks::enthalpy::specific::residual,
+                    utils::blocks::density::mass::value>,
+                operators::StellarEquilibriumJacobianCoupling<
+                    utils::blocks::enthalpy::specific::residual,
+                    utils::blocks::surface_deformation::parameters::value>>;
+        };
+
+        template <>
+        struct StellarStructureBackendHandledCouplings<
+            operators::PreparedStellarEquilibriumOperator,
+            models::FixedCentralDensity> {
+            using Type = utils::blocks::type_list<
+                operators::StellarEquilibriumJacobianCoupling<
+                    utils::blocks::enthalpy::specific::residual,
+                    utils::blocks::enthalpy::specific::value>>;
+        };
+
+        template <
+            typename Coupling,
+            typename Model,
+            typename PhysicalCore,
+            typename ModelSpecifications>
+        struct EveryCouplingContributionHandled;
+
+        template <
+            typename Coupling,
+            model::StellarModelType Model,
+            typename PhysicalCore,
+            models::ModelSpecification... Specifications>
+        struct EveryCouplingContributionHandled<
+            Coupling,
+            Model,
+            PhysicalCore,
+            models::detail::SpecificationSetStorage<Specifications...>> final {
+        private:
+            template <typename Specification>
+            static constexpr bool handled =
+                !utils::blocks::contains_type_v<
+                    Coupling,
+                    typename operators::StellarEquilibriumSpecificationCompilation<
+                        Specification>::JacobianCouplings> ||
+                (operators::stellarEquilibriumBackendRuntimeAuthorized<
+                     Specification,
+                     Model> &&
+                 utils::blocks::contains_type_v<
+                     Coupling,
+                     typename StellarStructureBackendHandledCouplings<
+                         PhysicalCore,
+                         Specification>::Type>) ||
+                operators::stellarEquilibriumSpecificationCouplingIsStructuralZero<
+                    Specification,
+                    Model,
+                    Coupling>;
+
+        public:
+            static constexpr bool value =
+                (handled<Specifications> && ...);
+        };
+
+        template <
+            typename Remaining,
+            typename Model,
+            typename PhysicalCore,
+            typename ModelSpecifications,
+            typename Unsupported>
+        struct CollectUnsupportedStellarStructureCouplings;
+
+        template <
+            typename Model,
+            typename PhysicalCore,
+            typename ModelSpecifications,
+            typename Unsupported>
+        struct CollectUnsupportedStellarStructureCouplings<
+            utils::blocks::type_list<>,
+            Model,
+            PhysicalCore,
+            ModelSpecifications,
+            Unsupported> {
+            using Type = Unsupported;
+        };
+
+        template <
+            typename Head,
+            typename... Tail,
+            typename Model,
+            typename PhysicalCore,
+            typename ModelSpecifications,
+            typename... Unsupported>
+        struct CollectUnsupportedStellarStructureCouplings<
+            utils::blocks::type_list<Head, Tail...>,
+            Model,
+            PhysicalCore,
+            ModelSpecifications,
+            utils::blocks::type_list<Unsupported...>> {
+        private:
+            static constexpr bool supported =
+                !isPurePhysicalStellarCoupling<Head> ||
+                EveryCouplingContributionHandled<
+                    Head,
+                    Model,
+                    PhysicalCore,
+                    ModelSpecifications>::value;
+            using Next = std::conditional_t<
+                supported,
+                utils::blocks::type_list<Unsupported...>,
+                utils::blocks::type_list<Unsupported..., Head>>;
+
+        public:
+            using Type = typename CollectUnsupportedStellarStructureCouplings<
+                utils::blocks::type_list<Tail...>,
+                Model,
+                PhysicalCore,
+                ModelSpecifications,
+                Next>::Type;
+        };
+
+        template <typename Candidate, typename = void>
+        struct DefaultStellarStructurePhysicalTopologyAudit {
+            using ContributionCouplings = utils::blocks::type_list<>;
+            using UnsupportedCouplings = utils::blocks::type_list<>;
+
+            static constexpr bool supported = false;
+        };
+
+        template <model::StellarModelType Model>
+            requires(
+                operators::StellarEquilibriumSystemCompilable<
+                    std::remove_cvref_t<Model>> &&
+                operators::hasStellarEquilibriumCoreRuntime<
+                    std::remove_cvref_t<Model>>)
+        struct DefaultStellarStructurePhysicalTopologyAudit<
+            Model,
+            std::void_t<
+                typename operators::CompiledStellarEquilibriumSystem<
+                    std::remove_cvref_t<Model>>::ContributionJacobianCouplings,
+                operators::StellarEquilibriumPhysicalCoreType<
+                    std::remove_cvref_t<Model>>>> {
+        private:
+            using Compilation = operators::CompiledStellarEquilibriumSystem<
+                std::remove_cvref_t<Model>>;
+            using PhysicalCore = operators::StellarEquilibriumPhysicalCoreType<
+                std::remove_cvref_t<Model>>;
+        public:
+            using ContributionCouplings =
+                typename Compilation::ContributionJacobianCouplings;
+            using UnsupportedCouplings =
+                typename CollectUnsupportedStellarStructureCouplings<
+                    ContributionCouplings,
+                    std::remove_cvref_t<Model>,
+                    PhysicalCore,
+                    typename std::remove_cvref_t<Model>::SpecificationTypes,
+                    utils::blocks::type_list<>>::Type;
+
+            static constexpr bool supported = UnsupportedCouplings::size == 0;
+        };
+    } // namespace detail
+
+    /* A generated-border edge is owned by specification-border machinery and
+     * is deliberately ignored here. Every pure physical edge contributed by a
+     * model must be owned by the selected numerical structure backend, or
+     * every non-backend provider of that edge must prove StructuralZero.  In
+     * particular, merely overlapping an existing base-Jacobian edge is not
+     * sufficient: a custom nonzero coefficient on that edge would otherwise
+     * disappear silently from the default preconditioner. This audit is
+     * detection-safe and therefore suitable for constraining factories. */
+    template <typename Candidate>
+    struct DefaultStellarStructurePhysicalTopologySupport
+        : detail::DefaultStellarStructurePhysicalTopologyAudit<
+              std::remove_cvref_t<Candidate>> { };
+
+    template <typename Candidate>
+    inline constexpr bool defaultStellarStructurePhysicalTopologySupported =
+        DefaultStellarStructurePhysicalTopologySupport<
+            std::remove_cvref_t<Candidate>>::supported;
+
+    template <typename Candidate>
+    concept DefaultStellarStructurePhysicalTopologySupportedFor =
+        defaultStellarStructurePhysicalTopologySupported<Candidate>;
+
     namespace backend {
         template <typename Component, typename Problem, typename Backend = typename Component::BackendType>
         class PreparedComponent;
@@ -174,56 +416,17 @@ export namespace mean_field::preconditioning {
     } // namespace backend
 
     namespace detail {
-        using DensityIdentity =
-            IdentityBlock<utils::blocks::density::mass::value, utils::blocks::density::mass::residual>;
-        using SurfaceIdentity = IdentityBlock<
-            utils::blocks::surface_deformation::parameters::value,
-            utils::blocks::surface_deformation::shape_equilibrium::residual>;
-        using GravityGradientIdentity =
-            IdentityBlock<utils::blocks::gravity::gradient::value, utils::blocks::gravity::gradient::residual>;
-        using GravityPotentialIdentity =
-            IdentityBlock<utils::blocks::gravity::poisson::value, utils::blocks::gravity::poisson::residual>;
-        using EnthalpyIdentity =
-            IdentityBlock<utils::blocks::enthalpy::specific::value, utils::blocks::enthalpy::specific::residual>;
-        using FixedMassIdentity = IdentityBlock<
-            utils::blocks::fixed_total_mass::mass_normalization::value,
-            utils::blocks::fixed_total_mass::mass_normalization::residual>;
-        using FixedCentralDensityIdentity = IdentityBlock<
-            utils::blocks::fixed_central_density::central_value::value,
-            utils::blocks::fixed_central_density::central_value::residual>;
-
         template <typename Form> struct IdentityPlanForForm;
 
-        template <> struct IdentityPlanForForm<utils::blocks::surface_deformed_stellar_equilibrium_form> {
+        template <typename... Values, typename... Residuals>
-            using Type = PreconditionerPlan<
+        struct IdentityPlanForForm<utils::blocks::block_form<
-                DensityIdentity,
+            utils::blocks::type_list<Values...>,
-                SurfaceIdentity,
+            utils::blocks::type_list<Residuals...>>> {
-                GravityGradientIdentity,
+            static_assert(sizeof...(Values) == sizeof...(Residuals));
-                GravityPotentialIdentity,
+            using Type = PreconditionerPlan<IdentityBlock<Values, Residuals>...>;
-                EnthalpyIdentity,
-                FixedMassIdentity>;
 
             [[nodiscard]] static constexpr Type Make() {
-                return Type{DensityIdentity{},          SurfaceIdentity{},  GravityGradientIdentity{},
+                return Type{IdentityBlock<Values, Residuals>{}...};
-                            GravityPotentialIdentity{}, EnthalpyIdentity{}, FixedMassIdentity{}};
-            }
-        };
-
-        template <> struct IdentityPlanForForm<utils::blocks::central_density_bordered_stellar_equilibrium_form> {
-            using Type = PreconditionerPlan<
-                DensityIdentity,
-                SurfaceIdentity,
-                GravityGradientIdentity,
-                GravityPotentialIdentity,
-                EnthalpyIdentity,
-                FixedMassIdentity,
-                FixedCentralDensityIdentity>;
-
-            [[nodiscard]] static constexpr Type Make() {
-                return Type{
-                    DensityIdentity{},  SurfaceIdentity{},   GravityGradientIdentity{},    GravityPotentialIdentity{},
-                    EnthalpyIdentity{}, FixedMassIdentity{}, FixedCentralDensityIdentity{}
-                };
             }
         };
 

libmeanfield/interface/preconditioning/stellar_structure.cppm

@@ -14,6 +14,7 @@ export module mean_field:preconditioning.stellar_structure;
 
 export import :preconditioning.gravity_field;
 export import :preconditioning.material_surface;
+export import :preconditioning.stellar_equilibrium;
 
 export namespace mean_field::preconditioning {
     struct IndependentStellarSubsystems final { };
@@ -626,28 +627,88 @@ export namespace mean_field::preconditioning {
         }
     };
 
+    /*
+     * Material/surface execution and stellar-structure execution are separate
+     * capabilities.  The latter also owns the physical cross-Jacobian and
+     * gravity-context wiring, which currently target the legacy prepared core.
+     * Add future cores here only together with matching cross-coupling and
+     * preparation implementations.
+     */
+    template <typename PhysicalCore>
+    struct StellarStructureExecutableRuntime {
+        static constexpr bool available = false;
+    };
+
+    template <>
+    struct StellarStructureExecutableRuntime<operators::PreparedStellarEquilibriumOperator> {
+        static constexpr bool available = true;
+    };
+
+    template <typename PhysicalCore>
+    concept ExecutableStellarStructureRuntimeFor = requires {
+        {
+            StellarStructureExecutableRuntime<std::remove_cvref_t<PhysicalCore>>::available
+        } -> std::convertible_to<bool>;
+        requires StellarStructureExecutableRuntime<std::remove_cvref_t<PhysicalCore>>::available;
+    };
+
+    template <typename Descriptor, typename PhysicalCore>
+    concept StellarStructureRuntimeFor =
+        MaterialSurfaceRuntimeFor<Descriptor, PhysicalCore> &&
+        ExecutableStellarStructureRuntimeFor<PhysicalCore>;
+
+    template <typename Candidate>
+    concept StellarStructurePreconditionerProblem =
+        equilibrium::DiscretizedStellarEquilibriumProblem<Candidate> &&
+        DefaultStellarStructurePhysicalTopologySupportedFor<
+            typename std::remove_cvref_t<Candidate>::ModelType> &&
+        requires {
+            requires StellarStructureRuntimeFor<
+                MaterialSurfaceDescriptorFor<std::remove_cvref_t<Candidate>>,
+                typename std::remove_cvref_t<Candidate>::PhysicalCoreType>;
+        };
+
     namespace detail {
-        template <equilibrium::StellarEquilibriumModel Model>
+        template <StellarStructurePreconditionerProblem Problem>
-        [[nodiscard]] const operators::PreparedStellarEquilibriumOperator &
+        [[nodiscard]] const auto &physicalOperator(const Problem &problem) {
-        physicalOperator(const equilibrium::StellarEquilibriumProblem<Model> &problem) {
+            return problem.GetPhysicalOperator();
-            if constexpr (equilibrium::StellarEquilibriumProblem<Model>::hasFixedCentralDensity) {
-                return problem.GetPreparedOperator().GetPhysicalOperator();
-            } else {
-                return problem.GetPreparedOperator();
-            }
         }
     } // namespace detail
 
+    template <typename Problem, typename MaterialComponent, typename GravityComponent>
+    concept StellarStructurePreparableFor =
+        StellarStructurePreconditionerProblem<std::remove_cvref_t<Problem>> &&
+        PreconditionerComponent<std::remove_cvref_t<MaterialComponent>> &&
+        PreconditionerComponent<std::remove_cvref_t<GravityComponent>> &&
+        requires(
+            const std::remove_cvref_t<Problem> &problem,
+            std::remove_cvref_t<MaterialComponent> materialComponent,
+            std::remove_cvref_t<GravityComponent> gravityComponent
+        ) {
+            preconditioning::prepare(problem, std::move(materialComponent));
+            preconditioning::prepare(
+                problem.GetPhysicalOperator().GetHydrostaticOperator().GetFEM(),
+                problem.GetPhysicalOperator().GetGravityContext().GetGeometryContext(),
+                std::move(gravityComponent)
+            );
+            StellarStructureCrossJacobianOperator{problem.GetPhysicalOperator()};
+        };
+
     template <
         equilibrium::StellarEquilibriumModel Model,
+        equilibrium::StellarDiscretizationType Discretization,
         typename MaterialComponent,
         backend::Registered GravityMassBackend,
         backend::ApplicationMode Mode,
         GravityFactorizationPolicy GravityPolicy,
         StellarStructureFactorizationPolicy StructurePolicy>
+        requires StellarStructurePreparableFor<
+            equilibrium::StellarEquilibriumProblem<Model, Discretization>,
+            MaterialComponent,
+            GravityFieldBlock<GravityMassBackend, backend::HypreBoomerAMG<Mode>, GravityPolicy>>
     class PreparedStellarStructureBlock final : public mfem::Solver {
     private:
-        using Problem          = equilibrium::StellarEquilibriumProblem<Model>;
+        using Problem          = equilibrium::StellarEquilibriumProblem<Model, Discretization>;
         using GravityComponent = GravityFieldBlock<GravityMassBackend, backend::HypreBoomerAMG<Mode>, GravityPolicy>;
         using Structure        = StellarStructureBlock<
             MaterialComponent,
@@ -769,7 +830,7 @@ export namespace mean_field::preconditioning {
     };
 
     template <
-        equilibrium::DiscretizedStellarEquilibriumProblem Problem,
+        StellarStructurePreconditionerProblem Problem,
         typename MaterialComponent,
         backend::Registered GravityMassBackend,
         backend::ApplicationMode Mode,
@@ -792,7 +853,7 @@ export namespace mean_field::preconditioning {
         };
     }
 
-    template <equilibrium::DiscretizedStellarEquilibriumProblem Problem>
+    template <StellarStructurePreconditionerProblem Problem>
     [[nodiscard]] constexpr auto stellarStructureBlock(const Problem &problem) {
         using FixedAMG = backend::HypreBoomerAMG<backend::FixedCycles>;
         auto material  = materialSurfaceBlock(problem);
@@ -805,25 +866,30 @@ export namespace mean_field::preconditioning {
 
     template <
         equilibrium::StellarEquilibriumModel Model,
+        equilibrium::StellarDiscretizationType Discretization,
         typename MaterialComponent,
         backend::Registered GravityMassBackend,
         backend::ApplicationMode Mode,
         GravityFactorizationPolicy GravityPolicy,
         StellarStructureFactorizationPolicy StructurePolicy>
+        requires StellarStructurePreparableFor<
+            equilibrium::StellarEquilibriumProblem<Model, Discretization>,
+            MaterialComponent,
+            GravityFieldBlock<GravityMassBackend, backend::HypreBoomerAMG<Mode>, GravityPolicy>>
     [[nodiscard]] auto prepare(
-        const equilibrium::StellarEquilibriumProblem<Model> &problem,
+        const equilibrium::StellarEquilibriumProblem<Model, Discretization> &problem,
         StellarStructureBlock<
             MaterialComponent,
             GravityFieldBlock<
                 GravityMassBackend,
                 backend::HypreBoomerAMG<Mode>,
                 GravityPolicy>,
-            typename equilibrium::StellarEquilibriumProblem<Model>::FormType,
+            typename equilibrium::StellarEquilibriumProblem<Model, Discretization>::FormType,
-            typename equilibrium::StellarEquilibriumProblem<Model>::JacobianFormType,
+            typename equilibrium::StellarEquilibriumProblem<Model, Discretization>::JacobianFormType,
             StructurePolicy> structure
     ) {
         return PreparedStellarStructureBlock<
-            Model, MaterialComponent, GravityMassBackend, Mode, GravityPolicy, StructurePolicy>{
+            Model, Discretization, MaterialComponent, GravityMassBackend, Mode, GravityPolicy, StructurePolicy>{
             problem, std::move(structure)
         };
     }

libmeanfield/interface/seed/stellar_equilibrium_projection.cppm

@@ -1,5 +1,7 @@
 module;
 
+#include <cmath>
+#include <cstddef>
 #include <concepts>
 #include <stdexcept>
 #include <type_traits>
@@ -25,6 +27,65 @@ export namespace mean_field::seed {
         mfem::Vector values;
     };
 
+    /*
+     * The public radial-projection extension boundary deliberately speaks in
+     * physical state names. A specification author supplies one small rule
+     * and opts in with
+     *
+     *   using RadialProjection = seed::projection::Use<MyProjectionPhysics>;
+     *
+     * There is no registry ordinal and no model-combination specialization.
+     */
+    struct RadialProjectionScales final {
+        dimensions::MassValue targetMass;
+        dimensions::LengthValue stellarRadius;
+        double bernoulliConstant;
+        double sphericalMomentOfInertia;
+        const mfem::Array<int> *surfaceCarrierRows;
+    };
+
+    struct RadialProjectionState final {
+        mfem::Vector density;
+        mfem::Vector surfaceShape;
+        mfem::Vector gravityGradient;
+        mfem::Vector gravityPotential;
+        mfem::Vector specificEnthalpy;
+    };
+
+    namespace projection {
+        template <typename Physics> struct Use final {
+            using PhysicsType = Physics;
+        };
+
+        /* An explicit opt-in for a specification that leaves a radial seed unchanged. */
+        struct NoStateChange {
+            static constexpr bool registered         = true;
+            static constexpr bool providesRadialMass = false;
+
+            template <typename Model>
+            static constexpr bool supports = true;
+
+            template <typename Specification, typename Model>
+            static void validate(
+                const Specification &,
+                const Model &,
+                const RadialProfile &,
+                const StellarEquilibriumProjectionOptions &
+            ) noexcept {
+            }
+
+            template <typename Specification, typename Model>
+            static void initialize(
+                const Specification &,
+                const Model &,
+                const RadialProjectionScales &,
+                RadialProjectionState &,
+                mfem::Vector
+            ) noexcept {
+            }
+        };
+    } // namespace projection
+
     namespace detail {
         struct ProjectedRadialFields final {
             mfem::Vector density;
@@ -32,13 +93,13 @@ export namespace mean_field::seed {
             mfem::Vector gravityPotential;
             mfem::Vector specificEnthalpy;
             double bernoulliConstant;
+            double sphericalMomentOfInertia;
         };
 
         [[nodiscard]] ProjectedRadialFields projectRadialFields(
-            const equilibrium::StellarDiscretization &discretization,
+            fem::FEM &finiteElementModel,
             const RadialProfile &profile,
             dimensions::MassValue targetMass,
-            dimensions::PressureValue targetSurfacePressure,
             const StellarEquilibriumProjectionOptions &options
         );
 
@@ -52,18 +113,402 @@ export namespace mean_field::seed {
             }
             destination = source;
         }
+
+        struct UnavailableRadialProjectionPhysics final {
+            static constexpr bool registered         = false;
+            static constexpr bool providesRadialMass = false;
+
+            template <typename Model>
+            static constexpr bool supports = false;
+        };
+
+        struct FixedTotalMassRadialProjectionPhysics final {
+            static constexpr bool registered         = true;
+            static constexpr bool providesRadialMass = true;
+
+            template <typename Model>
+            static constexpr bool supports = true;
+
+            [[nodiscard]] static dimensions::MassValue targetMass(const models::FixedTotalMass &specification) {
+                return specification.targetMass();
+            }
+
+            template <typename Model>
+            static void validate(
+                const models::FixedTotalMass &,
+                const Model &,
+                const RadialProfile &,
+                const StellarEquilibriumProjectionOptions &
+            ) noexcept {
+            }
+
+            template <typename Model>
+            static void initialize(
+                const models::FixedTotalMass &,
+                const Model &,
+                const RadialProjectionScales &scales,
+                RadialProjectionState &,
+                mfem::Vector coordinate
+            ) {
+                if (coordinate.Size() != 1) {
+                    throw std::invalid_argument(
+                        "A fixed-total-mass radial projection requires exactly one generated multiplier."
+                    );
+                }
+                coordinate(0) = scales.bernoulliConstant;
+            }
+        };
+
+        struct IsobaricRadialProjectionPhysics final {
+            static constexpr bool registered         = true;
+            static constexpr bool providesRadialMass = false;
+
+            template <typename Model>
+            static constexpr bool supports = requires(
+                const Model &model,
+                const surface::Isobaric &condition
+            ) {
+                {
+                    eos::evaluate<dimensions::quantity::SpecificEnthalpy>(
+                        model.equationOfState(), condition.targetPressure()
+                    )
+                } -> std::same_as<dimensions::SpecificEnthalpyValue>;
+            };
+
+            template <typename Model>
+            static void validate(
+                const surface::Isobaric &condition,
+                const Model &,
+                const RadialProfile &,
+                const StellarEquilibriumProjectionOptions &
+            ) {
+                if (condition.targetPressure().value() != 0.0) {
+                    throw std::invalid_argument("A Lane-Emden radial seed requires a zero-pressure isobaric surface.");
+                }
+            }
+
+            template <typename Model>
+            static void initialize(
+                const surface::Isobaric &condition,
+                const Model &model,
+                const RadialProjectionScales &scales,
+                RadialProjectionState &state,
+                mfem::Vector coordinate
+            ) {
+                if (coordinate.Size() != 0) {
+                    throw std::logic_error("An isobaric surface must not generate a radial-seed coordinate.");
+                }
+                if (scales.surfaceCarrierRows == nullptr) {
+                    throw std::logic_error("An isobaric radial projection requires compiled surface-carrier rows.");
+                }
+
+                const dimensions::SpecificEnthalpyValue requiredSurfaceEnthalpy =
+                    eos::evaluate<dimensions::quantity::SpecificEnthalpy>(
+                        model.equationOfState(), condition.targetPressure()
+                    );
+                for (const int surfaceRow : *scales.surfaceCarrierRows) {
+                    state.specificEnthalpy(surfaceRow) = requiredSurfaceEnthalpy.value();
+                }
+            }
+        };
+
+        struct FixedCentralDensityRadialProjectionPhysics final {
+            static constexpr bool registered         = true;
+            static constexpr bool providesRadialMass = false;
+
+            template <typename Model>
+            static constexpr bool supports = true;
+
+            template <typename Model>
+            static void validate(
+                const models::FixedCentralDensity &,
+                const Model &,
+                const RadialProfile &,
+                const StellarEquilibriumProjectionOptions &
+            ) noexcept {
+            }
+
+            template <typename Model>
+            static void initialize(
+                const models::FixedCentralDensity &,
+                const Model &,
+                const RadialProjectionScales &,
+                RadialProjectionState &,
+                mfem::Vector coordinate
+            ) {
+                if (coordinate.Size() != 1) {
+                    throw std::invalid_argument(
+                        "A fixed-central-density radial projection requires exactly one generated phase coordinate."
+                    );
+                }
+                coordinate(0) = 0.0;
+            }
+        };
+
+        struct FixedAngularMomentumRadialProjectionPhysics final {
+            static constexpr bool registered         = true;
+            static constexpr bool providesRadialMass = false;
+
+            template <typename Model>
+            static constexpr bool supports = true;
+
+            template <typename Model>
+            static void validate(
+                const models::FixedAngularMomentum &,
+                const Model &,
+                const RadialProfile &,
+                const StellarEquilibriumProjectionOptions &
+            ) noexcept {
+            }
+
+            template <typename Model>
+            static void initialize(
+                const models::FixedAngularMomentum &constraint,
+                const Model &,
+                const RadialProjectionScales &scales,
+                RadialProjectionState &,
+                mfem::Vector coordinate
+            ) {
+                if (coordinate.Size() != 1) {
+                    throw std::logic_error(
+                        "A fixed-angular-momentum radial projection requires one angular-velocity coordinate."
+                    );
+                }
+
+                double centerSquared   = 0.0;
+                double centerAlongAxis = 0.0;
+                for (std::size_t component = 0; component < constraint.center().size(); ++component) {
+                    centerSquared += constraint.center()[component] * constraint.center()[component];
+                    centerAlongAxis += constraint.center()[component] * constraint.axis()[component];
+                }
+                const double parallelAxisCorrection =
+                    scales.targetMass.value() * (centerSquared - centerAlongAxis * centerAlongAxis);
+                const double momentOfInertia = scales.sphericalMomentOfInertia + parallelAxisCorrection;
+                if (!std::isfinite(momentOfInertia) || momentOfInertia <= 0.0) {
+                    throw std::runtime_error("The radial seed has no finite, positive axial moment of inertia.");
+                }
+                coordinate(0) = constraint.targetAngularMomentum().value() / momentOfInertia;
+            }
+        };
+
+        template <typename Specification> struct BuiltinRadialProjectionPhysics {
+            using Type = UnavailableRadialProjectionPhysics;
+        };
+
+        template <> struct BuiltinRadialProjectionPhysics<eos::Polytrope> {
+            using Type = projection::NoStateChange;
+        };
+
+        template <> struct BuiltinRadialProjectionPhysics<surface::Isobaric> {
+            using Type = IsobaricRadialProjectionPhysics;
+        };
+
+        template <> struct BuiltinRadialProjectionPhysics<models::FixedTotalMass> {
+            using Type = FixedTotalMassRadialProjectionPhysics;
+        };
+
+        template <> struct BuiltinRadialProjectionPhysics<models::FixedCentralDensity> {
+            using Type = FixedCentralDensityRadialProjectionPhysics;
+        };
+
+        template <> struct BuiltinRadialProjectionPhysics<models::FixedAngularMomentum> {
+            using Type = FixedAngularMomentumRadialProjectionPhysics;
+        };
+
+        template <typename Candidate> struct UnwrapRadialProjectionPhysics {
+            using Type                   = UnavailableRadialProjectionPhysics;
+            static constexpr bool valid = false;
+        };
+
+        template <typename Physics> struct UnwrapRadialProjectionPhysics<projection::Use<Physics>> {
+            using Type                   = Physics;
+            static constexpr bool valid = true;
+        };
+
+        template <typename Specification, typename = void> struct SelectRadialProjectionPhysics {
+            using Type = typename BuiltinRadialProjectionPhysics<Specification>::Type;
+        };
+
+        template <typename Specification>
+        struct SelectRadialProjectionPhysics<Specification, std::void_t<typename Specification::RadialProjection>> {
+        private:
+            using Wrapped = UnwrapRadialProjectionPhysics<typename Specification::RadialProjection>;
+
+        public:
+            using Type = std::conditional_t<Wrapped::valid, typename Wrapped::Type, UnavailableRadialProjectionPhysics>;
+        };
+
+        template <typename Physics> [[nodiscard]] consteval bool radialProjectionPhysicsRegistered() {
+            if constexpr (requires {
+                              { Physics::registered } -> std::convertible_to<bool>;
+                          }) {
+                return static_cast<bool>(Physics::registered);
+            } else {
+                return false;
+            }
+        }
+
+        template <typename Physics> [[nodiscard]] consteval bool radialProjectionPhysicsProvidesMass() {
+            if constexpr (requires {
+                              { Physics::providesRadialMass } -> std::convertible_to<bool>;
+                          }) {
+                return static_cast<bool>(Physics::providesRadialMass);
+            } else {
+                return false;
+            }
+        }
+
+        template <typename Specification, typename Model>
+        [[nodiscard]] consteval bool radialProjectionPhysicsIsComplete() {
+            using Physics = typename SelectRadialProjectionPhysics<Specification>::Type;
+            if constexpr (!radialProjectionPhysicsRegistered<Physics>()) {
+                return false;
+            } else if constexpr (!requires {
+                                     { Physics::template supports<Model> } -> std::convertible_to<bool>;
+                                 }) {
+                return false;
+            } else if constexpr (!static_cast<bool>(Physics::template supports<Model>)) {
+                return false;
+            } else if constexpr (!requires(
+                                     const Specification &specification,
+                                     const Model &model,
+                                     const RadialProfile &profile,
+                                     const StellarEquilibriumProjectionOptions &options,
+                                     const RadialProjectionScales &scales,
+                                     RadialProjectionState &state,
+                                     mfem::Vector coordinate
+                                 ) {
+                                     Physics::validate(specification, model, profile, options);
+                                     Physics::initialize(specification, model, scales, state, coordinate);
+                                 }) {
+                return false;
+            } else if constexpr (radialProjectionPhysicsProvidesMass<Physics>()) {
+                return requires(const Specification &specification) {
+                    { Physics::targetMass(specification) } -> std::same_as<dimensions::MassValue>;
+                };
+            } else {
+                return true;
+            }
+        }
+
+        template <models::ModelSpecification Specification, models::GeneratedStateKind Kind>
+        struct RadialProjectionCoordinateTerm;
+
+        template <models::ModelSpecification Specification>
+        struct RadialProjectionCoordinateTerm<Specification, models::GeneratedStateKind::multiplier> final {
+            using value = utils::blocks::generated_value_block<models::MultiplierFor<Specification>>;
+        };
+
+        template <models::ModelSpecification Specification>
+        struct RadialProjectionCoordinateTerm<Specification, models::GeneratedStateKind::physical_coordinate> final {
+            using value = utils::blocks::generated_value_block<models::PhysicalCoordinateFor<Specification>>;
+        };
+
+        template <models::ModelSpecification Specification>
+        struct RadialProjectionCoordinateTerm<Specification, models::GeneratedStateKind::solver_border> final {
+            using value = utils::blocks::generated_value_block<models::BorderFor<Specification>>;
+        };
+
+        template <typename Model, typename SpecificationSet> struct CompileRadialProjection;
+
+        template <model::StellarModelType Model, models::ModelSpecification... Specifications>
+        struct CompileRadialProjection<Model, models::detail::SpecificationSetStorage<Specifications...>> {
+            using ModelType = std::remove_cvref_t<Model>;
+
+            static constexpr std::size_t radialMassProviderCount =
+                (std::size_t{0} + ... +
+                 (radialProjectionPhysicsProvidesMass<
+                      typename SelectRadialProjectionPhysics<Specifications>::Type>()
+                      ? std::size_t{1}
+                      : std::size_t{0}));
+            static constexpr bool complete = radialMassProviderCount == 1 &&
+                                             (radialProjectionPhysicsIsComplete<Specifications, ModelType>() && ...);
+
+            [[nodiscard]] static dimensions::MassValue targetMass(const ModelType &model) requires complete {
+                dimensions::MassValue result{0.0};
+                ([&] {
+                    using Physics = typename SelectRadialProjectionPhysics<Specifications>::Type;
+                    if constexpr (radialProjectionPhysicsProvidesMass<Physics>()) {
+                        result = Physics::targetMass(model.template specification<Specifications>());
+                    }
+                }(), ...);
+                return result;
+            }
+
+            static void validate(
+                const ModelType &model,
+                const RadialProfile &profile,
+                const StellarEquilibriumProjectionOptions &options
+            ) requires complete {
+                ([&] {
+                    using Physics = typename SelectRadialProjectionPhysics<Specifications>::Type;
+                    Physics::validate(model.template specification<Specifications>(), model, profile, options);
+                }(), ...);
+            }
+
+            template <typename StateView>
+            static void initialize(
+                const ModelType &model,
+                const RadialProjectionScales &scales,
+                RadialProjectionState &state,
+                const StateView &stateView
+            ) requires complete {
+                ([&] {
+                    using Contribution = models::SpecificationContribution<Specifications>;
+                    using Physics      = typename SelectRadialProjectionPhysics<Specifications>::Type;
+                    if constexpr (Contribution::generatedValueArity == 0) {
+                        Physics::initialize(
+                            model.template specification<Specifications>(), model, scales, state, mfem::Vector{}
+                        );
+                    } else {
+                        static_assert(
+                            Contribution::generatedValueArity == 1,
+                            "Radial projection currently requires each specification contribution to generate at "
+                            "most one scalar coordinate."
+                        );
+                        using Term = RadialProjectionCoordinateTerm<Specifications, Contribution::generatedStateKind>;
+                        Physics::initialize(
+                            model.template specification<Specifications>(), model, scales, state,
+                            stateView.block(Term{})
+                        );
+                    }
+                }(), ...);
+            }
+        };
+
+        template <typename Candidate, bool = model::StellarModelType<Candidate>>
+        struct RadialProjectionCompilationAudit {
+            static constexpr bool complete = false;
+        };
+
+        template <typename Candidate>
+        struct RadialProjectionCompilationAudit<Candidate, true>
+            : CompileRadialProjection<Candidate, typename Candidate::SpecificationTypes> { };
     } // namespace detail
 
-    template <equilibrium::StellarEquilibriumModel Model>
+    template <typename Candidate>
+    inline constexpr bool radialProjectionIsCompilable =
+        detail::RadialProjectionCompilationAudit<std::remove_cvref_t<Candidate>>::complete;
+
+    template <typename Candidate>
+    concept RadialProfileProjectableModel =
+        model::StellarModelType<Candidate> && radialProjectionIsCompilable<std::remove_cvref_t<Candidate>>;
+
+    template <equilibrium::StellarEquilibriumModel Model, equilibrium::StellarDiscretizationType Discretization>
+        requires RadialProfileProjectableModel<Model>
     [[nodiscard]] ProjectedEquilibriumState<Model> projectRadialProfile(
-        const equilibrium::StellarEquilibriumProblem<Model> &problem,
+        const equilibrium::StellarEquilibriumProblem<Model, Discretization> &problem,
         const RadialProfile &profile,
         const StellarEquilibriumProjectionOptions &options = {}
     ) {
+        using Projection = detail::CompileRadialProjection<
+            std::remove_cvref_t<Model>,
+            typename std::remove_cvref_t<Model>::SpecificationTypes>;
+        const auto &stellarModel = problem.GetStellarModel();
+        Projection::validate(stellarModel, profile, options);
+        const dimensions::MassValue targetMass = Projection::targetMass(stellarModel);
         const detail::ProjectedRadialFields fields = detail::projectRadialFields(
-            problem.GetDiscretization(), profile,
+            problem.GetDiscretization().finiteElementModel(), profile, targetMass, options
-            problem.GetStellarModel().template specification<models::FixedTotalMass>().targetMass(),
-            problem.GetStellarModel().template specification<surface::Isobaric>().targetPressure(), options
         );
 
         mfem::Vector values(problem.StateSize());
@@ -89,45 +534,40 @@ export namespace mean_field::seed {
         );
 
         /*
-         * Projection of a continuous spherical profile onto a faceted
+         * Each specification now initializes only its inferred contribution.
-         * reference mesh generally leaves a small trace error on the physical
+         * In particular, the surface rule imposes the exact carrier trace and
-         * surface. The pressure condition replaces these carrier rows in the
+         * generated constraints obtain their coordinate by type, not by a
-         * compiled equilibrium problem, so impose its required carrier value
+         * hard-coded whole-model layout.
-         * exactly after bulk projection instead of treating that geometric
-         * mismatch as part of the initial residual.
          */
-        mfem::Vector enthalpy = stateView.block(utils::blocks::enthalpy_field.specific_term);
+        RadialProjectionState projectedState{
-        const dimensions::SpecificEnthalpyValue requiredSurfaceEnthalpy =
+            .density          = stateView.block(utils::blocks::density_field.mass_term),
-            eos::evaluate<dimensions::quantity::SpecificEnthalpy>(
+            .surfaceShape     = stateView.block(utils::blocks::surface_deformation_field.parameters_term),
-                problem.GetStellarModel().template specification<eos::Polytrope>(),
+            .gravityGradient  = stateView.block(utils::blocks::gravity_field.gradient_term),
-                problem.GetStellarModel().template specification<surface::Isobaric>().targetPressure()
+            .gravityPotential = stateView.block(utils::blocks::gravity_field.poisson_term),
-            );
+            .specificEnthalpy = stateView.block(utils::blocks::enthalpy_field.specific_term)
-        for (const int surfaceRow : problem.GetPressureSurfaceRows().reduced_dofs()) {
+        };
-            enthalpy(surfaceRow) = requiredSurfaceEnthalpy.value();
+        const RadialProjectionScales scales{
-        }
+            .targetMass               = targetMass,
-
+            .stellarRadius            = profile.stellarRadius,
-        mfem::Vector fixedMassCoordinate =
+            .bernoulliConstant        = fields.bernoulliConstant,
-            stateView.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term);
+            .sphericalMomentOfInertia = fields.sphericalMomentOfInertia,
-        if (fixedMassCoordinate.Size() != 1) {
+            .surfaceCarrierRows       = &problem.GetPressureSurfaceRows().reduced_dofs()
-            throw std::invalid_argument("FixedTotalMass must generate exactly one equilibrium-state coordinate.");
+        };
-        }
+        Projection::initialize(stellarModel, scales, projectedState, stateView);
-        fixedMassCoordinate(0) = fields.bernoulliConstant;
-
-        if constexpr (std::remove_cvref_t<Model>::template containsSpecification<models::FixedCentralDensity>) {
-            stateView.block(utils::blocks::fixed_central_density_phase.central_value_term) = 0.0;
-        }
 
         return {.values = std::move(values)};
     }
 
     template <
         equilibrium::StellarEquilibriumModel Model,
+        equilibrium::StellarDiscretizationType Discretization,
         typename Strategy>
         requires RadialSeedStrategyFor<
             Strategy,
-            typename equilibrium::StellarEquilibriumProblem<Model>::ModelType>
+            typename equilibrium::StellarEquilibriumProblem<Model, Discretization>::ModelType> &&
+                 RadialProfileProjectableModel<Model>
     [[nodiscard]] ProjectedEquilibriumState<Model> makeProjectedEquilibriumState(
-        const equilibrium::StellarEquilibriumProblem<Model> &problem,
+        const equilibrium::StellarEquilibriumProblem<Model, Discretization> &problem,
         const Strategy &strategy,
         const StellarEquilibriumProjectionOptions &options = {}
     ) {

libmeanfield/interface/utils/blocks.cppm

@@ -136,6 +136,19 @@ export namespace mean_field::utils::blocks {
         static inline constexpr central_value central_value_term{};
     };
 
+    struct fixed_angular_momentum final : field {
+        using SpecificationType = models::FixedAngularMomentum;
+        using CoordinateType    = models::PhysicalCoordinateFor<SpecificationType>;
+        using ResidualType      = models::ResidualFor<SpecificationType>;
+
+        struct angular_velocity final : term {
+            using value    = generated_value_block<CoordinateType>;
+            using residual = generated_residual_block<ResidualType>;
+        };
+
+        static inline constexpr angular_velocity angular_velocity_term{};
+    };
+
     // Compatibility name for the current barotropic formulation. The scalar
     // is generated by FixedTotalMass; its realization in this formulation is
     // the historical C coordinate.
@@ -148,6 +161,7 @@ export namespace mean_field::utils::blocks {
     inline constexpr enthalpy enthalpy_field{};
     inline constexpr fixed_total_mass fixed_total_mass_constraint{};
     inline constexpr fixed_central_density fixed_central_density_phase{};
+    inline constexpr fixed_angular_momentum fixed_angular_momentum_constraint{};
     inline constexpr barotropic_constant barotropic_constant_field{};
 
     template <typename... Types> struct type_list {
@@ -516,6 +530,7 @@ export namespace mean_field::utils::blocks {
             enthalpy::specific::value,
             gravity::poisson::value,
             surface_deformation::parameters::value,
+            density::mass::value,
             barotropic_constant::mass_normalization::value>,
         block_row<
             barotropic_constant::mass_normalization::residual,
@@ -569,6 +584,7 @@ export namespace mean_field::utils::blocks {
             enthalpy::specific::value,
             gravity::poisson::value,
             surface_deformation::parameters::value,
+            density::mass::value,
             barotropic_constant::mass_normalization::value,
             fixed_central_density::central_value::value>,
         block_row<

tests/field/field_registry.cpp

@@ -85,6 +85,8 @@ TEST_CASE(
 
     STATIC_REQUIRE(field::FieldTag<field::BarotropicConstant>);
 
+    STATIC_REQUIRE(field::FieldTag<field::AngularVelocity>);
+
     STATIC_REQUIRE_FALSE(field::FieldTag<field_registry_test_utils::MissingSupportField>);
 
     STATIC_REQUIRE_FALSE(field::FieldTag<field_registry_test_utils::InvalidSupportField>);
@@ -113,6 +115,8 @@ TEST_CASE(
 
     STATIC_REQUIRE(field::NonSpatialField<field::BarotropicConstant>);
 
+    STATIC_REQUIRE(field::NonSpatialField<field::AngularVelocity>);
+
     STATIC_REQUIRE(std::same_as<field::FieldDomainT<field::Density>, domain::Stellar>);
 
     STATIC_REQUIRE(std::same_as<field::FieldDomainT<field::Enthalpy>, domain::Stellar>);
@@ -123,6 +127,8 @@ TEST_CASE(
 
     STATIC_REQUIRE(std::same_as<field::FieldSupportT<field::BarotropicConstant>, field::NonSpatialSupport>);
 
+    STATIC_REQUIRE(std::same_as<field::FieldSupportT<field::AngularVelocity>, field::NonSpatialSupport>);
+
     CHECK(true);
 }
 

tests/models/model_specifications.cpp

@@ -1,3 +1,5 @@
+#include <array>
+#include <cmath>
 #include <concepts>
 #include <limits>
 #include <stdexcept>
@@ -23,10 +25,18 @@ namespace {
         mean_field::eos::Polytrope,
         mean_field::models::FixedTotalMass>;
 
+    using AngularMomentumPolytropicMassSpecifications = mean_field::models::SpecificationSet<
+        mean_field::models::FixedAngularMomentum,
+        mean_field::surface::Isobaric,
+        mean_field::eos::Polytrope,
+        mean_field::models::FixedTotalMass>;
+
     using PolytropicMassModel         = mean_field::model::StellarModel<PolytropicMassSpecifications>;
     using PermutedPolytropicMassModel = mean_field::model::StellarModel<PermutedPolytropicMassSpecifications>;
     using CentralDensityPolytropicMassModel =
         mean_field::model::StellarModel<CentralDensityPolytropicMassSpecifications>;
+    using AngularMomentumPolytropicMassModel =
+        mean_field::model::StellarModel<AngularMomentumPolytropicMassSpecifications>;
 } // namespace
 
 TEST_CASE(
@@ -36,11 +46,13 @@ TEST_CASE(
     STATIC_CHECK(mean_field::models::ModelSpecification<mean_field::eos::Polytrope>);
     STATIC_CHECK(mean_field::models::ModelSpecification<mean_field::surface::ConstantPressureSurface>);
     STATIC_CHECK(mean_field::models::ModelSpecification<mean_field::models::FixedTotalMass>);
+    STATIC_CHECK(mean_field::models::ModelSpecification<mean_field::models::FixedAngularMomentum>);
     STATIC_CHECK(mean_field::models::ModelSpecification<mean_field::models::FixedCentralDensity>);
     STATIC_CHECK_FALSE(mean_field::models::ModelSpecification<NotAModelSpecification>);
     STATIC_CHECK(mean_field::models::ResolvedModelSpecification<mean_field::eos::Polytrope>);
     STATIC_CHECK(mean_field::models::ResolvedModelSpecification<mean_field::surface::ConstantPressureSurface>);
     STATIC_CHECK(mean_field::models::ResolvedModelSpecification<mean_field::models::FixedTotalMass>);
+    STATIC_CHECK(mean_field::models::ResolvedModelSpecification<mean_field::models::FixedAngularMomentum>);
     STATIC_CHECK(mean_field::models::ResolvedModelSpecification<mean_field::models::FixedCentralDensity>);
 
     STATIC_CHECK(
@@ -61,8 +73,10 @@ TEST_CASE(
 
     STATIC_CHECK(std::same_as<PolytropicMassModel, PermutedPolytropicMassModel>);
     STATIC_CHECK_FALSE(std::same_as<PolytropicMassModel, CentralDensityPolytropicMassModel>);
+    STATIC_CHECK_FALSE(std::same_as<PolytropicMassModel, AngularMomentumPolytropicMassModel>);
     STATIC_CHECK(mean_field::model::StellarModelType<PolytropicMassModel>);
     STATIC_CHECK(mean_field::model::StellarModelType<CentralDensityPolytropicMassModel>);
+    STATIC_CHECK(mean_field::model::StellarModelType<AngularMomentumPolytropicMassModel>);
 }
 
 TEST_CASE(
@@ -81,6 +95,22 @@ TEST_CASE(
             typename MassSignature::GeneratedValues>
     );
 
+    using AngularMomentumSignature = AngularMomentumPolytropicMassModel::OperatorSignature;
+
+    STATIC_CHECK(AngularMomentumSignature::generatedValueArity == 2);
+    STATIC_CHECK(AngularMomentumSignature::generatedResidualArity == 2);
+    STATIC_CHECK(AngularMomentumSignature::symbolicallySquare);
+    STATIC_CHECK(
+        mean_field::models::modelTypeListContains<
+            mean_field::models::PhysicalCoordinateFor<mean_field::models::FixedAngularMomentum>,
+            typename AngularMomentumSignature::GeneratedValues>
+    );
+    STATIC_CHECK(
+        mean_field::models::modelTypeListContains<
+            mean_field::models::ResidualFor<mean_field::models::FixedAngularMomentum>,
+            typename AngularMomentumSignature::GeneratedResiduals>
+    );
+
     STATIC_CHECK(
         mean_field::models::modelTypeListContains<
             mean_field::models::ResidualFor<mean_field::models::FixedTotalMass>,
@@ -106,6 +136,76 @@ TEST_CASE(
     );
 }
 
+TEST_CASE(
+    "Fixed Angular Momentum Compiles A Physical Angular Velocity And Invariant Row",
+    tags::model_specification_type_contract
+) {
+    using namespace mean_field;
+    using Request = models::FixedAngularMomentumLayoutRequest;
+    using Form = operators::CompiledStellarEquilibriumForm<AngularMomentumPolytropicMassModel>;
+    using Jacobian = operators::CompiledStellarEquilibriumJacobianForm<AngularMomentumPolytropicMassModel>;
+    using AngularValue = utils::blocks::fixed_angular_momentum::angular_velocity::value;
+    using AngularResidual = utils::blocks::fixed_angular_momentum::angular_velocity::residual;
+
+    STATIC_CHECK(models::ConstraintLayoutRequestType<Request>);
+    STATIC_CHECK(models::CompiledConstraint<models::CompiledFixedAngularMomentum>);
+    STATIC_CHECK(std::same_as<
+                 typename Request::GeneratedValueType,
+                 models::PhysicalCoordinateFor<models::FixedAngularMomentum>>);
+    STATIC_CHECK(std::same_as<
+                 typename AngularValue::GeneratedType,
+                 models::PhysicalCoordinateFor<models::FixedAngularMomentum>>);
+    STATIC_CHECK(std::same_as<
+                 typename models::CompiledFixedAngularMomentum::AngularVelocityField,
+                 field::AngularVelocity>);
+    STATIC_CHECK(Form::value_block_count == 7);
+    STATIC_CHECK(Form::residual_block_count == 7);
+    STATIC_CHECK(Request::valueBlock<Form>().index == 6);
+    STATIC_CHECK(Request::residualBlock<Form>().index == 6);
+    STATIC_CHECK(utils::blocks::valid_jacobian_form<Form, Jacobian>);
+    STATIC_CHECK(utils::blocks::has_jacobian_coupling_v<
+                 AngularResidual,
+                 utils::blocks::density::mass::value,
+                 Jacobian>);
+    STATIC_CHECK(utils::blocks::has_jacobian_coupling_v<
+                 AngularResidual,
+                 utils::blocks::surface_deformation::parameters::value,
+                 Jacobian>);
+    STATIC_CHECK(utils::blocks::has_jacobian_coupling_v<AngularResidual, AngularValue, Jacobian>);
+    STATIC_CHECK(utils::blocks::has_jacobian_coupling_v<
+                 utils::blocks::surface_deformation::shape_equilibrium::residual,
+                 AngularValue,
+                 Jacobian>);
+    STATIC_CHECK(utils::blocks::has_jacobian_coupling_v<
+                 utils::blocks::enthalpy::specific::residual,
+                 AngularValue,
+                 Jacobian>);
+    STATIC_CHECK_FALSE(utils::blocks::has_jacobian_coupling_v<
+                       utils::blocks::gravity::poisson::residual,
+                       AngularValue,
+                       Jacobian>);
+
+    const integral::FixedAngularMomentum specification({
+        .Jtotal = dimensions::AngularMomentumValue{2.75},
+        .axis = {0.0, 3.0, 4.0},
+        .center = {0.25, -0.5, 0.75}
+    });
+    const models::CompiledFixedAngularMomentum compiled = models::compileConstraint(specification);
+    CHECK(compiled.targetAngularMomentum() == dimensions::AngularMomentumValue{2.75});
+    CHECK(std::abs(compiled.specification().axis()[0]) < 1.0e-15);
+    CHECK(std::abs(compiled.specification().axis()[1] - 0.6) < 1.0e-15);
+    CHECK(std::abs(compiled.specification().axis()[2] - 0.8) < 1.0e-15);
+    CHECK(compiled.specification().center() == std::array<double, 3>{0.25, -0.5, 0.75});
+
+    const physics::RigidRotation rotation = compiled.makeRotation(1.5);
+    CHECK(std::abs(rotation.angular_velocity()(0)) < 1.0e-15);
+    CHECK(std::abs(rotation.angular_velocity()(1) - 0.9) < 1.0e-15);
+    CHECK(std::abs(rotation.angular_velocity()(2) - 1.2) < 1.0e-15);
+    CHECK(rotation.center()(0) == 0.25);
+    CHECK(rotation.center()(1) == -0.5);
+    CHECK(rotation.center()(2) == 0.75);
+}
+
 TEST_CASE(
     "Fixed Total Mass Compiles Its Generated Multiplier And Canonical Residual Row",
     tags::model_specification_type_contract
@@ -146,6 +246,8 @@ TEST_CASE(
     constexpr auto surface =
         mean_field::models::specificationDescriptor<mean_field::surface::ConstantPressureSurface>();
     constexpr auto mass = mean_field::models::specificationDescriptor<mean_field::models::FixedTotalMass>();
+    constexpr auto angularMomentum =
+        mean_field::models::specificationDescriptor<mean_field::models::FixedAngularMomentum>();
     constexpr auto centralDensity =
         mean_field::models::specificationDescriptor<mean_field::models::FixedCentralDensity>();
 
@@ -162,6 +264,11 @@ TEST_CASE(
     STATIC_CHECK(mass.generatedValueArity == 1);
     STATIC_CHECK(mass.generatedResidualArity == 1);
 
+    STATIC_CHECK(angularMomentum.name == "FixedAngularMomentum");
+    STATIC_CHECK(angularMomentum.role == mean_field::models::SpecificationRole::invariant);
+    STATIC_CHECK(angularMomentum.generatedValueArity == 1);
+    STATIC_CHECK(angularMomentum.generatedResidualArity == 1);
+
     STATIC_CHECK(centralDensity.name == "FixedCentralDensity");
     STATIC_CHECK(centralDensity.role == mean_field::models::SpecificationRole::phase_condition);
     STATIC_CHECK(centralDensity.generatedValueArity == 1);
@@ -174,9 +281,15 @@ TEST_CASE(
 ) {
     const mean_field::models::FixedTotalMass mass{mean_field::dimensions::MassValue{1.25}};
     const mean_field::models::FixedCentralDensity centralDensity{mean_field::eos::DensityValue{2.5}};
+    const mean_field::models::FixedAngularMomentum angularMomentum{
+        mean_field::dimensions::AngularMomentumValue{0.75}
+    };
 
     CHECK(mass.targetMass() == mean_field::dimensions::MassValue{1.25});
     CHECK(centralDensity.targetDensity() == mean_field::eos::DensityValue{2.5});
+    CHECK(angularMomentum.targetAngularMomentum() == mean_field::dimensions::AngularMomentumValue{0.75});
+    CHECK(angularMomentum.axis() == std::array<double, 3>{0.0, 0.0, 1.0});
+    CHECK(angularMomentum.center() == std::array<double, 3>{0.0, 0.0, 0.0});
 
     CHECK_THROWS_AS(mean_field::models::FixedTotalMass{mean_field::dimensions::MassValue{0.0}}, std::invalid_argument);
     CHECK_THROWS_AS(mean_field::models::FixedTotalMass{mean_field::dimensions::MassValue{-1.0}}, std::invalid_argument);
@@ -184,6 +297,31 @@ TEST_CASE(
         mean_field::models::FixedTotalMass{mean_field::dimensions::MassValue{std::numeric_limits<double>::infinity()}},
         std::invalid_argument
     );
+    CHECK_THROWS_AS(
+        mean_field::models::FixedAngularMomentum{mean_field::dimensions::AngularMomentumValue{-1.0}},
+        std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        mean_field::models::FixedAngularMomentum{
+            mean_field::dimensions::AngularMomentumValue{std::numeric_limits<double>::infinity()}
+        },
+        std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        mean_field::models::FixedAngularMomentum({
+            .Jtotal = mean_field::dimensions::AngularMomentumValue{1.0},
+            .axis = {0.0, 0.0, 0.0}
+        }),
+        std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        mean_field::models::FixedAngularMomentum({
+            .Jtotal = mean_field::dimensions::AngularMomentumValue{1.0},
+            .axis = {0.0, 0.0, 1.0},
+            .center = {0.0, std::numeric_limits<double>::quiet_NaN(), 0.0}
+        }),
+        std::invalid_argument
+    );
 
     CHECK_THROWS_AS(mean_field::models::FixedCentralDensity{mean_field::eos::DensityValue{0.0}}, std::invalid_argument);
     CHECK_THROWS_AS(

tests/models/typed_stellar_model.cpp

@@ -1,5 +1,6 @@
 #include <concepts>
 #include <limits>
+#include <memory>
 #include <stdexcept>
 #include <type_traits>
 
@@ -19,6 +20,101 @@ namespace {
         mean_field::eos::Polytrope,
         mean_field::surface::Isobaric,
         mean_field::integral::FixedTotalMass>>;
+
+    class MoveOnlyEquationOfState final {
+    public:
+        struct Parameters final {
+            int marker;
+        };
+
+        using ModelDefinition =
+            mean_field::eos::ConstitutiveLaw<MoveOnlyEquationOfState, "MoveOnlyEquationOfState">;
+
+        explicit MoveOnlyEquationOfState(const Parameters parameters)
+            : m_marker(std::make_unique<int>(parameters.marker)) {
+        }
+
+        MoveOnlyEquationOfState(const MoveOnlyEquationOfState &) = delete;
+        MoveOnlyEquationOfState &operator=(const MoveOnlyEquationOfState &) = delete;
+        MoveOnlyEquationOfState(MoveOnlyEquationOfState &&) noexcept = default;
+        MoveOnlyEquationOfState &operator=(MoveOnlyEquationOfState &&) noexcept = default;
+
+        [[nodiscard]] const std::unique_ptr<int> &marker() const noexcept {
+            return m_marker;
+        }
+
+    private:
+        std::unique_ptr<int> m_marker;
+    };
+
+    class MoveOnlySurfaceCondition final {
+    public:
+        struct Parameters final {
+            int marker;
+        };
+
+        using ModelDefinition =
+            mean_field::surface::BoundaryCondition<MoveOnlySurfaceCondition, "MoveOnlySurfaceCondition">;
+
+        explicit MoveOnlySurfaceCondition(const Parameters parameters)
+            : m_marker(std::make_unique<int>(parameters.marker)) {
+        }
+
+        MoveOnlySurfaceCondition(const MoveOnlySurfaceCondition &) = delete;
+        MoveOnlySurfaceCondition &operator=(const MoveOnlySurfaceCondition &) = delete;
+        MoveOnlySurfaceCondition(MoveOnlySurfaceCondition &&) noexcept = default;
+        MoveOnlySurfaceCondition &operator=(MoveOnlySurfaceCondition &&) noexcept = default;
+
+        [[nodiscard]] const std::unique_ptr<int> &marker() const noexcept {
+            return m_marker;
+        }
+
+    private:
+        std::unique_ptr<int> m_marker;
+    };
+
+    class MoveOnlyIntegralConstraint final {
+    public:
+        struct Parameters final {
+            int marker;
+        };
+
+        using ScalarDescription = mean_field::stellar::ScalarConstraint<
+            mean_field::dimensions::quantity::Mass,
+            mean_field::dimensions::quantity::SpecificEnergy,
+            mean_field::dimensions::quantity::Mass,
+            "move_only_integral.multiplier",
+            "C_move",
+            "move_only_integral.residual",
+            "R_move">;
+        using TargetValue = typename ScalarDescription::TargetValue;
+        using ModelDefinition = mean_field::integral::FixedScalarWithMultiplier<
+            MoveOnlyIntegralConstraint,
+            "MoveOnlyIntegralConstraint",
+            mean_field::stellar::Reads<mean_field::stellar::state::Density>,
+            mean_field::stellar::Changes<mean_field::stellar::equation::HydrostaticBalance>,
+            ScalarDescription>;
+
+        explicit MoveOnlyIntegralConstraint(const Parameters parameters)
+            : m_marker(std::make_unique<int>(parameters.marker)) {
+        }
+
+        MoveOnlyIntegralConstraint(const MoveOnlyIntegralConstraint &) = delete;
+        MoveOnlyIntegralConstraint &operator=(const MoveOnlyIntegralConstraint &) = delete;
+        MoveOnlyIntegralConstraint(MoveOnlyIntegralConstraint &&) noexcept = default;
+        MoveOnlyIntegralConstraint &operator=(MoveOnlyIntegralConstraint &&) noexcept = default;
+
+        [[nodiscard]] const std::unique_ptr<int> &marker() const noexcept {
+            return m_marker;
+        }
+
+        [[nodiscard]] TargetValue target() const noexcept {
+            return TargetValue{static_cast<double>(*m_marker)};
+        }
+
+    private:
+        std::unique_ptr<int> m_marker;
+    };
 } // namespace
 
 TEST_CASE(
@@ -38,8 +134,8 @@ TEST_CASE(
     STATIC_CHECK(models::SpecifiedModelType<decltype(stellarModel)>);
     STATIC_CHECK(CanonicalModel::specificationCount == 4);
     STATIC_CHECK(CanonicalModel::symbolicallySquare);
-    STATIC_CHECK(CanonicalModel::hasCompleteEquilibriumCompiler);
+    STATIC_CHECK(CanonicalModel::hasCompleteEquilibriumDeclaration);
-    STATIC_CHECK(CanonicalModel::compilationClass == models::EquilibriumSystemCompilation::complete_equilibrium_system);
+    STATIC_CHECK(operators::StellarEquilibriumSystemCompilable<CanonicalModel>);
 
     CHECK(stellarModel.specification<eos::Polytrope>().polytropic_index() == 3.0);
     CHECK(stellarModel.specification<eos::Polytrope>().polytropic_constant() == 0.25);
@@ -50,6 +146,44 @@ TEST_CASE(
     );
 }
 
+TEST_CASE(
+    "Incomplete Stellar Model Spellings Fail Capability Probes Without Diagnostics",
+    tags::stellar_model_specification_api
+) {
+    using namespace mean_field;
+    using IncompleteModel = model::StellarModel<int>;
+
+    STATIC_CHECK_FALSE(model::StellarModelType<IncompleteModel>);
+    STATIC_CHECK_FALSE(models::SpecifiedModelType<IncompleteModel>);
+    STATIC_CHECK(model::specificationRoleCount<models::SpecificationRole::constitutive_law, IncompleteModel> == 0);
+    STATIC_CHECK_FALSE(model::HasEquationOfState<IncompleteModel>);
+    STATIC_CHECK_FALSE(model::HasSurfaceCondition<IncompleteModel>);
+    STATIC_CHECK_FALSE(operators::StellarEquilibriumSystemCompilable<IncompleteModel>);
+    STATIC_CHECK_FALSE(equilibrium::StellarEquilibriumModel<IncompleteModel>);
+}
+
+TEST_CASE(
+    "Specification Sets Canonicalize Cvref-Qualified Physics Types",
+    tags::stellar_model_specification_api
+) {
+    using namespace mean_field;
+    using Qualified = models::SpecificationSet<
+        const eos::Polytrope &,
+        volatile surface::Isobaric &&,
+        const integral::FixedTotalMass>;
+    using Plain = models::SpecificationSet<
+        eos::Polytrope,
+        surface::Isobaric,
+        integral::FixedTotalMass>;
+
+    STATIC_CHECK(std::same_as<Qualified, Plain>);
+    STATIC_CHECK(models::ValidModelSpecificationPack<
+                 const eos::Polytrope &,
+                 volatile surface::Isobaric &&,
+                 const integral::FixedTotalMass>);
+    STATIC_CHECK(model::StellarModelType<model::StellarModel<Qualified>>);
+}
+
 TEST_CASE(
     "Stellar Model Deduction Canonicalizes Unordered Specifications",
     tags::stellar_model_specification_api
@@ -85,6 +219,40 @@ TEST_CASE(
     CHECK(descriptors[3].specification.name == "FixedCentralDensity");
 }
 
+TEST_CASE(
+    "Canonical Model Construction Preserves Every Move-Only Physics Specification",
+    tags::stellar_model_specification_api
+) {
+    using namespace mean_field;
+
+    auto stellarModel = model::StellarModel(
+        MoveOnlyIntegralConstraint({.marker = 307}),
+        MoveOnlySurfaceCondition({.marker = 211}),
+        MoveOnlyEquationOfState({.marker = 101})
+    );
+    using Model = std::remove_cvref_t<decltype(stellarModel)>;
+    using Expected = model::StellarModel<models::SpecificationSet<
+        MoveOnlyEquationOfState,
+        MoveOnlySurfaceCondition,
+        MoveOnlyIntegralConstraint>>;
+
+    STATIC_CHECK(std::same_as<Model, Expected>);
+    STATIC_CHECK_FALSE(std::copy_constructible<MoveOnlyEquationOfState>);
+    STATIC_CHECK_FALSE(std::copy_constructible<MoveOnlySurfaceCondition>);
+    STATIC_CHECK_FALSE(std::copy_constructible<MoveOnlyIntegralConstraint>);
+
+    const auto &equationOfState = stellarModel.specification<MoveOnlyEquationOfState>();
+    const auto &surfaceCondition = stellarModel.specification<MoveOnlySurfaceCondition>();
+    const auto &integralConstraint = stellarModel.specification<MoveOnlyIntegralConstraint>();
+
+    REQUIRE(equationOfState.marker() != nullptr);
+    REQUIRE(surfaceCondition.marker() != nullptr);
+    REQUIRE(integralConstraint.marker() != nullptr);
+    CHECK(*equationOfState.marker() == 101);
+    CHECK(*surfaceCondition.marker() == 211);
+    CHECK(*integralConstraint.marker() == 307);
+}
+
 TEST_CASE(
     "Stellar Specification Parameter Constructors Preserve Validation",
     tags::stellar_model_specification_api

tests/mpi/distributed_execution.cpp

@@ -1,8 +1,12 @@
 #include <algorithm>
 #include <catch2/catch_test_macros.hpp>
 #include <cmath>
+#include <cstdint>
 #include <limits>
 #include <mfem.hpp>
+#include <numbers>
+#include <stdexcept>
+#include <type_traits>
 #include <vector>
 
 import mean_field;
@@ -30,12 +34,41 @@ namespace {
         return vector;
     }
 
+    void require_all_ranks(
+        const bool localCondition,
+        const MPI_Comm communicator,
+        const char *description
+    ) {
+        int rank = 0;
+        int size = 0;
+        MPI_Comm_rank(communicator, &rank);
+        MPI_Comm_size(communicator, &size);
+
+        const int localFailure = localCondition ? size : rank;
+        int firstFailure = size;
+        const int result = MPI_Allreduce(
+            &localFailure,
+            &firstFailure,
+            1,
+            MPI_INT,
+            MPI_MIN,
+            communicator
+        );
+        REQUIRE(result == MPI_SUCCESS);
+        CAPTURE(description, localCondition, firstFailure);
+        REQUIRE(firstFailure == size);
+    }
+
     double global_dot(
         const mfem::Vector &left,
         const mfem::Vector &right,
         const MPI_Comm communicator
     ) {
-        REQUIRE(left.Size() == right.Size());
+        require_all_ranks(
+            left.Size() == right.Size(),
+            communicator,
+            "global dot-product vector sizes"
+        );
         const double local = left * right;
         double global      = 0.0;
         REQUIRE(MPI_Allreduce(&local, &global, 1, MPI_DOUBLE, MPI_SUM, communicator) == MPI_SUCCESS);
@@ -48,6 +81,39 @@ namespace {
     ) {
         return std::sqrt(global_dot(vector, vector, communicator));
     }
+
+    [[nodiscard]] mean_field::operators::StellarEquilibriumDependencies
+    make_stellar_dependencies(const std::uint64_t revision = 1) {
+        return {
+            .discretization     = {.identity = 16101, .revision = 1},
+            .density            = {.identity = 16103, .revision = revision},
+            .surfaceDeformation = {.identity = 16111, .revision = revision},
+            .gravityGradient    = {.identity = 16127, .revision = revision},
+            .gravityPotential   = {.identity = 16139, .revision = revision},
+            .enthalpy           = {.identity = 16141, .revision = revision},
+            .bernoulliConstant  = {.identity = 16183, .revision = revision},
+            .rotation           = {.identity = 16187, .revision = revision},
+            .targetMass         = {.identity = 16189, .revision = 1}
+        };
+    }
+
+    void check_rank_consistent_scalar(
+        const double value,
+        const MPI_Comm communicator,
+        const double relativeTolerance = 2.0e-13
+    ) {
+        double minimum = 0.0;
+        double maximum = 0.0;
+        REQUIRE(MPI_Allreduce(&value, &minimum, 1, MPI_DOUBLE, MPI_MIN, communicator) == MPI_SUCCESS);
+        REQUIRE(MPI_Allreduce(&value, &maximum, 1, MPI_DOUBLE, MPI_MAX, communicator) == MPI_SUCCESS);
+        CAPTURE(value, minimum, maximum);
+        CHECK(std::isfinite(minimum));
+        CHECK(std::isfinite(maximum));
+        CHECK(
+            std::abs(maximum - minimum) <=
+            relativeTolerance * std::max({1.0, std::abs(minimum), std::abs(maximum)})
+        );
+    }
 } // namespace
 
 TEST_CASE(
@@ -97,6 +163,750 @@ TEST_CASE(
     CHECK(f.logicalReferenceMesh->GetNE() == f.mesh->GetNE());
 }
 
+TEST_CASE(
+    "MPI Fixed Angular Momentum Produces One Consistent Global Invariant Row",
+    "[mpi][distributed][fixed-angular-momentum][physics][jacobian]"
+) {
+    using namespace mean_field;
+
+    const auto args = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(args.mesh_file, args, 0);
+    require_all_ranks(
+        finiteElements.okay(),
+        MPI_COMM_WORLD,
+        "fixed-angular-momentum FEM setup"
+    );
+    const MPI_Comm communicator = finiteElements.mesh->GetComm();
+
+    mfem::ParGridFunction densityField(finiteElements.densityFes.get());
+    mfem::ConstantCoefficient densityCoefficient(1.23);
+    densityField.ProjectCoefficient(densityCoefficient);
+    mfem::Vector densityTrue;
+    densityField.GetTrueDofs(densityTrue);
+
+    mfem::Vector displacementTrue(finiteElements.displacementFes->GetTrueVSize());
+    displacementTrue = 0.0;
+    finiteElements.displacement->SetFromTrueDofs(displacementTrue);
+    mfem::Vector gravityGradientTrue(finiteElements.gravityFluxFes->GetTrueVSize());
+    mfem::Vector gravityPotentialTrue(finiteElements.gravityPotentialFes->GetTrueVSize());
+    gravityGradientTrue = 0.0;
+    gravityPotentialTrue = 0.0;
+
+    operators::context::gravity_field::GravityFieldLinearizationContext gravityContext(
+        finiteElements,
+        *finiteElements.domainMapperStateless
+    );
+    gravityContext.Prepare(
+        {.density = gravityContext.GetDensityMap().gather(densityTrue),
+         .displacement = gravityContext.GetDisplacementMap().gather(displacementTrue),
+         .gravity_gradient = gravityContext.GetGravityGradientMap().gather(gravityGradientTrue),
+         .gravity_potential = gravityContext.GetGravityPotentialMap().gather(gravityPotentialTrue)},
+        {.discretization = {.value = 2},
+         .displacement = {.value = 3},
+         .density = {.value = 5},
+         .gravity_gradient = {.value = 7},
+         .gravity_potential = {.value = 11}}
+    );
+
+    constexpr double targetAngularMomentum = 0.37;
+    constexpr double angularVelocity = 0.61;
+    operators::PreparedAngularMomentumOperator invariant(
+        finiteElements,
+        *finiteElements.domainMapperStateless,
+        gravityContext,
+        models::compileConstraint(
+            integral::FixedAngularMomentum({.Jtotal = dimensions::AngularMomentumValue{targetAngularMomentum}})
+        )
+    );
+    const operators::AngularMomentumDependencies dependencies{
+        .discretization = {.identity = 101, .revision = 2},
+        .density = {.identity = 103, .revision = 5},
+        .displacement = {.identity = 107, .revision = 3},
+        .rotation = {.identity = 109, .revision = 13}
+    };
+    const auto preparation = invariant.Prepare(angularVelocity, dependencies);
+    CHECK(preparation.rebuiltStaticPlan);
+    CHECK(preparation.refreshedGeometry);
+    CHECK(preparation.refreshedDensity);
+    CHECK(preparation.updatedAngularVelocity);
+    CHECK(preparation.assembledResidual);
+
+    const double independentMoment = analysis::get_moment_of_inertia(finiteElements, densityField);
+    const double preparedMoment = invariant.GetMomentOfInertia();
+    const double comparisonScale = std::max({std::abs(independentMoment), std::abs(preparedMoment), 1.0e-300});
+    CHECK(std::abs(preparedMoment - independentMoment) / comparisonScale <= 3.0e-13);
+
+    double minimumMoment = 0.0;
+    double maximumMoment = 0.0;
+    MPI_Allreduce(&preparedMoment, &minimumMoment, 1, MPI_DOUBLE, MPI_MIN, finiteElements.mesh->GetComm());
+    MPI_Allreduce(&preparedMoment, &maximumMoment, 1, MPI_DOUBLE, MPI_MAX, finiteElements.mesh->GetComm());
+    CHECK(std::abs(maximumMoment - minimumMoment) / comparisonScale <= 2.0e-15);
+
+    mfem::Vector residual;
+    invariant.BuildResidual(residual);
+    require_all_ranks(
+        residual.Size() == 1,
+        communicator,
+        "fixed-angular-momentum residual size"
+    );
+    CHECK(
+        std::abs(residual(0) - (angularVelocity * independentMoment - targetAngularMomentum)) /
+            std::max({std::abs(residual(0)), std::abs(angularVelocity * independentMoment), 1.0}) <=
+        3.0e-13
+    );
+
+    mfem::Vector densityAction;
+    invariant.ApplyDensityJacobianAction(gravityContext.GetDensityMap().gather(densityTrue), densityAction);
+    require_all_ranks(
+        densityAction.Size() == 1,
+        communicator,
+        "fixed-angular-momentum density action size"
+    );
+    CHECK(std::abs(densityAction(0) - angularVelocity * preparedMoment) / comparisonScale <= 3.0e-13);
+
+    constexpr double angularVelocityVariation = -0.29;
+    mfem::Vector angularVelocityAction;
+    invariant.ApplyAngularVelocityJacobianAction(angularVelocityVariation, angularVelocityAction);
+    require_all_ranks(
+        angularVelocityAction.Size() == 1,
+        communicator,
+        "fixed-angular-momentum rotation action size"
+    );
+    CHECK(
+        std::abs(angularVelocityAction(0) - angularVelocityVariation * preparedMoment) / comparisonScale <=
+        2.0e-15
+    );
+}
+
+TEST_CASE(
+    "MPI Density Volume Context Forwards The Prepared Global Mass Integral",
+    "[mpi][distributed][integral-context][physics-extension]"
+) {
+    using namespace mean_field;
+
+    const utils::Args arguments = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(arguments.mesh_file, arguments, 0);
+    require_all_ranks(
+        finiteElements.okay(),
+        MPI_COMM_WORLD,
+        "density-volume context FEM setup"
+    );
+    const MPI_Comm communicator = finiteElements.mesh->GetComm();
+
+    constexpr double densityValue = 1.23;
+    mfem::ParGridFunction densityField(finiteElements.densityFes.get());
+    mfem::ConstantCoefficient densityCoefficient(densityValue);
+    densityField.ProjectCoefficient(densityCoefficient);
+    mfem::Vector densityTrue;
+    densityField.GetTrueDofs(densityTrue);
+
+    mfem::Vector displacementTrue(finiteElements.displacementFes->GetTrueVSize());
+    displacementTrue = 0.0;
+    finiteElements.displacement->SetFromTrueDofs(displacementTrue);
+    mfem::Vector gravityGradientTrue(finiteElements.gravityFluxFes->GetTrueVSize());
+    mfem::Vector gravityPotentialTrue(finiteElements.gravityPotentialFes->GetTrueVSize());
+    gravityGradientTrue = 0.0;
+    gravityPotentialTrue = 0.0;
+
+    operators::context::gravity_field::GravityFieldLinearizationContext gravityContext(
+        finiteElements,
+        *finiteElements.domainMapperStateless
+    );
+    const mfem::Vector reducedDensity = gravityContext.GetDensityMap().gather(
+        densityTrue
+    );
+    gravityContext.Prepare(
+        {.density = reducedDensity,
+         .displacement = gravityContext.GetDisplacementMap().gather(displacementTrue),
+         .gravity_gradient = gravityContext.GetGravityGradientMap().gather(gravityGradientTrue),
+         .gravity_potential = gravityContext.GetGravityPotentialMap().gather(gravityPotentialTrue)},
+        {.discretization = {.value = 2},
+         .displacement = {.value = 3},
+         .density = {.value = 5},
+         .gravity_gradient = {.value = 7},
+         .gravity_potential = {.value = 11}}
+    );
+
+    operators::PreparedMassNormalizationOperator massIntegral(
+        finiteElements,
+        *finiteElements.domainMapperStateless,
+        gravityContext
+    );
+    massIntegral.Prepare(
+        models::compileConstraint(
+            integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}})
+        ),
+        {.discretization = {.identity = 101, .revision = 2},
+         .density = {.identity = 103, .revision = 5},
+         .displacement = {.identity = 107, .revision = 3},
+         .targetMass = {.identity = 109, .revision = 1}}
+    );
+
+    class DistributedMassIntegralCore final {
+    public:
+        explicit DistributedMassIntegralCore(
+            const operators::PreparedMassNormalizationOperator &mass
+        ) noexcept
+            : m_mass(&mass) {
+        }
+
+        [[nodiscard]] double ApplyDensityVolumeIntegralDensityAction(
+            const mfem::Vector &direction
+        ) const {
+            mfem::Vector action;
+            m_mass->ApplyDensityJacobianAction(direction, action);
+            return action(0);
+        }
+
+        [[nodiscard]] double ApplyDensityVolumeIntegralSurfaceShapeAction(
+            const mfem::Vector &
+        ) const noexcept {
+            return 0.0;
+        }
+
+    private:
+        const operators::PreparedMassNormalizationOperator *m_mass;
+    };
+
+    const DistributedMassIntegralCore testCore{massIntegral};
+
+    const stellar::DensityVolumeIntegralContext<integral::FixedTotalMass>
+        densityIntegral{testCore};
+    const double integratedMass =
+        densityIntegral.integrateDensity(reducedDensity).value();
+    const double linearizedDensityMass =
+        densityIntegral.linearizeDensityIntegral(reducedDensity).value();
+    const double preparedMass = massIntegral.GetCurrentMass();
+    const double independentMass =
+        densityValue * analysis::get_mesh_volume(finiteElements);
+
+    check_rank_consistent_scalar(integratedMass, communicator);
+    check_rank_consistent_scalar(linearizedDensityMass, communicator);
+    check_rank_consistent_scalar(preparedMass, communicator);
+    check_rank_consistent_scalar(independentMass, communicator);
+    const double massScale = std::max(
+        {1.0, std::abs(integratedMass), std::abs(independentMass)}
+    );
+    CHECK(std::abs(integratedMass - independentMass) <= 3.0e-12 * massScale);
+    CHECK(std::abs(integratedMass - preparedMass) <= 3.0e-13 * massScale);
+    CHECK(std::abs(linearizedDensityMass - integratedMass) <=
+          3.0e-13 * massScale);
+}
+
+TEST_CASE(
+    "MPI Assembled Variadic Stellar Root Normalizes And Applies Its Inferred Preconditioner",
+    "[mpi][distributed][stellar-equilibrium][normalization][preconditioning][integration]"
+) {
+    using namespace mean_field;
+
+    const utils::Args arguments = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(arguments.mesh_file, arguments, 0);
+    require_all_ranks(
+        finiteElements.okay(),
+        MPI_COMM_WORLD,
+        "variadic stellar-root FEM setup"
+    );
+    const MPI_Comm communicator = finiteElements.mesh->GetComm();
+
+    constexpr double radius = utils::RADIUS;
+    constexpr double mass = utils::MASS;
+    const double polytropicConstant =
+        2.0 * utils::G * radius * radius / std::numbers::pi_v<double>;
+    const double centralDensity =
+        std::numbers::pi_v<double> * mass / (4.0 * radius * radius * radius);
+
+    auto model = model::StellarModel(
+        eos::Polytrope({.n = 1.0, .K = polytropicConstant}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{mass}}),
+        integral::FixedAngularMomentum({
+            .Jtotal = dimensions::AngularMomentumValue{0.05},
+            .axis = {0.0, 0.0, 1.0}
+        }),
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{centralDensity}})
+    );
+    auto problem = equilibrium::discretize(
+        model,
+        equilibrium::makeStellarDiscretization(
+            finiteElements,
+            normalization::PhysicalRieszDiagonal{
+                dimensions::LengthValue{radius},
+                utils::G
+            }
+        )
+    );
+    auto projected = seed::makeProjectedEquilibriumState(
+        problem,
+        seed::LaneEmden({
+            .centralDensity = dimensions::DensityValue{centralDensity},
+            .radialSampleCount = 512
+        })
+    );
+    auto normalized = normalization::makeNormalizedStellarEquilibriumOperator(problem);
+    using Problem = std::remove_cvref_t<decltype(problem)>;
+    using Form = typename Problem::FormType;
+    constexpr auto massResidualBlock = utils::blocks::get_residual_block<Form>(
+        utils::blocks::fixed_total_mass_constraint.mass_normalization_term
+    );
+    constexpr auto angularMomentumResidualBlock = utils::blocks::get_residual_block<Form>(
+        utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term
+    );
+    constexpr auto centralDensityResidualBlock = utils::blocks::get_residual_block<Form>(
+        utils::blocks::fixed_central_density_phase.central_value_term
+    );
+    require_all_ranks(
+        problem.StateSize() == problem.EquationSize(),
+        communicator,
+        "variadic stellar-root square layout"
+    );
+
+    mfem::Vector normalizedState;
+    normalized.NormalizeState(projected.values, normalizedState);
+    const auto preparation = normalized.Prepare(
+        normalizedState,
+        make_stellar_dependencies(1)
+    );
+    require_all_ranks(
+        preparation.DidAnyWork() &&
+            preparation.generatedPhysicalControl &&
+            preparation.template specification<models::FixedAngularMomentum>().generatedRotation &&
+            problem.IsPrepared() &&
+            normalized.IsPrepared(),
+        communicator,
+        "initial variadic stellar-root preparation"
+    );
+
+    // The astronomy-facing integral handle must delegate to the same mapped
+    // physical-volume quadrature and collective reduction as the prepared
+    // mass equation, without exposing the FEM/core objects to extension
+    // physics.  Linearity in density gives an independent check of both
+    // public operations on every rank.
+    const auto physicalState = problem.GetManifest().stateView(projected.values);
+    const auto physicalDensity = physicalState.block(
+        utils::blocks::density_field.mass_term
+    );
+    const stellar::DensityVolumeIntegralContext<integral::FixedTotalMass>
+        densityIntegral{problem.GetPhysicalOperator()};
+    const double integratedMass = densityIntegral.integrateDensity(
+        physicalDensity
+    ).value();
+    const double linearizedMass = densityIntegral.linearizeDensityIntegral(
+        physicalDensity
+    ).value();
+    const double preparedMass = problem.GetPhysicalOperator()
+                                    .GetFixedMassReport()
+                                    .achieved;
+    check_rank_consistent_scalar(integratedMass, communicator);
+    check_rank_consistent_scalar(linearizedMass, communicator);
+    const double massComparisonScale = std::max(
+        {std::abs(integratedMass), std::abs(preparedMass), 1.0e-300}
+    );
+    CHECK(std::abs(integratedMass - preparedMass) / massComparisonScale <=
+          3.0e-13);
+    CHECK(std::abs(linearizedMass - preparedMass) / massComparisonScale <=
+          3.0e-13);
+
+    mfem::Vector normalizedResidual;
+    normalized.BuildResidual(normalizedResidual);
+    require_all_ranks(
+        normalizedResidual.Size() == problem.EquationSize(),
+        communicator,
+        "normalized variadic residual size"
+    );
+
+    auto residualView = problem.GetManifest().residualView(normalizedResidual);
+    const auto &layout = problem.GetManifest().layout();
+    const auto &residualFactors = normalized.GetNormalization().ResidualFactors();
+    const auto massReport = problem.GetPreparedOperator().GetFixedMassReport();
+    const auto angularMomentumReport =
+        problem.GetPreparedOperator().GetAngularMomentumReport();
+    const auto centralDensityReport =
+        problem.GetPreparedOperator().GetCentralDensityReport();
+    const auto checkReportedScalarResidual = [&](const mfem::Vector &block,
+                                                  const double expected) {
+        require_all_ranks(
+            block.Size() == 1,
+            communicator,
+            "reported scalar residual block size"
+        );
+        const double actual = block(0);
+        CAPTURE(actual, expected);
+        CHECK(std::isfinite(expected));
+        CHECK(
+            std::abs(actual - expected) <=
+            5.0e-13 * std::max({1.0, std::abs(actual), std::abs(expected)})
+        );
+        check_rank_consistent_scalar(actual, finiteElements.mesh->GetComm());
+    };
+    checkReportedScalarResidual(
+        residualView.block(
+            utils::blocks::fixed_total_mass_constraint.mass_normalization_term
+        ),
+        massReport.dimensionalResidual *
+            residualFactors(layout.offset(massResidualBlock))
+    );
+    checkReportedScalarResidual(
+        residualView.block(
+            utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term
+        ),
+        angularMomentumReport.dimensionalResidual *
+            residualFactors(layout.offset(angularMomentumResidualBlock))
+    );
+    checkReportedScalarResidual(
+        residualView.block(
+            utils::blocks::fixed_central_density_phase.central_value_term
+        ),
+        centralDensityReport.enthalpyResidual *
+            residualFactors(layout.offset(centralDensityResidualBlock))
+    );
+
+    mfem::Vector normalizedDirection = make_deterministic_vector(problem.StateSize(), 0.37);
+    const auto directionState = problem.GetManifest().stateView(normalizedDirection);
+    mfem::Vector massDirection = directionState.block(
+        utils::blocks::fixed_total_mass_constraint.mass_normalization_term
+    );
+    mfem::Vector angularVelocityDirection = directionState.block(
+        utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term
+    );
+    mfem::Vector phaseDirection = directionState.block(
+        utils::blocks::fixed_central_density_phase.central_value_term
+    );
+    require_all_ranks(
+        massDirection.Size() == 1 &&
+            angularVelocityDirection.Size() == 1 &&
+            phaseDirection.Size() == 1,
+        communicator,
+        "generated scalar direction block sizes"
+    );
+    massDirection(0) = 0.17;
+    angularVelocityDirection(0) = -0.23;
+    phaseDirection(0) = 0.31;
+    massDirection.SyncAliasMemory(normalizedDirection);
+    angularVelocityDirection.SyncAliasMemory(normalizedDirection);
+    phaseDirection.SyncAliasMemory(normalizedDirection);
+    normalizedDirection /= global_norm(normalizedDirection, finiteElements.mesh->GetComm());
+
+    mfem::Vector normalizedAction;
+    normalized.Mult(normalizedDirection, normalizedAction);
+    require_all_ranks(
+        normalizedAction.Size() == problem.EquationSize(),
+        communicator,
+        "normalized variadic Jacobian-action size"
+    );
+
+    /* Different dependency revisions are intentional: the state changes in
+     * each difference evaluation, so the distributed physical contexts must
+     * be rebuilt even though all persistent identities remain the same. */
+    constexpr double differenceStep = 1.0e-5;
+    mfem::Vector plusState(normalizedState);
+    plusState.Add(differenceStep, normalizedDirection);
+    const auto plusPreparation = normalized.Prepare(
+        plusState,
+        make_stellar_dependencies(2)
+    );
+    require_all_ranks(
+        plusPreparation.DidAnyWork(),
+        communicator,
+        "positive finite-difference preparation"
+    );
+    mfem::Vector plusResidual;
+    normalized.BuildResidual(plusResidual);
+
+    mfem::Vector minusState(normalizedState);
+    minusState.Add(-differenceStep, normalizedDirection);
+    const auto minusPreparation = normalized.Prepare(
+        minusState,
+        make_stellar_dependencies(3)
+    );
+    require_all_ranks(
+        minusPreparation.DidAnyWork(),
+        communicator,
+        "negative finite-difference preparation"
+    );
+    mfem::Vector minusResidual;
+    normalized.BuildResidual(minusResidual);
+
+    mfem::Vector finiteDifference(plusResidual);
+    finiteDifference -= minusResidual;
+    finiteDifference /= 2.0 * differenceStep;
+    const auto restoredPreparation = normalized.Prepare(
+        normalizedState,
+        make_stellar_dependencies(4)
+    );
+    require_all_ranks(
+        restoredPreparation.DidAnyWork() && normalized.IsPrepared(),
+        communicator,
+        "restored finite-difference preparation"
+    );
+    mfem::Vector restoredResidual;
+    normalized.BuildResidual(restoredResidual);
+    mfem::Vector restoredResidualDifference(restoredResidual);
+    restoredResidualDifference -= normalizedResidual;
+    const double restoredResidualError = global_norm(
+        restoredResidualDifference,
+        finiteElements.mesh->GetComm()
+    ) / std::max({
+        global_norm(restoredResidual, finiteElements.mesh->GetComm()),
+        global_norm(normalizedResidual, finiteElements.mesh->GetComm()),
+        std::numeric_limits<double>::epsilon()
+    });
+    CAPTURE(restoredResidualError);
+    CHECK(restoredResidualError <= 2.0e-12);
+
+    mfem::Vector finiteDifferenceError(normalizedAction);
+    finiteDifferenceError -= finiteDifference;
+    const double actionNorm = global_norm(
+        normalizedAction,
+        finiteElements.mesh->GetComm()
+    );
+    const double finiteDifferenceNorm = global_norm(
+        finiteDifference,
+        finiteElements.mesh->GetComm()
+    );
+    const double completeDifferenceError = global_norm(
+        finiteDifferenceError,
+        finiteElements.mesh->GetComm()
+    ) / std::max({
+        actionNorm,
+        finiteDifferenceNorm,
+        std::numeric_limits<double>::epsilon()
+    });
+    CAPTURE(actionNorm, finiteDifferenceNorm, completeDifferenceError);
+    CHECK(actionNorm > std::numeric_limits<double>::min());
+    CHECK(finiteDifferenceNorm > std::numeric_limits<double>::min());
+    CHECK(completeDifferenceError <= 8.0e-5);
+
+    const int locallyFinite =
+        vector_is_finite(normalizedResidual) &&
+        vector_is_finite(normalizedAction) &&
+        vector_is_finite(finiteDifference) ? 1 : 0;
+    int globallyFinite = 0;
+    REQUIRE(MPI_Allreduce(
+        &locallyFinite,
+        &globallyFinite,
+        1,
+        MPI_INT,
+        MPI_MIN,
+        finiteElements.mesh->GetComm()
+    ) == MPI_SUCCESS);
+    CHECK(globallyFinite == 1);
+    CHECK(global_norm(normalizedResidual, finiteElements.mesh->GetComm()) > 0.0);
+    CHECK(global_norm(normalizedAction, finiteElements.mesh->GetComm()) > 0.0);
+
+    auto actionView = problem.GetManifest().residualView(normalizedAction);
+    auto finiteDifferenceView = problem.GetManifest().residualView(finiteDifference);
+    const auto checkGlobalRow = [&](const auto &term, const char *rowName) {
+        const mfem::Vector residualBlock = residualView.block(term);
+        const mfem::Vector actionBlock = actionView.block(term);
+        const mfem::Vector differenceBlock = finiteDifferenceView.block(term);
+        require_all_ranks(
+            residualBlock.Size() == 1 &&
+                actionBlock.Size() == 1 &&
+                differenceBlock.Size() == 1,
+            communicator,
+            "global scalar residual/Jacobian block sizes"
+        );
+        check_rank_consistent_scalar(residualBlock(0), finiteElements.mesh->GetComm());
+        check_rank_consistent_scalar(actionBlock(0), finiteElements.mesh->GetComm());
+        check_rank_consistent_scalar(differenceBlock(0), finiteElements.mesh->GetComm());
+        const double rowMagnitude = std::max(
+            std::abs(actionBlock(0)),
+            std::abs(differenceBlock(0))
+        );
+        const double rowDifferenceError =
+            std::abs(actionBlock(0) - differenceBlock(0)) /
+            std::max(rowMagnitude, std::numeric_limits<double>::epsilon());
+        CAPTURE(rowName, actionBlock(0), differenceBlock(0), rowMagnitude,
+                rowDifferenceError);
+        CHECK(rowMagnitude > 1.0e-10);
+        CHECK(rowDifferenceError <= 8.0e-5);
+    };
+    checkGlobalRow(
+        utils::blocks::fixed_total_mass_constraint.mass_normalization_term,
+        "fixed-total-mass"
+    );
+    checkGlobalRow(
+        utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term,
+        "fixed-angular-momentum"
+    );
+    checkGlobalRow(
+        utils::blocks::fixed_central_density_phase.central_value_term,
+        "fixed-central-density phase"
+    );
+
+    auto component = preconditioning::makePreconditioner(problem);
+    STATIC_CHECK(decltype(component)::borderValueArity == 3);
+    STATIC_CHECK(decltype(component)::borderResidualArity == 3);
+    auto physicalInverse = preconditioning::prepare(problem, component);
+    require_all_ranks(
+        physicalInverse.IsCurrent(),
+        communicator,
+        "prepared physical preconditioner currentness"
+    );
+    auto scaledInverse = normalized.MakeScaledPreconditioner(physicalInverse);
+    require_all_ranks(
+        scaledInverse.IsCurrent(),
+        communicator,
+        "prepared normalized preconditioner currentness"
+    );
+
+    mfem::Vector normalizedRightHandSide =
+        make_deterministic_vector(problem.EquationSize(), 0.73);
+    auto rightHandSideView = problem.GetManifest().residualView(normalizedRightHandSide);
+    mfem::Vector massRightHandSide = rightHandSideView.block(
+        utils::blocks::fixed_total_mass_constraint.mass_normalization_term
+    );
+    mfem::Vector angularMomentumRightHandSide = rightHandSideView.block(
+        utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term
+    );
+    mfem::Vector phaseRightHandSide = rightHandSideView.block(
+        utils::blocks::fixed_central_density_phase.central_value_term
+    );
+    massRightHandSide(0) = 0.11;
+    angularMomentumRightHandSide(0) = -0.19;
+    phaseRightHandSide(0) = 0.29;
+    massRightHandSide.SyncAliasMemory(normalizedRightHandSide);
+    angularMomentumRightHandSide.SyncAliasMemory(normalizedRightHandSide);
+    phaseRightHandSide.SyncAliasMemory(normalizedRightHandSide);
+    normalizedRightHandSide /= global_norm(
+        normalizedRightHandSide,
+        finiteElements.mesh->GetComm()
+    );
+
+    mfem::Vector firstCorrection(scaledInverse.Height());
+    mfem::Vector repeatedCorrection(scaledInverse.Height());
+    firstCorrection = 0.0;
+    repeatedCorrection = 0.0;
+    scaledInverse.Mult(normalizedRightHandSide, firstCorrection);
+    scaledInverse.Mult(normalizedRightHandSide, repeatedCorrection);
+
+    mfem::Vector repeatDifference(repeatedCorrection);
+    repeatDifference -= firstCorrection;
+    const double correctionNorm = global_norm(firstCorrection, finiteElements.mesh->GetComm());
+    const double repeatError = global_norm(repeatDifference, finiteElements.mesh->GetComm()) /
+        std::max(correctionNorm, std::numeric_limits<double>::epsilon());
+    CAPTURE(correctionNorm, repeatError);
+    const int locallyFiniteCorrections =
+        vector_is_finite(firstCorrection) && vector_is_finite(repeatedCorrection) ? 1 : 0;
+    int globallyFiniteCorrections = 0;
+    REQUIRE(MPI_Allreduce(
+        &locallyFiniteCorrections,
+        &globallyFiniteCorrections,
+        1,
+        MPI_INT,
+        MPI_MIN,
+        finiteElements.mesh->GetComm()
+    ) == MPI_SUCCESS);
+    CHECK(globallyFiniteCorrections == 1);
+    CHECK(std::isfinite(correctionNorm));
+    CHECK(correctionNorm > 0.0);
+    CHECK(repeatError <= 2.0e-12);
+    require_all_ranks(
+        physicalInverse.IsCurrent() && scaledInverse.IsCurrent(),
+        communicator,
+        "preconditioner currentness after repeated application"
+    );
+
+    const mfem::Vector &readOnlyCorrection = firstCorrection;
+    const auto correctionView = problem.GetManifest().stateView(readOnlyCorrection);
+    const double massCorrection = correctionView.block(
+        utils::blocks::fixed_total_mass_constraint.mass_normalization_term
+    )(0);
+    const double angularVelocityCorrection = correctionView.block(
+        utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term
+    )(0);
+    const double phaseCorrection = correctionView.block(
+        utils::blocks::fixed_central_density_phase.central_value_term
+    )(0);
+    CAPTURE(massCorrection, angularVelocityCorrection, phaseCorrection);
+    check_rank_consistent_scalar(
+        massCorrection,
+        finiteElements.mesh->GetComm()
+    );
+    check_rank_consistent_scalar(
+        angularVelocityCorrection,
+        finiteElements.mesh->GetComm()
+    );
+    check_rank_consistent_scalar(
+        phaseCorrection,
+        finiteElements.mesh->GetComm()
+    );
+
+    /* A Newton iteration reparses the same normalized state under fresh
+     * dependency revisions.  Every rank must observe the stale inverse, and
+     * refresh must reconstruct the inferred border actions and Schur data. */
+    const auto secondPreparation = normalized.Prepare(
+        normalizedState,
+        make_stellar_dependencies(5)
+    );
+    require_all_ranks(
+        secondPreparation.DidAnyWork() && normalized.IsPrepared(),
+        communicator,
+        "second variadic stellar-root preparation"
+    );
+    mfem::Vector secondPreparedResidual;
+    normalized.BuildResidual(secondPreparedResidual);
+    mfem::Vector secondPreparedResidualDifference(secondPreparedResidual);
+    secondPreparedResidualDifference -= normalizedResidual;
+    const double secondPreparedResidualError = global_norm(
+        secondPreparedResidualDifference,
+        finiteElements.mesh->GetComm()
+    ) / std::max({
+        global_norm(secondPreparedResidual, finiteElements.mesh->GetComm()),
+        global_norm(normalizedResidual, finiteElements.mesh->GetComm()),
+        std::numeric_limits<double>::epsilon()
+    });
+    CAPTURE(secondPreparedResidualError);
+    CHECK(secondPreparedResidualError <= 2.0e-12);
+    require_all_ranks(
+        !physicalInverse.IsCurrent() && !scaledInverse.IsCurrent(),
+        communicator,
+        "preconditioners become stale together"
+    );
+    CHECK_THROWS_AS(
+        scaledInverse.Mult(normalizedRightHandSide, repeatedCorrection),
+        std::logic_error
+    );
+
+    const auto refresh = physicalInverse.Refresh();
+    CHECK(refresh.specificationActionsRefreshed);
+    CHECK(refresh.rebuiltSchurComplement);
+    CHECK(refresh.DidAnyWork());
+    require_all_ranks(
+        physicalInverse.IsCurrent() && scaledInverse.IsCurrent(),
+        communicator,
+        "refreshed preconditioner currentness"
+    );
+    const auto noOpRefresh = physicalInverse.Refresh();
+    CHECK_FALSE(noOpRefresh.DidAnyWork());
+
+    mfem::Vector refreshedCorrection(scaledInverse.Height());
+    refreshedCorrection = 0.0;
+    scaledInverse.Mult(normalizedRightHandSide, refreshedCorrection);
+    mfem::Vector refreshDifference(refreshedCorrection);
+    refreshDifference -= firstCorrection;
+    const double refreshError = global_norm(
+        refreshDifference,
+        finiteElements.mesh->GetComm()
+    ) / std::max(
+        correctionNorm,
+        std::numeric_limits<double>::epsilon()
+    );
+    const int locallyFiniteRefresh = vector_is_finite(refreshedCorrection) ? 1 : 0;
+    int globallyFiniteRefresh = 0;
+    REQUIRE(MPI_Allreduce(
+        &locallyFiniteRefresh,
+        &globallyFiniteRefresh,
+        1,
+        MPI_INT,
+        MPI_MIN,
+        finiteElements.mesh->GetComm()
+    ) == MPI_SUCCESS);
+    CAPTURE(refreshError);
+    CHECK(globallyFiniteRefresh == 1);
+    CHECK(refreshError <= 2.0e-10);
+}
+
 TEST_CASE(
     "MPI Prepared Gravity Operators Preserve Global Algebraic Identities",
     "[mpi][distributed][gravity][operators][unit]"

tests/normalization/physical_riesz.cpp

@@ -0,0 +1,828 @@
+#include <algorithm>
+#include <array>
+#include <cmath>
+#include <concepts>
+#include <limits>
+#include <span>
+#include <stdexcept>
+#include <utility>
+
+#include <catch2/catch_approx.hpp>
+#include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    namespace blocks = mean_field::utils::blocks;
+    namespace normalization = mean_field::normalization;
+    namespace models = mean_field::models;
+
+    struct ModelWithoutFixedTotalMass final { };
+
+    template <typename Model>
+    concept SupportsModelDerivedStellarScales = requires(
+        const normalization::PhysicalRieszDiagonal<> &policy,
+        const Model &model
+    ) {
+        {
+            normalization::deriveStellarCharacteristicScales(policy, model)
+        } -> std::same_as<normalization::StellarCharacteristicScales>;
+    };
+
+    struct TestValue final : blocks::value_block_base { };
+    struct TestResidual final : blocks::residual_block_base { };
+    using TestForm = blocks::block_form<
+        blocks::type_list<TestValue>,
+        blocks::type_list<TestResidual>>;
+
+    using GlobalSpecificEnergyNormalization = models::CoordinateNormalization<
+        models::RieszTopology::global_scalar,
+        models::PhysicalScaleLaw::specific_energy>;
+    using VolumeSpecificEnergyNormalization = models::CoordinateNormalization<
+        models::RieszTopology::scalar_volume_l2,
+        models::PhysicalScaleLaw::specific_energy>;
+
+    class SelfDescribingMagneticSpecificEnergy final {
+    public:
+        struct Parameters final {
+            double target;
+        };
+
+        using ModelDefinition = mean_field::integral::FixedWithPhysicalCoordinate<
+            SelfDescribingMagneticSpecificEnergy,
+            "NormalizationMockMagneticSpecificEnergy",
+            models::DependsOn<blocks::density::mass::value>,
+            models::Affects<blocks::enthalpy::specific::residual>,
+            models::GlobalScalarNormalization<
+                models::PhysicalScaleLaw::dimensionless,
+                models::PhysicalScaleLaw::specific_energy>>;
+
+        explicit constexpr SelfDescribingMagneticSpecificEnergy(const Parameters parameters) noexcept
+            : m_target(parameters.target) {
+        }
+
+    private:
+        double m_target;
+    };
+
+    class MissingGeneratedNormalization final {
+    public:
+        struct Parameters final {
+            double target;
+        };
+
+        using ModelDefinition = mean_field::integral::FixedWithMultiplier<
+            MissingGeneratedNormalization,
+            "NormalizationMockMissing",
+            models::DependsOn<blocks::density::mass::value>,
+            models::Affects<blocks::enthalpy::specific::residual>>;
+
+        explicit constexpr MissingGeneratedNormalization(const Parameters parameters) noexcept
+            : m_target(parameters.target) {
+        }
+
+    private:
+        double m_target;
+    };
+
+    class GeneratedVolumeCoordinateWithoutMetricSource final {
+    public:
+        struct Parameters final {
+            double target;
+        };
+
+        using ModelDefinition = mean_field::integral::FixedWithPhysicalCoordinate<
+            GeneratedVolumeCoordinateWithoutMetricSource,
+            "NormalizationMockVolumeCoordinate",
+            models::DependsOn<blocks::density::mass::value>,
+            models::Affects<blocks::enthalpy::specific::residual>,
+            models::GeneratedNormalization<
+                VolumeSpecificEnergyNormalization,
+                GlobalSpecificEnergyNormalization>>;
+
+        explicit constexpr GeneratedVolumeCoordinateWithoutMetricSource(const Parameters parameters) noexcept
+            : m_target(parameters.target) {
+        }
+
+    private:
+        double m_target;
+    };
+
+    struct MalformedGeneratedNormalization final { };
+
+    class MalformedGeneratedNormalizationConstraint final {
+    public:
+        struct Parameters final {
+            double target;
+        };
+
+        using ModelDefinition = mean_field::integral::FixedWithMultiplier<
+            MalformedGeneratedNormalizationConstraint,
+            "NormalizationMockMalformed",
+            models::DependsOn<blocks::density::mass::value>,
+            models::Affects<blocks::enthalpy::specific::residual>,
+            MalformedGeneratedNormalization>;
+
+        explicit constexpr MalformedGeneratedNormalizationConstraint(const Parameters parameters) noexcept
+            : m_target(parameters.target) {
+        }
+
+    private:
+        double m_target;
+    };
+
+    template <typename Specification>
+    using GeneratedValueBlock = blocks::generated_value_block<
+        models::PhysicalCoordinateFor<Specification>>;
+
+    template <typename Specification>
+    using GeneratedMultiplierBlock = blocks::generated_value_block<
+        models::MultiplierFor<Specification>>;
+
+    template <typename Specification>
+    using GeneratedResidualBlock = blocks::generated_residual_block<
+        models::ResidualFor<Specification>>;
+
+    using SelfDescribingValue = GeneratedValueBlock<SelfDescribingMagneticSpecificEnergy>;
+    using SelfDescribingResidual = GeneratedResidualBlock<SelfDescribingMagneticSpecificEnergy>;
+    using SelfDescribingForm = blocks::block_form<
+        blocks::type_list<SelfDescribingValue>,
+        blocks::type_list<SelfDescribingResidual>>;
+
+    using MissingValue = GeneratedMultiplierBlock<MissingGeneratedNormalization>;
+    using MissingResidual = GeneratedResidualBlock<MissingGeneratedNormalization>;
+    using MissingNormalizationForm = blocks::block_form<
+        blocks::type_list<MissingValue>,
+        blocks::type_list<MissingResidual>>;
+
+    using UnpreparedVolumeValue = GeneratedValueBlock<GeneratedVolumeCoordinateWithoutMetricSource>;
+    using UnpreparedVolumeResidual = GeneratedResidualBlock<GeneratedVolumeCoordinateWithoutMetricSource>;
+    using UnpreparedVolumeForm = blocks::block_form<
+        blocks::type_list<UnpreparedVolumeValue>,
+        blocks::type_list<UnpreparedVolumeResidual>>;
+
+    class DenseOperator final : public mfem::Operator {
+    public:
+        explicit DenseOperator(const mfem::DenseMatrix &matrix)
+            : mfem::Operator(matrix.Height(), matrix.Width()),
+              m_matrix(matrix) {
+        }
+
+        void Mult(
+            const mfem::Vector &input,
+            mfem::Vector &output
+        ) const override {
+            m_matrix.Mult(input, output);
+        }
+
+    private:
+        mfem::DenseMatrix m_matrix;
+    };
+
+    class DenseInverseSolver final : public mfem::Solver {
+    public:
+        explicit DenseInverseSolver(const mfem::DenseMatrix &inverse)
+            : mfem::Solver(inverse.Height(), inverse.Width()),
+              m_inverse(inverse) {
+        }
+
+        void SetOperator(const mfem::Operator &operation) override {
+            if (operation.Height() != Height() || operation.Width() != Width()) {
+                throw std::invalid_argument("The dense inverse received an incompatible operator.");
+            }
+            m_boundOperator = &operation;
+            ++m_bindings;
+        }
+
+        void Mult(
+            const mfem::Vector &input,
+            mfem::Vector &output
+        ) const override {
+            if (m_boundOperator == nullptr) {
+                throw std::logic_error("The dense inverse must be bound before application.");
+            }
+            m_inverse.Mult(input, output);
+        }
+
+        [[nodiscard]] const mfem::Operator *BoundOperator() const noexcept {
+            return m_boundOperator;
+        }
+
+        [[nodiscard]] int Bindings() const noexcept {
+            return m_bindings;
+        }
+
+    private:
+        mfem::DenseMatrix m_inverse;
+        const mfem::Operator *m_boundOperator{nullptr};
+        int m_bindings{0};
+    };
+
+    [[nodiscard]] mfem::Vector vector(std::initializer_list<double> values) {
+        mfem::Vector result(static_cast<int>(values.size()));
+        int index = 0;
+        for (const double value : values) {
+            result(index++) = value;
+        }
+        return result;
+    }
+
+    void checkVector(
+        const mfem::Vector &actual,
+        const mfem::Vector &expected,
+        const double epsilon = 2.0e-13
+    ) {
+        REQUIRE(actual.Size() == expected.Size());
+        for (int index = 0; index < actual.Size(); ++index) {
+            CHECK(actual(index) == Catch::Approx(expected(index)).epsilon(epsilon).margin(1.0e-300));
+        }
+    }
+} // namespace
+
+TEST_CASE(
+    "Pointer-Retaining Normalization Operators Reject Temporary Dependencies",
+    "[normalization][type][lifetime]"
+) {
+    using Map = normalization::DiagonalNormalization;
+
+    STATIC_CHECK(std::constructible_from<
+                 normalization::ScaledJacobianOperator,
+                 const DenseOperator &,
+                 const Map &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       normalization::ScaledJacobianOperator,
+                       DenseOperator &&,
+                       const Map &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       normalization::ScaledJacobianOperator,
+                       const DenseOperator &&,
+                       const Map &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       normalization::ScaledJacobianOperator,
+                       const DenseOperator &,
+                       Map &&>);
+
+    STATIC_CHECK(std::constructible_from<
+                 normalization::ScaledInverseOperator,
+                 const DenseOperator &,
+                 const Map &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       normalization::ScaledInverseOperator,
+                       DenseOperator &&,
+                       const Map &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       normalization::ScaledInverseOperator,
+                       const DenseOperator &,
+                       Map &&>);
+
+    STATIC_CHECK(std::constructible_from<
+                 normalization::ScaledPreconditioner,
+                 DenseInverseSolver &,
+                 const DenseOperator &,
+                 const DenseOperator &,
+                 const Map &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       normalization::ScaledPreconditioner,
+                       DenseInverseSolver &&,
+                       const DenseOperator &,
+                       const DenseOperator &,
+                       const Map &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       normalization::ScaledPreconditioner,
+                       DenseInverseSolver &,
+                       DenseOperator &&,
+                       const DenseOperator &,
+                       const Map &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       normalization::ScaledPreconditioner,
+                       DenseInverseSolver &,
+                       const DenseOperator &,
+                       DenseOperator &&,
+                       const Map &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       normalization::ScaledPreconditioner,
+                       DenseInverseSolver &,
+                       const DenseOperator &,
+                       const DenseOperator &,
+                       Map &&>);
+}
+
+TEST_CASE("Characteristic Stellar Scales Satisfy Gravity Virial And Rotation Identities", "[normalization][physics]") {
+    using namespace mean_field;
+
+    constexpr double mass = 7.0;
+    constexpr double radius = 3.0;
+    constexpr double gravity = 5.0;
+    const auto scales = normalization::deriveStellarCharacteristicScales(
+        dimensions::MassValue{mass}, dimensions::LengthValue{radius}, gravity
+    );
+
+    CHECK(scales.density == Catch::Approx(mass / std::pow(radius, 3)));
+    CHECK(scales.acceleration == Catch::Approx(gravity * mass / std::pow(radius, 2)));
+    CHECK(scales.specificEnergy == Catch::Approx(gravity * mass / radius));
+    CHECK(scales.pressure == Catch::Approx(gravity * mass * mass / std::pow(radius, 4)));
+    CHECK(scales.angularVelocity == Catch::Approx(std::sqrt(gravity * mass / std::pow(radius, 3))));
+    CHECK(scales.angularMomentum == Catch::Approx(mass * std::sqrt(gravity * mass * radius)));
+
+    // Hydrostatic/virial energy scales agree: P R^3 = M Phi = F R.
+    const double virial = scales.pressure * std::pow(radius, 3);
+    CHECK(virial == Catch::Approx(mass * scales.specificEnergy).epsilon(2.0e-15));
+    CHECK(virial == Catch::Approx(scales.force * radius).epsilon(2.0e-15));
+
+    // Omega_0 is the Kepler/break-up scale and J_0 = M R^2 Omega_0.
+    CHECK(scales.angularVelocity * scales.angularVelocity * radius ==
+          Catch::Approx(scales.acceleration).epsilon(2.0e-15));
+    CHECK(scales.angularMomentum ==
+          Catch::Approx(mass * radius * radius * scales.angularVelocity).epsilon(2.0e-15));
+}
+
+TEST_CASE("Characteristic Scales Obey The Expected Stellar Homology Exponents", "[normalization][physics]") {
+    using namespace mean_field;
+
+    const auto reference = normalization::deriveStellarCharacteristicScales(
+        dimensions::MassValue{2.5}, dimensions::LengthValue{4.0}, 3.0
+    );
+    constexpr double massFactor = 11.0;
+    constexpr double radiusFactor = 0.2;
+    constexpr double gravityFactor = 7.0;
+    const auto transformed = normalization::deriveStellarCharacteristicScales(
+        dimensions::MassValue{2.5 * massFactor},
+        dimensions::LengthValue{4.0 * radiusFactor},
+        3.0 * gravityFactor
+    );
+
+    CHECK(transformed.density / reference.density ==
+          Catch::Approx(massFactor / std::pow(radiusFactor, 3)).epsilon(4.0e-15));
+    CHECK(transformed.acceleration / reference.acceleration ==
+          Catch::Approx(gravityFactor * massFactor / std::pow(radiusFactor, 2)).epsilon(4.0e-15));
+    CHECK(transformed.inverseTimeSquared / reference.inverseTimeSquared ==
+          Catch::Approx(gravityFactor * massFactor / std::pow(radiusFactor, 3)).epsilon(4.0e-15));
+    CHECK(transformed.specificEnergy / reference.specificEnergy ==
+          Catch::Approx(gravityFactor * massFactor / radiusFactor).epsilon(4.0e-15));
+    CHECK(transformed.pressure / reference.pressure ==
+          Catch::Approx(gravityFactor * massFactor * massFactor / std::pow(radiusFactor, 4)).epsilon(4.0e-15));
+    CHECK(transformed.angularVelocity / reference.angularVelocity == Catch::Approx(
+              std::sqrt(gravityFactor * massFactor / std::pow(radiusFactor, 3))
+          ).epsilon(4.0e-15));
+    CHECK(transformed.angularMomentum / reference.angularMomentum == Catch::Approx(
+              massFactor * std::sqrt(gravityFactor * massFactor * radiusFactor)
+          ).epsilon(4.0e-15));
+}
+
+TEST_CASE("Physical Block Scales Distinguish Invariants From Numerical Phase Conditions", "[normalization][physics]") {
+    using namespace mean_field;
+
+    const auto scales = normalization::deriveStellarCharacteristicScales(
+        dimensions::MassValue{9.0}, dimensions::LengthValue{2.0}, 4.0
+    );
+    CHECK(normalization::physicalScale<blocks::density::mass::value>(scales) == scales.density);
+    CHECK(normalization::physicalScale<blocks::gravity::gradient::value>(scales) == scales.acceleration);
+    CHECK(normalization::physicalScale<blocks::gravity::poisson::residual>(scales) == scales.inverseTimeSquared);
+    CHECK(normalization::physicalScale<blocks::fixed_total_mass::mass_normalization::residual>(scales) == 9.0);
+    CHECK(normalization::physicalScale<blocks::fixed_angular_momentum::angular_velocity::value>(scales) ==
+          scales.angularVelocity);
+    CHECK(normalization::physicalScale<blocks::fixed_angular_momentum::angular_velocity::residual>(scales) ==
+          scales.angularMomentum);
+
+    // The central-density condition is implemented as h(0)-h_target, so its residual scale is energy/mass,
+    // despite the physical target being expressed as a density.
+    CHECK(normalization::physicalScale<blocks::fixed_central_density::central_value::residual>(scales) ==
+          scales.specificEnergy);
+    CHECK(normalization::physicalScale<blocks::fixed_central_density::central_value::residual>(scales) !=
+          scales.density);
+}
+
+TEST_CASE(
+    "Generated Physical Riesz Laws Come From A Self-Describing Physics Specification",
+    "[normalization][type][extension]"
+) {
+    using namespace mean_field;
+    using ValueTraits = normalization::PhysicalRieszBlockTraits<SelfDescribingValue>;
+    using ResidualTraits = normalization::PhysicalRieszBlockTraits<SelfDescribingResidual>;
+
+    STATIC_CHECK(models::SelfDescribingModelSpecification<SelfDescribingMagneticSpecificEnergy>);
+    STATIC_CHECK(models::CompleteGeneratedNormalizationFor<SelfDescribingMagneticSpecificEnergy>);
+    STATIC_CHECK(operators::StellarEquilibriumSpecificationCompilable<SelfDescribingMagneticSpecificEnergy>);
+    STATIC_CHECK(normalization::GeneratedValuePhysicalRieszNormalizable<
+                 models::PhysicalCoordinateFor<SelfDescribingMagneticSpecificEnergy>>);
+    STATIC_CHECK(normalization::GeneratedResidualPhysicalRieszNormalizable<
+                 models::ResidualFor<SelfDescribingMagneticSpecificEnergy>>);
+    STATIC_CHECK(normalization::CompleteGeneratedPhysicalRieszNormalizationFor<
+                 SelfDescribingMagneticSpecificEnergy>);
+    STATIC_CHECK(normalization::CompilableNormalizationFor<
+                 normalization::PhysicalRieszDiagonal<>,
+                 SelfDescribingForm>);
+    STATIC_CHECK(normalization::RegisteredStellarSpecificationNormalization<
+                 SelfDescribingMagneticSpecificEnergy>);
+    STATIC_CHECK(normalization::CompleteStellarSpecificationNormalizationFor<
+                 SelfDescribingMagneticSpecificEnergy,
+                 SelfDescribingForm>);
+
+    STATIC_CHECK(ValueTraits::Method::topology == normalization::RieszTopology::global_scalar);
+    STATIC_CHECK(ValueTraits::Method::scale == normalization::PhysicalScaleKind::dimensionless);
+    STATIC_CHECK(ResidualTraits::Method::topology == normalization::RieszTopology::global_scalar);
+    STATIC_CHECK(ResidualTraits::Method::scale == normalization::PhysicalScaleKind::specific_energy);
+
+    const auto scales = normalization::deriveStellarCharacteristicScales(
+        dimensions::MassValue{9.0}, dimensions::LengthValue{2.0}, 4.0
+    );
+    const blocks::form_layout<SelfDescribingForm> layout({1}, {1});
+    normalization::DiagonalNormalizationBuilder<SelfDescribingForm> builder(layout);
+    normalization::StellarSpecificationNormalizationContribution<
+        SelfDescribingMagneticSpecificEnergy>::Apply(builder, scales);
+    const normalization::DiagonalNormalization map = std::move(builder).Build();
+
+    REQUIRE(map.StateFactors().Size() == 1);
+    REQUIRE(map.ResidualFactors().Size() == 1);
+    CHECK(map.StateFactors()(0) == Catch::Approx(1.0).epsilon(2.0e-15));
+    CHECK(map.ResidualFactors()(0) == Catch::Approx(1.0 / scales.specificEnergy).epsilon(2.0e-15));
+}
+
+TEST_CASE(
+    "Generated Normalization Completeness Is SFINAE Safe And Rejects Missing Runtime Metrics",
+    "[normalization][type][validation]"
+) {
+    using namespace mean_field;
+
+    STATIC_CHECK_FALSE(normalization::CompleteGeneratedPhysicalRieszNormalizationFor<int>);
+    STATIC_CHECK_FALSE(normalization::RegisteredStellarSpecificationNormalization<int>);
+    STATIC_CHECK_FALSE(normalization::CompleteStellarNormalizationFor<int, SelfDescribingForm>);
+
+    STATIC_CHECK(models::ModelSpecification<MissingGeneratedNormalization>);
+    STATIC_CHECK_FALSE(models::CompleteGeneratedNormalizationFor<MissingGeneratedNormalization>);
+    STATIC_CHECK_FALSE(normalization::CompleteGeneratedPhysicalRieszNormalizationFor<
+                       MissingGeneratedNormalization>);
+    STATIC_CHECK_FALSE(normalization::CompilableNormalizationFor<
+                       normalization::PhysicalRieszDiagonal<>,
+                       MissingNormalizationForm>);
+    STATIC_CHECK_FALSE(normalization::RegisteredStellarSpecificationNormalization<
+                       MissingGeneratedNormalization>);
+    STATIC_CHECK_FALSE(normalization::CompleteStellarSpecificationNormalizationFor<
+                       MissingGeneratedNormalization,
+                       MissingNormalizationForm>);
+
+    STATIC_CHECK(models::ModelSpecification<MalformedGeneratedNormalizationConstraint>);
+    STATIC_CHECK_FALSE(models::CompleteGeneratedNormalizationFor<
+                       MalformedGeneratedNormalizationConstraint>);
+    STATIC_CHECK_FALSE(normalization::CompleteGeneratedPhysicalRieszNormalizationFor<
+                       MalformedGeneratedNormalizationConstraint>);
+    STATIC_CHECK_FALSE(normalization::RegisteredStellarSpecificationNormalization<
+                       MalformedGeneratedNormalizationConstraint>);
+
+    // The declaration itself is a valid Riesz law, but runtime stellar
+    // preparation has no finite-element Gram source for a generated volume
+    // field.  The stronger runtime concept must therefore reject it.
+    STATIC_CHECK(normalization::CompleteGeneratedPhysicalRieszNormalizationFor<
+                 GeneratedVolumeCoordinateWithoutMetricSource>);
+    STATIC_CHECK(normalization::CompilableNormalizationFor<
+                 normalization::PhysicalRieszDiagonal<>,
+                 UnpreparedVolumeForm>);
+    STATIC_CHECK_FALSE(normalization::RegisteredStellarSpecificationNormalization<
+                       GeneratedVolumeCoordinateWithoutMetricSource>);
+    STATIC_CHECK_FALSE(normalization::CompleteStellarSpecificationNormalizationFor<
+                       GeneratedVolumeCoordinateWithoutMetricSource,
+                       UnpreparedVolumeForm>);
+}
+
+TEST_CASE("Characteristic Scale Construction Rejects Invalid Or Overflowing References", "[normalization][validation]") {
+    using namespace mean_field;
+
+    CHECK_THROWS_AS(
+        normalization::deriveStellarCharacteristicScales(
+            dimensions::MassValue{0.0}, dimensions::LengthValue{1.0}, 1.0
+        ),
+        std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        normalization::deriveStellarCharacteristicScales(
+            dimensions::MassValue{1.0}, dimensions::LengthValue{-1.0}, 1.0
+        ),
+        std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        (normalization::PhysicalRieszDiagonal{dimensions::LengthValue{1.0},
+                                               std::numeric_limits<double>::quiet_NaN()}),
+        std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        normalization::deriveStellarCharacteristicScales(
+            dimensions::MassValue{1.0e300}, dimensions::LengthValue{1.0e-200}, 1.0e100
+        ),
+        std::overflow_error
+    );
+}
+
+TEST_CASE("Diagonal Riesz Maps Reproduce Primal And Dual Norms Across Extreme Metrics", "[normalization][math]") {
+    const blocks::form_layout<TestForm> layout({3}, {3});
+    normalization::DiagonalNormalizationBuilder<TestForm> builder(layout);
+    const mfem::Vector gram = vector({1.0e-20, 4.0, 9.0e20});
+    constexpr double stateScale = 10.0;
+    constexpr double residualScale = 0.25;
+    builder.SetValueBlock<TestValue>(stateScale, gram);
+    builder.SetResidualBlock<TestResidual>(residualScale, gram);
+    const normalization::DiagonalNormalization map = std::move(builder).Build();
+
+    const mfem::Vector state = vector({3.0e10, -2.0, 4.0e-10});
+    const mfem::Vector residual = vector({2.0e-10, -3.0, 5.0e10});
+    double expectedPrimalNormSquared = 0.0;
+    double expectedDualNormSquared = 0.0;
+    for (int index = 0; index < gram.Size(); ++index) {
+        expectedPrimalNormSquared += gram(index) * state(index) * state(index) /
+                                     (stateScale * stateScale);
+        expectedDualNormSquared += residual(index) * residual(index) /
+                                   (gram(index) * residualScale * residualScale);
+    }
+    CHECK(map.LocalStateNormSquared(state) == Catch::Approx(expectedPrimalNormSquared).epsilon(3.0e-15));
+    CHECK(map.LocalResidualNormSquared(residual) == Catch::Approx(expectedDualNormSquared).epsilon(3.0e-15));
+
+    mfem::Vector normalizedState;
+    mfem::Vector recoveredState;
+    mfem::Vector normalizedResidual;
+    mfem::Vector recoveredResidual;
+    map.NormalizeState(state, normalizedState);
+    map.DenormalizeState(normalizedState, recoveredState);
+    map.NormalizeResidual(residual, normalizedResidual);
+    map.DenormalizeResidual(normalizedResidual, recoveredResidual);
+    checkVector(recoveredState, state, 3.0e-15);
+    checkVector(recoveredResidual, residual, 3.0e-15);
+}
+
+TEST_CASE("Hybrid Riesz Rows Replace Missing Volume Metrics With Point Metrics", "[normalization][math]") {
+    using HybridForm = blocks::block_form<
+        blocks::type_list<blocks::enthalpy::specific::value>,
+        blocks::type_list<blocks::enthalpy::specific::residual>>;
+    const blocks::form_layout<HybridForm> layout({4}, {4});
+    normalization::DiagonalNormalizationBuilder<HybridForm> builder(layout);
+    builder.SetValueBlock<blocks::enthalpy::specific::value>(2.0, vector({2.0, 3.0, 5.0, 7.0}));
+
+    // Replaced isobaric rows may have zero bulk mass because they are no longer volume weak rows.
+    const mfem::Vector bulkMetric = vector({4.0, 0.0, 16.0, 0.0});
+    const std::array<int, 2> pointRows{1, 3};
+    builder.SetHybridResidualBlock<blocks::enthalpy::specific::residual>(
+        5.0, bulkMetric, std::span<const int>{pointRows}, 1.0
+    );
+    const auto map = std::move(builder).Build();
+    CHECK(map.ResidualFactors()(0) == Catch::Approx(1.0 / 10.0));
+    CHECK(map.ResidualFactors()(1) == Catch::Approx(1.0 / 5.0));
+    CHECK(map.ResidualFactors()(2) == Catch::Approx(1.0 / 20.0));
+    CHECK(map.ResidualFactors()(3) == Catch::Approx(1.0 / 5.0));
+
+    normalization::DiagonalNormalizationBuilder<HybridForm> duplicateRows(layout);
+    duplicateRows.SetValueGlobal<blocks::enthalpy::specific::value>(1.0);
+    const std::array<int, 2> duplicates{1, 1};
+    CHECK_THROWS_AS(
+        duplicateRows.SetHybridResidualBlock<blocks::enthalpy::specific::residual>(
+            1.0, vector({1.0, 1.0, 1.0, 1.0}), std::span<const int>{duplicates}
+        ),
+        std::invalid_argument
+    );
+}
+
+TEST_CASE("Runtime Normalization Assembly Rejects Missing Duplicate And Invalid Data", "[normalization][validation]") {
+    const blocks::form_layout<TestForm> layout({2}, {2});
+
+    normalization::DiagonalNormalizationBuilder<TestForm> missing(layout);
+    missing.SetValueBlock<TestValue>(1.0, vector({1.0, 1.0}));
+    CHECK_THROWS_AS(std::move(missing).Build(), std::logic_error);
+
+    normalization::DiagonalNormalizationBuilder<TestForm> duplicate(layout);
+    duplicate.SetValueBlock<TestValue>(1.0, vector({1.0, 1.0}));
+    CHECK_THROWS_AS(duplicate.SetValueBlock<TestValue>(1.0, vector({1.0, 1.0})), std::logic_error);
+
+    normalization::DiagonalNormalizationBuilder<TestForm> zeroMetric(layout);
+    CHECK_THROWS_AS(zeroMetric.SetValueBlock<TestValue>(1.0, vector({1.0, 0.0})), std::invalid_argument);
+
+    normalization::DiagonalNormalizationBuilder<TestForm> wrongSize(layout);
+    CHECK_THROWS_AS(wrongSize.SetResidualBlock<TestResidual>(1.0, vector({1.0})), std::invalid_argument);
+
+    CHECK_THROWS_AS(
+        normalization::DiagonalNormalization(vector({1.0, std::numeric_limits<double>::infinity()}), vector({1.0})),
+        std::invalid_argument
+    );
+}
+
+TEST_CASE("Scaled Jacobian And Inverse Implement The Exact Coordinate Change", "[normalization][linear-algebra]") {
+    const mfem::Vector stateFactors = vector({1.0e-9, 2.0e7});
+    const mfem::Vector residualFactors = vector({5.0e8, 3.0e-6});
+    const normalization::DiagonalNormalization map(stateFactors, residualFactors);
+
+    // Start from a well-conditioned normalized Jacobian A_hat and form the dimensional
+    // J = L^{-1} A_hat R^{-1}. Its entries span the physical unit ranges, while L J R
+    // must recover A_hat rather than an artificially ill-conditioned dense matrix.
+    constexpr double normalizedMatrix[2][2]{{4.0, 1.0}, {2.0, 3.0}};
+    constexpr double normalizedInverse[2][2]{{0.3, -0.1}, {-0.2, 0.4}};
+    mfem::DenseMatrix matrix(2);
+    mfem::DenseMatrix inverse(2);
+    for (int row = 0; row < 2; ++row) {
+        for (int column = 0; column < 2; ++column) {
+            matrix(row, column) = normalizedMatrix[row][column] * stateFactors(column) /
+                                  residualFactors(row);
+            inverse(row, column) = normalizedInverse[row][column] * residualFactors(column) /
+                                   stateFactors(row);
+        }
+    }
+    const DenseOperator physicalJacobian(matrix);
+    const DenseOperator physicalInverse(inverse);
+    const normalization::ScaledJacobianOperator scaledJacobian(physicalJacobian, map);
+    const normalization::ScaledInverseOperator scaledInverse(physicalInverse, map);
+
+    const mfem::Vector direction = vector({0.75, -1.25});
+    mfem::Vector action;
+    scaledJacobian.Mult(direction, action);
+    mfem::Vector expected(2);
+    expected(0) = 4.0 * direction(0) + direction(1);
+    expected(1) = 2.0 * direction(0) + 3.0 * direction(1);
+    checkVector(action, expected, 4.0e-15);
+
+    mfem::Vector recovered;
+    scaledInverse.Mult(action, recovered);
+    checkVector(recovered, direction, 2.0e-13);
+}
+
+TEST_CASE("Scaled Preconditioning Routes An Exact Physical Inverse Through FGMRES", "[normalization][solver]") {
+    const mfem::Vector stateFactors = vector({1.0e-9, 2.0e7});
+    const mfem::Vector residualFactors = vector({5.0e8, 3.0e-6});
+    const normalization::DiagonalNormalization map(stateFactors, residualFactors);
+
+    constexpr double normalizedMatrix[2][2]{{4.0, 1.0}, {2.0, 3.0}};
+    constexpr double normalizedInverse[2][2]{{0.3, -0.1}, {-0.2, 0.4}};
+    mfem::DenseMatrix physicalMatrix(2);
+    mfem::DenseMatrix physicalInverseMatrix(2);
+    for (int row = 0; row < 2; ++row) {
+        for (int column = 0; column < 2; ++column) {
+            physicalMatrix(row, column) = normalizedMatrix[row][column] * stateFactors(column) /
+                                          residualFactors(row);
+            physicalInverseMatrix(row, column) = normalizedInverse[row][column] * residualFactors(column) /
+                                                  stateFactors(row);
+        }
+    }
+
+    const DenseOperator physicalJacobian(physicalMatrix);
+    const normalization::ScaledJacobianOperator scaledJacobian(physicalJacobian, map);
+    DenseInverseSolver physicalInverse(physicalInverseMatrix);
+    normalization::ScaledPreconditioner scaledPreconditioner(
+        physicalInverse, physicalJacobian, scaledJacobian, map
+    );
+
+    CHECK(physicalInverse.BoundOperator() == &physicalJacobian);
+    CHECK(&scaledPreconditioner.GetPhysicalJacobian() == &physicalJacobian);
+    CHECK(&scaledPreconditioner.GetNormalizedJacobian() == &scaledJacobian);
+
+    const mfem::Vector rightHandSide = vector({1.5, -0.75});
+    mfem::Vector directCorrection(2);
+    scaledPreconditioner.Mult(rightHandSide, directCorrection);
+    const mfem::Vector expected = vector({0.525, -0.6});
+    checkVector(directCorrection, expected, 3.0e-13);
+
+    mfem::FGMRESSolver krylov(MPI_COMM_WORLD);
+    krylov.SetRelTol(1.0e-13);
+    krylov.SetAbsTol(1.0e-15);
+    krylov.SetMaxIter(4);
+    krylov.SetKDim(2);
+    krylov.SetPrintLevel(0);
+    krylov.SetPreconditioner(scaledPreconditioner);
+    krylov.SetOperator(scaledJacobian);
+
+    mfem::Vector solution(2);
+    solution = 0.0;
+    krylov.Mult(rightHandSide, solution);
+    CHECK(krylov.GetConverged());
+    CHECK(krylov.GetNumIterations() <= 1);
+    checkVector(solution, expected, 3.0e-13);
+    CHECK(physicalInverse.BoundOperator() == &physicalJacobian);
+    CHECK(physicalInverse.Bindings() >= 2);
+    CHECK(scaledPreconditioner.GetStatistics().operatorBindings >= 2);
+    CHECK(scaledPreconditioner.GetStatistics().applications >= 2);
+
+    mfem::IdentityOperator differentNormalizedJacobian(2);
+    CHECK_THROWS_AS(
+        scaledPreconditioner.SetOperator(differentNormalizedJacobian),
+        std::invalid_argument
+    );
+    mfem::IdentityOperator wrongSize(3);
+    CHECK_THROWS_AS(scaledPreconditioner.SetOperator(wrongSize), std::invalid_argument);
+    mfem::Vector wrongCorrection(1);
+    CHECK_THROWS_AS(scaledPreconditioner.Mult(rightHandSide, wrongCorrection), std::invalid_argument);
+}
+
+TEST_CASE("Physical Riesz Scaling Collapses A Forty-Eight-Decade Diagonal Imbalance", "[normalization][numerics]") {
+    const mfem::Vector stateFactors = vector({1.0e-12, 1.0, 1.0e12});
+    const mfem::Vector residualFactors = vector({1.0e12, 1.0, 1.0e-12});
+    const normalization::DiagonalNormalization map(stateFactors, residualFactors);
+
+    mfem::DenseMatrix physicalMatrix(3);
+    physicalMatrix = 0.0;
+    for (int index = 0; index < 3; ++index) {
+        physicalMatrix(index, index) = stateFactors(index) / residualFactors(index);
+    }
+    CHECK(physicalMatrix(2, 2) / physicalMatrix(0, 0) == Catch::Approx(1.0e48));
+
+    const DenseOperator physicalJacobian(physicalMatrix);
+    const normalization::ScaledJacobianOperator scaledJacobian(physicalJacobian, map);
+    const mfem::Vector direction = vector({-2.0, 3.5, 0.125});
+    mfem::Vector action;
+    scaledJacobian.Mult(direction, action);
+    checkVector(action, direction, 4.0e-15);
+}
+
+TEST_CASE("A Compiled Stellar Problem Prepares Reference Physical Riesz Coordinates", "[normalization][integration]") {
+    using namespace mean_field;
+
+    utils::Args args = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(args.mesh_file, args, 0);
+    REQUIRE(finiteElements.okay());
+
+    constexpr double targetMass = 2.0;
+    constexpr double referenceRadius = 1.25;
+    constexpr double gravitationalConstant = 3.0;
+    const normalization::PhysicalRieszDiagonal policy{
+        dimensions::LengthValue{referenceRadius}, gravitationalConstant
+    };
+    const auto discretization = equilibrium::makeStellarDiscretization(finiteElements, policy);
+    auto problem = equilibrium::discretize(
+        model::StellarModel(
+            eos::Polytrope({.n = 3.0, .K = 0.25}),
+            surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+            integral::FixedTotalMass({.Mtotal = dimensions::MassValue{targetMass}}),
+            constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{1.0}})
+        ),
+        discretization
+    );
+
+    using ProblemType = std::remove_cvref_t<decltype(problem)>;
+    using Form = typename ProblemType::FormType;
+    using ModelType = std::remove_cvref_t<decltype(problem.GetStellarModel())>;
+    STATIC_CHECK(SupportsModelDerivedStellarScales<ModelType>);
+    STATIC_CHECK_FALSE(SupportsModelDerivedStellarScales<ModelWithoutFixedTotalMass>);
+    STATIC_CHECK(std::same_as<
+                 typename ProblemType::NormalizationPrescriptionType,
+                 std::remove_cvref_t<decltype(policy)>>);
+    CHECK(problem.GetNormalizationPrescription().referenceRadius() == dimensions::LengthValue{referenceRadius});
+
+    const normalization::DiagonalNormalization map = normalization::prepareNormalization(problem);
+    REQUIRE(map.StateSize() == problem.StateSize());
+    REQUIRE(map.ResidualSize() == problem.EquationSize());
+    for (int index = 0; index < map.StateSize(); ++index) {
+        CHECK(std::isfinite(map.StateFactors()(index)));
+        CHECK(map.StateFactors()(index) > 0.0);
+    }
+    for (int index = 0; index < map.ResidualSize(); ++index) {
+        CHECK(std::isfinite(map.ResidualFactors()(index)));
+        CHECK(map.ResidualFactors()(index) > 0.0);
+    }
+
+    const auto scales = normalization::deriveStellarCharacteristicScales(policy, problem.GetStellarModel());
+    const auto &layout = problem.GetManifest().layout();
+    constexpr int massValueBlock = blocks::type_index_v<
+        blocks::fixed_total_mass::mass_normalization::value,
+        typename Form::value_blocks>;
+    constexpr int massResidualBlock = blocks::type_index_v<
+        blocks::fixed_total_mass::mass_normalization::residual,
+        typename Form::residual_blocks>;
+    constexpr int phaseValueBlock = blocks::type_index_v<
+        blocks::fixed_central_density::central_value::value,
+        typename Form::value_blocks>;
+    constexpr int phaseResidualBlock = blocks::type_index_v<
+        blocks::fixed_central_density::central_value::residual,
+        typename Form::residual_blocks>;
+    constexpr int enthalpyResidualBlock = blocks::type_index_v<
+        blocks::enthalpy::specific::residual,
+        typename Form::residual_blocks>;
+
+    CHECK(map.StateFactors()(layout.value_offsets()[massValueBlock]) ==
+          Catch::Approx(1.0 / scales.specificEnergy).epsilon(2.0e-15));
+    CHECK(map.ResidualFactors()(layout.residual_offsets()[massResidualBlock]) ==
+          Catch::Approx(1.0 / targetMass).epsilon(2.0e-15));
+    CHECK(map.StateFactors()(layout.value_offsets()[phaseValueBlock]) ==
+          Catch::Approx(1.0 / scales.specificEnergy).epsilon(2.0e-15));
+    CHECK(map.ResidualFactors()(layout.residual_offsets()[phaseResidualBlock]) ==
+          Catch::Approx(1.0 / scales.specificEnergy).epsilon(2.0e-15));
+
+    const auto &surfaceRows = problem.GetPressureSurfaceRows().reduced_dofs();
+    REQUIRE(surfaceRows.Size() > 0);
+    for (const int row : surfaceRows) {
+        const int rootRow = layout.residual_offsets()[enthalpyResidualBlock] + row;
+        CHECK(map.ResidualFactors()(rootRow) ==
+              Catch::Approx(1.0 / scales.specificEnergy).epsilon(2.0e-15));
+    }
+
+    mfem::Vector physicalState(problem.StateSize());
+    for (int index = 0; index < physicalState.Size(); ++index) {
+        physicalState(index) = std::sin(0.37 * static_cast<double>(index + 1));
+    }
+    mfem::Vector normalizedState;
+    mfem::Vector recoveredState;
+    map.NormalizeState(physicalState, normalizedState);
+    map.DenormalizeState(normalizedState, recoveredState);
+    checkVector(recoveredState, physicalState, 4.0e-15);
+
+    const normalization::ScaledJacobianOperator scaledJacobian(problem.GetLinearizationOperator(), map);
+    CHECK(scaledJacobian.Width() == problem.StateSize());
+    CHECK(scaledJacobian.Height() == problem.EquationSize());
+
+    // The existing provisional structure preconditioner remains available for this distinct problem type.
+    const auto structureBlock = preconditioning::stellarStructureBlock(problem);
+    STATIC_CHECK(preconditioning::PreconditionerComponent<std::remove_cvref_t<decltype(structureBlock)>>);
+}

tests/normalization/plan.cpp

@@ -0,0 +1,228 @@
+#include <concepts>
+#include <type_traits>
+#include <utility>
+
+#include <catch2/catch_test_macros.hpp>
+
+import mean_field;
+
+namespace {
+    namespace blocks = mean_field::utils::blocks;
+    namespace normalization = mean_field::normalization;
+
+    using PhysicalForm = blocks::surface_deformed_stellar_equilibrium_form;
+    using PhaseForm = blocks::central_density_bordered_stellar_equilibrium_form;
+    using PhysicalPlan = normalization::PhysicalRieszNormalizationPlanFor<PhysicalForm>;
+    using PhasePlan = normalization::PhysicalRieszNormalizationPlanFor<PhaseForm>;
+
+    struct ValueA final : blocks::value_block_base { };
+    struct ValueB final : blocks::value_block_base { };
+    struct ResidualA final : blocks::residual_block_base { };
+    struct ResidualB final : blocks::residual_block_base { };
+    struct ForeignValue final : blocks::value_block_base { };
+    struct ForeignResidual final : blocks::residual_block_base { };
+
+    using SmallForm = blocks::block_form<
+        blocks::type_list<ValueA, ValueB>,
+        blocks::type_list<ResidualA, ResidualB>>;
+    using ValueAIdentity = normalization::CoordinateComponent<
+        normalization::CoordinateKind::value,
+        blocks::type_list<ValueA>,
+        normalization::IdentityCoordinate>;
+    using ValueBIdentity = normalization::CoordinateComponent<
+        normalization::CoordinateKind::value,
+        blocks::type_list<ValueB>,
+        normalization::IdentityCoordinate>;
+    using ResidualAIdentity = normalization::CoordinateComponent<
+        normalization::CoordinateKind::residual,
+        blocks::type_list<ResidualA>,
+        normalization::IdentityCoordinate>;
+    using ResidualBIdentity = normalization::CoordinateComponent<
+        normalization::CoordinateKind::residual,
+        blocks::type_list<ResidualB>,
+        normalization::IdentityCoordinate>;
+    using ForeignIdentity = normalization::CoordinateComponent<
+        normalization::CoordinateKind::value,
+        blocks::type_list<ForeignValue>,
+        normalization::IdentityCoordinate>;
+    using ForeignResidualIdentity = normalization::CoordinateComponent<
+        normalization::CoordinateKind::residual,
+        blocks::type_list<ForeignResidual>,
+        normalization::IdentityCoordinate>;
+
+    using CompleteSmallPlan = normalization::NormalizationPlan<
+        ValueAIdentity,
+        ValueBIdentity,
+        ResidualAIdentity,
+        ResidualBIdentity>;
+    using MissingSmallPlan = normalization::NormalizationPlan<
+        ValueAIdentity,
+        ResidualAIdentity,
+        ResidualBIdentity>;
+    using DuplicateSmallPlan = normalization::NormalizationPlan<
+        ValueAIdentity,
+        ValueAIdentity,
+        ValueBIdentity,
+        ResidualAIdentity,
+        ResidualBIdentity>;
+    using ForeignSmallPlan = normalization::NormalizationPlan<
+        ValueAIdentity,
+        ValueBIdentity,
+        ForeignIdentity,
+        ResidualAIdentity,
+        ResidualBIdentity,
+        ForeignResidualIdentity>;
+
+    struct MalformedComponent final {
+        using Blocks = blocks::type_list<ValueA>;
+        using Method = normalization::IdentityCoordinate;
+    };
+
+    struct IncoherentComponent final {
+        using Blocks                         = blocks::type_list<ValueA>;
+        using Method                         = normalization::IdentityCoordinate;
+        using ValueBlocks                    = blocks::type_list<ValueB>;
+        using ResidualBlocks                 = blocks::type_list<>;
+        static constexpr auto kind           = normalization::CoordinateKind::value;
+    };
+
+    using WrongDensityTopology = normalization::CoordinateComponent<
+        normalization::CoordinateKind::value,
+        blocks::type_list<blocks::density::mass::value>,
+        normalization::PhysicalRieszCoordinate<
+            normalization::RieszTopology::vector_volume_l2,
+            normalization::PhysicalScaleKind::density>>;
+
+    struct FutureInvariantValue final : blocks::value_block_base { };
+    struct FutureInvariantResidual final : blocks::residual_block_base { };
+    using UnregisteredFutureForm = blocks::block_form<
+        blocks::type_list<FutureInvariantValue>,
+        blocks::type_list<FutureInvariantResidual>>;
+
+    using BaseModel = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::integral::FixedTotalMass>>;
+    using RieszPolicy = normalization::PhysicalRieszDiagonal<>;
+    using RieszDiscretization = mean_field::equilibrium::StellarDiscretizationFor<RieszPolicy>;
+    using BaselineProblem = mean_field::equilibrium::StellarEquilibriumProblem<BaseModel>;
+    using RieszProblem = mean_field::equilibrium::StellarEquilibriumProblem<BaseModel, RieszDiscretization>;
+    using AngularModel = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::integral::FixedTotalMass,
+        mean_field::integral::FixedAngularMomentum>>;
+    using AngularForm = mean_field::operators::CompiledStellarEquilibriumForm<AngularModel>;
+
+    template <typename Mapper>
+    concept CanMakeRieszDiscretization = requires(
+        mean_field::fem::FEM &finiteElements,
+        Mapper &&mapper,
+        RieszPolicy policy
+    ) {
+        mean_field::equilibrium::makeStellarDiscretization(
+            finiteElements,
+            std::forward<Mapper>(mapper),
+            policy
+        );
+    };
+} // namespace
+
+TEST_CASE("Normalization Plans Prove Exact Ownership Of Every Compiled Coordinate", "[normalization][type]") {
+    STATIC_CHECK(normalization::NormalizationPlanType<PhysicalPlan>);
+    STATIC_CHECK(normalization::CompleteNormalizationFor<PhysicalPlan, PhysicalForm>);
+    STATIC_CHECK(normalization::CompleteNormalizationFor<PhasePlan, PhaseForm>);
+    STATIC_CHECK(normalization::CompilableNormalizationFor<normalization::Unnormalized, PhysicalForm>);
+    STATIC_CHECK(normalization::CompilableNormalizationFor<RieszPolicy, PhysicalForm>);
+    STATIC_CHECK(normalization::CompilableNormalizationFor<RieszPolicy, PhaseForm>);
+    STATIC_CHECK(normalization::CompilableNormalizationFor<RieszPolicy, AngularForm>);
+    STATIC_CHECK(normalization::StellarSpecificationNormalizationContribution<
+                 mean_field::integral::FixedAngularMomentum>::registered);
+
+    STATIC_CHECK(normalization::CompleteNormalizationFor<CompleteSmallPlan, SmallForm>);
+    STATIC_CHECK_FALSE(normalization::CompleteNormalizationFor<MissingSmallPlan, SmallForm>);
+    STATIC_CHECK_FALSE(normalization::CompleteNormalizationFor<DuplicateSmallPlan, SmallForm>);
+    STATIC_CHECK_FALSE(normalization::CompleteNormalizationFor<ForeignSmallPlan, SmallForm>);
+
+    using Missing = normalization::NormalizationCoverage<SmallForm, MissingSmallPlan>;
+    using Duplicate = normalization::NormalizationCoverage<SmallForm, DuplicateSmallPlan>;
+    using Foreign = normalization::NormalizationCoverage<SmallForm, ForeignSmallPlan>;
+    STATIC_CHECK(Missing::MissingValueBlocks::size == 1);
+    STATIC_CHECK(blocks::contains_type_v<ValueB, typename Missing::MissingValueBlocks>);
+    STATIC_CHECK(Duplicate::RepeatedValueBlocks::size == 1);
+    STATIC_CHECK(blocks::contains_type_v<ValueA, typename Duplicate::RepeatedValueBlocks>);
+    STATIC_CHECK(Foreign::UnexpectedValueBlocks::size == 1);
+    STATIC_CHECK(Foreign::UnexpectedResidualBlocks::size == 1);
+}
+
+TEST_CASE("Physical Riesz Methods Reject Incompatible Or Unregistered Field Topologies", "[normalization][type]") {
+    STATIC_CHECK_FALSE(normalization::NormalizationComponent<MalformedComponent>);
+    STATIC_CHECK_FALSE(normalization::NormalizationComponent<IncoherentComponent>);
+    STATIC_CHECK_FALSE(normalization::NormalizationComponent<WrongDensityTopology>);
+    STATIC_CHECK_FALSE(normalization::CompilableNormalizationFor<RieszPolicy, UnregisteredFutureForm>);
+    STATIC_CHECK(normalization::CompilableNormalizationFor<normalization::Unnormalized, UnregisteredFutureForm>);
+
+    using Density = normalization::PhysicalRieszBlockTraits<blocks::density::mass::value>;
+    using Gravity = normalization::PhysicalRieszBlockTraits<blocks::gravity::gradient::value>;
+    using Surface = normalization::PhysicalRieszBlockTraits<blocks::surface_deformation::parameters::value>;
+    using EnthalpyResidual = normalization::PhysicalRieszBlockTraits<blocks::enthalpy::specific::residual>;
+    using MassResidual = normalization::PhysicalRieszBlockTraits<
+        blocks::fixed_total_mass::mass_normalization::residual>;
+
+    STATIC_CHECK(Density::Method::topology == normalization::RieszTopology::scalar_volume_l2);
+    STATIC_CHECK(Gravity::Method::topology == normalization::RieszTopology::vector_volume_l2);
+    STATIC_CHECK(Surface::Method::topology == normalization::RieszTopology::scalar_boundary_l2);
+    STATIC_CHECK(
+        EnthalpyResidual::Method::topology == normalization::RieszTopology::hybrid_scalar_volume_point_rows
+    );
+    STATIC_CHECK(MassResidual::Method::topology == normalization::RieszTopology::global_scalar);
+    STATIC_CHECK(MassResidual::Method::scale == normalization::PhysicalScaleKind::mass);
+}
+
+TEST_CASE("Normalization Is Part Of The Compile-Time Discretization And Problem Type", "[normalization][type]") {
+    STATIC_CHECK(mean_field::equilibrium::StellarDiscretizationType<RieszDiscretization>);
+    STATIC_CHECK_FALSE(std::same_as<RieszDiscretization, mean_field::equilibrium::StellarDiscretization>);
+    STATIC_CHECK_FALSE(std::same_as<RieszProblem, BaselineProblem>);
+    STATIC_CHECK(std::same_as<typename BaselineProblem::NormalizationPrescriptionType, normalization::Unnormalized>);
+    STATIC_CHECK(std::same_as<typename RieszProblem::NormalizationPrescriptionType, RieszPolicy>);
+    STATIC_CHECK(mean_field::equilibrium::DiscretizedStellarEquilibriumProblem<RieszProblem>);
+    STATIC_CHECK(std::constructible_from<
+                 RieszDiscretization,
+                 mean_field::fem::FEM &,
+                 const mean_field::mapping::DomainMapper &,
+                 RieszPolicy>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       RieszDiscretization,
+                       mean_field::fem::FEM &,
+                       mean_field::mapping::DomainMapper &&,
+                       RieszPolicy>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       RieszDiscretization,
+                       mean_field::fem::FEM &,
+                       const mean_field::mapping::DomainMapper &&,
+                       RieszPolicy>);
+    STATIC_CHECK(std::constructible_from<
+                 mean_field::equilibrium::StellarDiscretization,
+                 mean_field::fem::FEM &,
+                 const mean_field::mapping::DomainMapper &>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       mean_field::equilibrium::StellarDiscretization,
+                       mean_field::fem::FEM &,
+                       mean_field::mapping::DomainMapper &&>);
+    STATIC_CHECK_FALSE(std::constructible_from<
+                       mean_field::equilibrium::StellarDiscretization,
+                       mean_field::fem::FEM &,
+                       const mean_field::mapping::DomainMapper &&>);
+    STATIC_CHECK(CanMakeRieszDiscretization<
+                 mean_field::mapping::DomainMapper &>);
+    STATIC_CHECK_FALSE(CanMakeRieszDiscretization<
+                       mean_field::mapping::DomainMapper>);
+    STATIC_CHECK_FALSE(CanMakeRieszDiscretization<
+                       const mean_field::mapping::DomainMapper>);
+
+    using SmallLayout = blocks::form_layout<SmallForm>;
+    using SmallBuilder = normalization::DiagonalNormalizationBuilder<SmallForm>;
+    STATIC_CHECK(std::constructible_from<SmallBuilder, const SmallLayout &>);
+    STATIC_CHECK_FALSE(std::constructible_from<SmallBuilder, SmallLayout &&>);
+    STATIC_CHECK_FALSE(std::constructible_from<SmallBuilder, const SmallLayout &&>);
+}

tests/operators/prepared_angular_momentum.cpp

@@ -0,0 +1,531 @@
+#include <algorithm>
+#include <array>
+#include <cmath>
+#include <cstdint>
+#include <limits>
+#include <type_traits>
+
+#include <catch2/catch_approx.hpp>
+#include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace angular_momentum_test_utils {
+    [[nodiscard]] mean_field::operators::AngularMomentumDependencies makeDependencies() {
+        return {
+            .discretization = {.identity = 15013, .revision = 3},
+            .density = {.identity = 15017, .revision = 5},
+            .displacement = {.identity = 15031, .revision = 7},
+            .rotation = {.identity = 15053, .revision = 11}
+        };
+    }
+
+    [[nodiscard]] mean_field::operators::context::gravity_field::GravityFieldRevisions makeGravityRevisions(
+        const mean_field::operators::AngularMomentumDependencies &dependencies,
+        const std::uint64_t gravityGradientRevision = 13,
+        const std::uint64_t gravityPotentialRevision = 17
+    ) {
+        return {
+            .discretization = {.value = dependencies.discretization.revision},
+            .displacement = {.value = dependencies.displacement.revision},
+            .density = {.value = dependencies.density.revision},
+            .gravity_gradient = {.value = gravityGradientRevision},
+            .gravity_potential = {.value = gravityPotentialRevision}
+        };
+    }
+
+    void prepareGravityContext(
+        mean_field::operators::context::gravity_field::GravityFieldLinearizationContext &context,
+        const mean_field::fem::FEM &finiteElements,
+        const mfem::Vector &density,
+        const mfem::Vector &displacement,
+        const mean_field::operators::AngularMomentumDependencies &dependencies,
+        const std::uint64_t gravityGradientRevision = 13,
+        const std::uint64_t gravityPotentialRevision = 17
+    ) {
+        mfem::Vector gravityGradient(finiteElements.gravityFluxFes->GetTrueVSize());
+        mfem::Vector gravityPotential(finiteElements.gravityPotentialFes->GetTrueVSize());
+        gravityGradient = 0.0;
+        gravityPotential = 0.0;
+        context.Prepare(
+            {.density = context.GetDensityMap().gather(density),
+             .displacement = context.GetDisplacementMap().gather(displacement),
+             .gravity_gradient = context.GetGravityGradientMap().gather(gravityGradient),
+             .gravity_potential = context.GetGravityPotentialMap().gather(gravityPotential)},
+            makeGravityRevisions(dependencies, gravityGradientRevision, gravityPotentialRevision)
+        );
+    }
+
+    [[nodiscard]] mfem::Vector projectDensity(
+        const mean_field::fem::FEM &finiteElements,
+        const double phase
+    ) {
+        mfem::ParGridFunction field(finiteElements.densityFes.get());
+        mfem::FunctionCoefficient coefficient([phase](const mfem::Vector &position) {
+            return 0.94 + 0.08 * std::sin(0.71 * position(0) + phase) +
+                   0.05 * std::cos(0.63 * position(1) - phase) + 0.03 * position(2) * position(2);
+        });
+        field.ProjectCoefficient(coefficient);
+        mfem::Vector result;
+        field.GetTrueDofs(result);
+        return result;
+    }
+
+    [[nodiscard]] mfem::Vector projectDensityDirection(
+        const mean_field::fem::FEM &finiteElements,
+        const double phase
+    ) {
+        mfem::ParGridFunction field(finiteElements.densityFes.get());
+        mfem::FunctionCoefficient coefficient([phase](const mfem::Vector &position) {
+            return 0.17 * std::sin(0.83 * position(0) + phase) -
+                   0.12 * std::cos(0.79 * position(1) - phase) + 0.06 * position(2);
+        });
+        field.ProjectCoefficient(coefficient);
+        mfem::Vector result;
+        field.GetTrueDofs(result);
+        return result;
+    }
+
+    [[nodiscard]] mfem::Vector projectConstantDensity(
+        const mean_field::fem::FEM &finiteElements,
+        const double value,
+        mfem::ParGridFunction *fieldOutput = nullptr
+    ) {
+        mfem::ParGridFunction field(finiteElements.densityFes.get());
+        mfem::ConstantCoefficient coefficient(value);
+        field.ProjectCoefficient(coefficient);
+        if (fieldOutput != nullptr) {
+            *fieldOutput = field;
+        }
+        mfem::Vector result;
+        field.GetTrueDofs(result);
+        return result;
+    }
+
+    [[nodiscard]] mfem::Vector projectAffineDisplacement(
+        const mean_field::fem::FEM &finiteElements,
+        const double scale
+    ) {
+        mfem::ParGridFunction field(finiteElements.displacementFes.get());
+        mfem::VectorFunctionCoefficient coefficient(
+            finiteElements.mesh->Dimension(),
+            [scale](const mfem::Vector &position, mfem::Vector &value) {
+                value.SetSize(position.Size());
+                for (int component = 0; component < position.Size(); ++component) {
+                    value(component) = scale * position(component);
+                }
+            }
+        );
+        field.ProjectCoefficient(coefficient);
+        mfem::Vector result;
+        field.GetTrueDofs(result);
+        return result;
+    }
+
+    [[nodiscard]] mfem::Vector projectDisplacementDirection(
+        const mean_field::fem::FEM &finiteElements,
+        const double scale
+    ) {
+        mfem::ParGridFunction field(finiteElements.displacementFes.get());
+        mfem::VectorFunctionCoefficient coefficient(
+            finiteElements.mesh->Dimension(),
+            [scale](const mfem::Vector &position, mfem::Vector &value) {
+                value.SetSize(3);
+                value(0) = scale * (0.07 * position(0) + 0.018 * position(1) * position(2));
+                value(1) = scale * (-0.05 * position(1) + 0.013 * position(0) * position(2));
+                value(2) = scale * (0.04 * position(2) - 0.011 * position(0) * position(1));
+            }
+        );
+        field.ProjectCoefficient(coefficient);
+        mfem::Vector result;
+        field.GetTrueDofs(result);
+        return result;
+    }
+
+    [[nodiscard]] double residual(const mean_field::operators::PreparedAngularMomentumOperator &operation) {
+        mfem::Vector value;
+        operation.BuildResidual(value);
+        REQUIRE(value.Size() == 1);
+        return value(0);
+    }
+
+    [[nodiscard]] double relativeError(const double actual, const double expected) {
+        return std::abs(actual - expected) /
+               std::max({std::abs(actual), std::abs(expected), 100.0 * std::numeric_limits<double>::epsilon()});
+    }
+} // namespace angular_momentum_test_utils
+
+TEST_CASE(
+    "Prepared Angular Momentum Satisfies Moment Scaling And The Parallel Axis Theorem",
+    "[fixed-angular-momentum][physics][analytic]"
+) {
+    using namespace mean_field;
+    using Catch::Approx;
+    using Operator = operators::PreparedAngularMomentumOperator;
+
+    STATIC_CHECK_FALSE(std::is_copy_constructible_v<Operator>);
+    STATIC_CHECK_FALSE(std::is_move_constructible_v<Operator>);
+
+    utils::Args arguments = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(arguments.mesh_file, arguments, 0);
+    REQUIRE(finiteElements.okay());
+
+    constexpr double densityValue = 1.37;
+    constexpr double angularVelocity = 0.73;
+    constexpr double targetAngularMomentum = 0.41;
+    mfem::ParGridFunction densityField(finiteElements.densityFes.get());
+    const mfem::Vector density = angular_momentum_test_utils::projectConstantDensity(
+        finiteElements,
+        densityValue,
+        &densityField
+    );
+    mfem::Vector displacement(finiteElements.displacementFes->GetTrueVSize());
+    displacement = 0.0;
+    finiteElements.displacement->SetFromTrueDofs(displacement);
+
+    auto dependencies = angular_momentum_test_utils::makeDependencies();
+    operators::context::gravity_field::GravityFieldLinearizationContext gravityContext(
+        finiteElements,
+        *finiteElements.domainMapperStateless
+    );
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        displacement,
+        dependencies
+    );
+
+    const models::CompiledFixedAngularMomentum originConstraint = models::compileConstraint(
+        integral::FixedAngularMomentum({
+            .Jtotal = dimensions::AngularMomentumValue{targetAngularMomentum},
+            .axis = {0.0, 0.0, 4.0}
+        })
+    );
+    Operator origin(
+        finiteElements,
+        *finiteElements.domainMapperStateless,
+        gravityContext,
+        originConstraint
+    );
+    const auto initial = origin.Prepare(angularVelocity, dependencies);
+    CHECK(initial.rebuiltStaticPlan);
+    CHECK(initial.refreshedGeometry);
+    CHECK(initial.refreshedDensity);
+    CHECK(initial.updatedAngularVelocity);
+    CHECK(initial.assembledResidual);
+
+    const double independentMoment = analysis::get_moment_of_inertia(finiteElements, densityField);
+    CHECK(angular_momentum_test_utils::relativeError(origin.GetMomentOfInertia(), independentMoment) < 2.0e-13);
+    CHECK(origin.GetCurrentAngularMomentum() ==
+          Approx(angularVelocity * origin.GetMomentOfInertia()).epsilon(2.0e-15));
+    CHECK(angular_momentum_test_utils::residual(origin) ==
+          Approx(angularVelocity * origin.GetMomentOfInertia() - targetAngularMomentum).epsilon(2.0e-15));
+
+    const auto report = origin.GetConstraintReport();
+    CHECK(report.targetAngularMomentum == targetAngularMomentum);
+    CHECK(report.achievedAngularMomentum == origin.GetCurrentAngularMomentum());
+    CHECK(report.momentOfInertia == origin.GetMomentOfInertia());
+    CHECK(report.angularVelocity == angularVelocity);
+    const physics::RigidRotation rotation = origin.GetRotation();
+    CHECK(rotation.angular_velocity()(0) == 0.0);
+    CHECK(rotation.angular_velocity()(1) == 0.0);
+    CHECK(rotation.angular_velocity()(2) == angularVelocity);
+
+    constexpr double affineScale = 0.086;
+    const mfem::Vector affineDisplacement =
+        angular_momentum_test_utils::projectAffineDisplacement(finiteElements, affineScale);
+    ++dependencies.displacement.revision;
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        affineDisplacement,
+        dependencies
+    );
+    const auto affine = origin.Prepare(angularVelocity, dependencies);
+    CHECK(affine.refreshedGeometry);
+    CHECK_FALSE(affine.refreshedDensity);
+    const double expectedAffineRatio = std::pow(1.0 + affineScale, 5);
+    const double measuredAffineRatio = origin.GetMomentOfInertia() / independentMoment;
+    INFO("Expected homothetic I ratio = " << expectedAffineRatio);
+    INFO("Measured homothetic I ratio = " << measuredAffineRatio);
+    CHECK(angular_momentum_test_utils::relativeError(measuredAffineRatio, expectedAffineRatio) < 7.0e-7);
+
+    displacement = 0.0;
+    ++dependencies.displacement.revision;
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        displacement,
+        dependencies
+    );
+    origin.Prepare(angularVelocity, dependencies);
+    constexpr std::array<double, 3> shiftedCenter{0.27, -0.19, 0.31};
+    Operator shifted(
+        finiteElements,
+        *finiteElements.domainMapperStateless,
+        gravityContext,
+        models::compileConstraint(integral::FixedAngularMomentum({
+            .Jtotal = dimensions::AngularMomentumValue{targetAngularMomentum},
+            .axis = {0.0, 0.0, 1.0},
+            .center = shiftedCenter
+        }))
+    );
+    shifted.Prepare(angularVelocity, dependencies);
+
+    const double mass = analysis::domain_integrate_grid_function(
+        finiteElements,
+        densityField,
+        utils::DOMAINS::STELLAR,
+        mapping::COORDINATE_SPACE::PHYSICAL
+    );
+    const mfem::Vector centerOfMass = analysis::get_com(finiteElements, densityField);
+    const double expectedShiftedMoment = origin.GetMomentOfInertia() +
+                                         mass * (shiftedCenter[0] * shiftedCenter[0] +
+                                                 shiftedCenter[1] * shiftedCenter[1]) -
+                                         2.0 * mass * (shiftedCenter[0] * centerOfMass(0) +
+                                                       shiftedCenter[1] * centerOfMass(1));
+    CHECK(angular_momentum_test_utils::relativeError(shifted.GetMomentOfInertia(), expectedShiftedMoment) < 3.0e-13);
+}
+
+TEST_CASE(
+    "Prepared Angular Momentum Jacobian Matches Density Geometry And Angular Velocity Differences",
+    "[fixed-angular-momentum][jacobian][accuracy]"
+) {
+    using namespace mean_field;
+
+    utils::Args arguments = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(arguments.mesh_file, arguments, 0);
+    REQUIRE(finiteElements.okay());
+
+    const mfem::Vector density = angular_momentum_test_utils::projectDensity(finiteElements, 0.31);
+    const mfem::Vector densityDirection =
+        angular_momentum_test_utils::projectDensityDirection(finiteElements, 0.67);
+    const mfem::Vector displacement =
+        angular_momentum_test_utils::projectDisplacementDirection(finiteElements, 0.43);
+    const mfem::Vector displacementDirection =
+        angular_momentum_test_utils::projectDisplacementDirection(finiteElements, -0.79);
+    constexpr double angularVelocity = 0.63;
+    constexpr double angularVelocityDirection = -0.37;
+
+    auto dependencies = angular_momentum_test_utils::makeDependencies();
+    operators::context::gravity_field::GravityFieldLinearizationContext gravityContext(
+        finiteElements,
+        *finiteElements.domainMapperStateless
+    );
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        displacement,
+        dependencies
+    );
+    operators::PreparedAngularMomentumOperator operation(
+        finiteElements,
+        *finiteElements.domainMapperStateless,
+        gravityContext,
+        models::compileConstraint(
+            integral::FixedAngularMomentum({.Jtotal = dimensions::AngularMomentumValue{0.81}})
+        )
+    );
+    operation.Prepare(angularVelocity, dependencies);
+
+    const mfem::Vector reducedDensityDirection = gravityContext.GetDensityMap().gather(densityDirection);
+    const mfem::Vector reducedDisplacementDirection =
+        gravityContext.GetDisplacementMap().gather(displacementDirection);
+    mfem::Vector densityAction;
+    mfem::Vector geometryAction;
+    mfem::Vector angularVelocityAction;
+    mfem::Vector completeAction;
+    operation.ApplyDensityJacobianAction(reducedDensityDirection, densityAction);
+    operation.ApplyDisplacementJacobianAction(reducedDisplacementDirection, geometryAction);
+    operation.ApplyAngularVelocityJacobianAction(angularVelocityDirection, angularVelocityAction);
+    operation.ApplyCompleteJacobianAction(
+        reducedDensityDirection,
+        reducedDisplacementDirection,
+        angularVelocityDirection,
+        completeAction
+    );
+    CHECK(angular_momentum_test_utils::relativeError(
+              completeAction(0),
+              densityAction(0) + geometryAction(0) + angularVelocityAction(0)
+          ) < 3.0e-15);
+    CHECK(angularVelocityAction(0) ==
+          Catch::Approx(operation.GetMomentOfInertia() * angularVelocityDirection).epsilon(2.0e-15));
+
+    constexpr double angularStep = 1.0e-6;
+    ++dependencies.rotation.revision;
+    operation.Prepare(angularVelocity + angularStep * angularVelocityDirection, dependencies);
+    const double angularPlus = angular_momentum_test_utils::residual(operation);
+    ++dependencies.rotation.revision;
+    operation.Prepare(angularVelocity - angularStep * angularVelocityDirection, dependencies);
+    const double angularMinus = angular_momentum_test_utils::residual(operation);
+    const double angularDifference = (angularPlus - angularMinus) / (2.0 * angularStep);
+    CHECK(angular_momentum_test_utils::relativeError(angularVelocityAction(0), angularDifference) < 2.0e-10);
+
+    constexpr double densityStep = 1.0e-3;
+    mfem::Vector densityPlus(density);
+    densityPlus.Add(densityStep, densityDirection);
+    ++dependencies.density.revision;
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        densityPlus,
+        displacement,
+        dependencies
+    );
+    operation.Prepare(angularVelocity, dependencies);
+    const double densityPlusResidual = angular_momentum_test_utils::residual(operation);
+    mfem::Vector densityMinus(density);
+    densityMinus.Add(-densityStep, densityDirection);
+    ++dependencies.density.revision;
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        densityMinus,
+        displacement,
+        dependencies
+    );
+    operation.Prepare(angularVelocity, dependencies);
+    const double densityMinusResidual = angular_momentum_test_utils::residual(operation);
+    const double densityDifference = (densityPlusResidual - densityMinusResidual) / (2.0 * densityStep);
+    CHECK(angular_momentum_test_utils::relativeError(densityAction(0), densityDifference) < 4.0e-8);
+
+    constexpr double geometryStep = 1.0e-6;
+    mfem::Vector displacementPlus(displacement);
+    displacementPlus.Add(geometryStep, displacementDirection);
+    ++dependencies.density.revision;
+    ++dependencies.displacement.revision;
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        displacementPlus,
+        dependencies
+    );
+    operation.Prepare(angularVelocity, dependencies);
+    const double geometryPlusResidual = angular_momentum_test_utils::residual(operation);
+    mfem::Vector displacementMinus(displacement);
+    displacementMinus.Add(-geometryStep, displacementDirection);
+    ++dependencies.displacement.revision;
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        displacementMinus,
+        dependencies
+    );
+    operation.Prepare(angularVelocity, dependencies);
+    const double geometryMinusResidual = angular_momentum_test_utils::residual(operation);
+    const double geometryDifference = (geometryPlusResidual - geometryMinusResidual) / (2.0 * geometryStep);
+    INFO("Density angular-momentum derivative error = " <<
+         angular_momentum_test_utils::relativeError(densityAction(0), densityDifference));
+    INFO("Geometry angular-momentum derivative error = " <<
+         angular_momentum_test_utils::relativeError(geometryAction(0), geometryDifference));
+    CHECK(angular_momentum_test_utils::relativeError(geometryAction(0), geometryDifference) < 4.0e-7);
+}
+
+TEST_CASE(
+    "Prepared Angular Momentum Refreshes Only Changed Runtime Data",
+    "[fixed-angular-momentum][prepared][lifecycle]"
+) {
+    using namespace mean_field;
+
+    utils::Args arguments = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(arguments.mesh_file, arguments, 0);
+    REQUIRE(finiteElements.okay());
+    mfem::Vector density = angular_momentum_test_utils::projectDensity(finiteElements, 0.29);
+    mfem::Vector displacement =
+        angular_momentum_test_utils::projectDisplacementDirection(finiteElements, 0.41);
+    auto dependencies = angular_momentum_test_utils::makeDependencies();
+    std::uint64_t gravityPotentialRevision = 17;
+
+    operators::context::gravity_field::GravityFieldLinearizationContext gravityContext(
+        finiteElements,
+        *finiteElements.domainMapperStateless
+    );
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        displacement,
+        dependencies,
+        13,
+        gravityPotentialRevision
+    );
+    operators::PreparedAngularMomentumOperator operation(
+        finiteElements,
+        *finiteElements.domainMapperStateless,
+        gravityContext,
+        models::compileConstraint(
+            integral::FixedAngularMomentum({.Jtotal = dimensions::AngularMomentumValue{0.71}})
+        )
+    );
+    operation.Prepare(0.52, dependencies);
+    const auto preparationCount = operation.GetPreparationCount();
+    const double moment = operation.GetMomentOfInertia();
+
+    const auto repeated = operation.Prepare(0.52, dependencies);
+    CHECK_FALSE(repeated.DidAnyWork());
+    CHECK(operation.GetPreparationCount() == preparationCount);
+
+    ++gravityPotentialRevision;
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        displacement,
+        dependencies,
+        13,
+        gravityPotentialRevision
+    );
+    const auto unrelatedPotential = operation.Prepare(0.52, dependencies);
+    CHECK_FALSE(unrelatedPotential.DidAnyWork());
+
+    const double residualBeforeRotation = angular_momentum_test_utils::residual(operation);
+    ++dependencies.rotation.revision;
+    const auto rotationOnly = operation.Prepare(0.81, dependencies);
+    CHECK(rotationOnly.updatedAngularVelocity);
+    CHECK(rotationOnly.assembledResidual);
+    CHECK_FALSE(rotationOnly.refreshedDensity);
+    CHECK_FALSE(rotationOnly.refreshedGeometry);
+    CHECK(operation.GetMomentOfInertia() == moment);
+    CHECK(angular_momentum_test_utils::residual(operation) - residualBeforeRotation ==
+          Catch::Approx((0.81 - 0.52) * moment).epsilon(3.0e-15));
+
+    density = angular_momentum_test_utils::projectDensity(finiteElements, 0.83);
+    ++dependencies.density.revision;
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        displacement,
+        dependencies,
+        13,
+        gravityPotentialRevision
+    );
+    const auto densityOnly = operation.Prepare(0.81, dependencies);
+    CHECK(densityOnly.refreshedDensity);
+    CHECK_FALSE(densityOnly.refreshedGeometry);
+    CHECK_FALSE(densityOnly.updatedAngularVelocity);
+
+    displacement = angular_momentum_test_utils::projectDisplacementDirection(finiteElements, 0.87);
+    ++dependencies.displacement.revision;
+    angular_momentum_test_utils::prepareGravityContext(
+        gravityContext,
+        finiteElements,
+        density,
+        displacement,
+        dependencies,
+        13,
+        gravityPotentialRevision
+    );
+    const auto geometryOnly = operation.Prepare(0.81, dependencies);
+    CHECK(geometryOnly.refreshedGeometry);
+    CHECK_FALSE(geometryOnly.refreshedDensity);
+    CHECK_FALSE(geometryOnly.updatedAngularVelocity);
+}

tests/operators/prepared_central_density_stellar_equilibrium.cpp

@@ -44,20 +44,11 @@ namespace {
 } // namespace
 
 TEST_CASE(
-    "Central Density Bordered Root Preserves The Physical Operator Prefix",
+    "Central Density Contribution Composes Through The Variadic Root",
     tags::central_density_phase_integration
 ) {
     using namespace mean_field;
 
-    STATIC_CHECK_FALSE(
-        std::same_as<
-            operators::PreparedStellarEquilibriumOperator, operators::PreparedCentralDensityStellarEquilibriumOperator>
-    );
-    STATIC_CHECK(
-        operators::CentralDensityStellarEquilibriumSpecificationModel::compilationClass ==
-        models::ModelCompilationClass::isolated_root
-    );
-
     utils::Args args = test_utils::setup_args();
     fem::FEM f       = fem::setup_fem(args.mesh_file, args, 0);
     REQUIRE(f.okay());
@@ -80,7 +71,8 @@ TEST_CASE(
     STATIC_CHECK(
         std::same_as<
             typename std::remove_cvref_t<decltype(equilibriumProblem)>::PreparedOperatorType,
-            operators::PreparedCentralDensityStellarEquilibriumOperator>
+            operators::PreparedVariadicStellarEquilibriumOperator<
+                typename std::remove_cvref_t<decltype(equilibriumProblem)>::ModelType>>
     );
     CHECK(
         equilibriumProblem.GetStellarModel().specification<constraint::FixedCentralDensity>().targetDensity() ==
@@ -95,19 +87,21 @@ TEST_CASE(
     CHECK(borderedOperator.GetRootManifest().constraints().size() == 3);
     CHECK(borderedOperator.GetRootManifest().specificationDescriptors().size() == 4);
 
-    const auto constraints = borderedOperator.GetRootManifest().constraints();
+    const auto &centralDescriptor =
-    CHECK(constraints[2].stableId == "FixedCentralDensity");
+        borderedOperator.GetRootManifest().specification<constraint::FixedCentralDensity>();
-    CHECK(constraints[2].role == models::SpecificationRole::phase_condition);
+    CHECK(centralDescriptor.stableId == "FixedCentralDensity");
-    CHECK(constraints[2].columnPolicy == operators::RootColumnPolicy::solver_border);
+    CHECK(centralDescriptor.role == models::SpecificationRole::phase_condition);
-    CHECK(constraints[2].target == 1.0);
+    CHECK(centralDescriptor.columnPolicy == operators::RootColumnPolicy::solver_border);
-    REQUIRE(constraints[2].carrierTarget.has_value());
+    CHECK(centralDescriptor.target == 1.0);
-    CHECK(*constraints[2].carrierTarget == 1.0);
+    REQUIRE(centralDescriptor.carrierTarget.has_value());
-    CHECK(constraints[2].targetUnits == "density");
+    CHECK(*centralDescriptor.carrierTarget == 1.0);
-    CHECK(constraints[2].residualUnits == "specific_enthalpy");
+    CHECK(centralDescriptor.targetUnits == "density");
+    CHECK(centralDescriptor.residualUnits == "specific_enthalpy");
 
     mfem::Vector physicalState(physicalOperator.Width());
     physicalState                                                   = 0.0;
-    const auto physicalStateView                                    = physicalOperator.GetRootStateView(physicalState);
+    const auto physicalStateView =
+        physicalOperator.GetRootManifest().stateView(physicalState);
     physicalStateView.block(utils::blocks::density_field.mass_term) = 1.0;
     physicalStateView.block(utils::blocks::enthalpy_field.specific_term) = 1.0;
 
@@ -118,10 +112,9 @@ TEST_CASE(
     const operators::StellarEquilibriumDependencies dependencies = make_dependencies();
     const physics::RigidRotation rotation                        = make_zero_rotation();
     physicalOperator.Prepare(physicalState, dependencies, rotation);
-    const operators::PreparedCentralDensityStellarEquilibriumReport initialReport =
+    const auto initialReport = equilibriumProblem.Prepare(borderedState, dependencies, rotation);
-        equilibriumProblem.Prepare(borderedState, dependencies, rotation);
     CHECK(initialReport.physical.assembledResidual);
-    CHECK(initialReport.phase.assembledResidual);
+    CHECK(initialReport.specification<constraint::FixedCentralDensity>().constraint.assembledResidual);
     CHECK(initialReport.assembledResidual);
 
     mfem::Vector physicalResidual;
@@ -164,13 +157,13 @@ TEST_CASE(
 
     const auto repeatedReport = borderedOperator.Prepare(borderedState, dependencies, rotation);
     CHECK_FALSE(repeatedReport.physical.DidAnyWork());
-    CHECK_FALSE(repeatedReport.phase.DidAnyWork());
+    CHECK_FALSE(repeatedReport.specification<constraint::FixedCentralDensity>().DidAnyWork());
-    CHECK_FALSE(repeatedReport.assembledResidual);
+    CHECK(repeatedReport.assembledResidual);
 
     borderedState(borderedState.Size() - 1) = 0.375;
     const auto borderReport                 = borderedOperator.Prepare(borderedState, dependencies, rotation);
     CHECK_FALSE(borderReport.physical.DidAnyWork());
-    CHECK(borderReport.phase.refreshedBorder);
+    CHECK(borderReport.specification<constraint::FixedCentralDensity>().constraint.refreshedBorder);
     CHECK(borderReport.assembledResidual);
 
     mfem::Vector borderOnlyDirection(borderedOperator.Width());

tests/operators/stellar_equilibrium_system.cpp

@@ -1,6 +1,7 @@
 #include <algorithm>
 #include <cmath>
 #include <concepts>
+#include <numbers>
 #include <type_traits>
 #include <utility>
 
@@ -10,6 +11,83 @@
 import mean_field;
 import test_helpers;
 
+namespace outer_manifest_report_test {
+    template <mean_field::model::StellarModelType Model>
+    class PreparedEarlierMultiplier;
+
+    class EarlierMultiplier final {
+    public:
+        struct Parameters final {
+            mean_field::dimensions::SpecificEnergyValue target;
+        };
+
+        using ModelDefinition = mean_field::integral::FixedWithMultiplier<
+            EarlierMultiplier,
+            "AardvarkOuterManifestMultiplier",
+            mean_field::models::DependsOn<mean_field::models::stellar::state::Density>,
+            mean_field::models::Affects<mean_field::models::stellar::equation::HydrostaticBalance>,
+            mean_field::models::GlobalScalarNormalization<
+                mean_field::models::PhysicalScaleLaw::specific_energy,
+                mean_field::models::PhysicalScaleLaw::specific_energy>,
+            mean_field::models::GeneratedManifest<
+                "aardvark_outer_manifest.value",
+                "a",
+                "aardvark_outer_manifest.residual",
+                "R_a",
+                "specific_energy",
+                "specific_energy">>;
+        using EquilibriumPhysics =
+            mean_field::operators::SpecificationEquilibriumPhysics<
+                PreparedEarlierMultiplier>;
+
+        explicit EarlierMultiplier(const Parameters parameters) noexcept
+            : m_target(parameters.target) {
+        }
+
+        [[nodiscard]] mean_field::dimensions::SpecificEnergyValue target() const noexcept {
+            return m_target;
+        }
+
+    private:
+        mean_field::dimensions::SpecificEnergyValue m_target;
+    };
+
+    template <mean_field::model::StellarModelType Model>
+    class PreparedEarlierMultiplier final {
+    public:
+        using Report = mean_field::operators::EmptySpecificationPreparationReport;
+
+        explicit PreparedEarlierMultiplier(const EarlierMultiplier &) noexcept {
+        }
+
+        template <typename StateView>
+        [[nodiscard]] Report PrepareAfterPhysical(const StateView &) noexcept {
+            return {};
+        }
+
+        template <typename Equation, typename Row>
+        [[nodiscard]] mean_field::stellar::StructuralZero AddResidual(
+            Equation,
+            Row &
+        ) const noexcept {
+            return mean_field::stellar::structuralZero;
+        }
+
+        template <typename Equation, typename State, typename Direction, typename Row>
+        [[nodiscard]] mean_field::stellar::StructuralZero AddJacobianAction(
+            mean_field::stellar::Derivative<Equation, State>,
+            const Direction &,
+            Row &
+        ) const noexcept {
+            return mean_field::stellar::zeroDerivative;
+        }
+
+        [[nodiscard]] bool IsPrepared() const noexcept {
+            return true;
+        }
+    };
+} // namespace outer_manifest_report_test
+
 namespace {
     using BaseModel           = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
         mean_field::eos::Polytrope,
@@ -22,9 +100,30 @@ namespace {
         mean_field::integral::FixedTotalMass,
         mean_field::constraint::FixedCentralDensity>>;
 
+    using AngularMomentumModel = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::integral::FixedTotalMass,
+        mean_field::integral::FixedAngularMomentum>>;
+
+    using AngularMomentumCentralDensityModel =
+        mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+            mean_field::eos::Polytrope,
+            mean_field::surface::Isobaric,
+            mean_field::integral::FixedTotalMass,
+            mean_field::integral::FixedAngularMomentum,
+            mean_field::constraint::FixedCentralDensity>>;
+
     using IncompleteModel =
         mean_field::model::StellarModel<mean_field::models::SpecificationSet<mean_field::eos::Polytrope>>;
 
+    using EarlierMultiplierModel =
+        mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+            mean_field::eos::Polytrope,
+            mean_field::surface::Isobaric,
+            outer_manifest_report_test::EarlierMultiplier,
+            mean_field::integral::FixedTotalMass>>;
+
     template <typename Candidate>
     concept HasLegacyNumericalModelAdapter = requires { typename Candidate::NumericalModelAdapter; };
 
@@ -58,6 +157,7 @@ namespace {
         difference -= right;
         return difference.Norml2() / std::max({1.0, left.Norml2(), right.Norml2()});
     }
+
 } // namespace
 
 TEST_CASE(
@@ -68,26 +168,65 @@ TEST_CASE(
 
     using BaseProblem           = equilibrium::StellarEquilibriumProblem<BaseModel>;
     using CentralDensityProblem = equilibrium::StellarEquilibriumProblem<CentralDensityModel>;
+    using AngularMomentumProblem = equilibrium::StellarEquilibriumProblem<AngularMomentumModel>;
+    using AngularMomentumCentralDensityProblem =
+        equilibrium::StellarEquilibriumProblem<AngularMomentumCentralDensityModel>;
 
     STATIC_CHECK(equilibrium::StellarEquilibriumModel<BaseModel>);
     STATIC_CHECK(equilibrium::StellarEquilibriumModel<CentralDensityModel>);
+    STATIC_CHECK(equilibrium::StellarEquilibriumModel<AngularMomentumModel>);
+    STATIC_CHECK(equilibrium::StellarEquilibriumModel<AngularMomentumCentralDensityModel>);
+    STATIC_CHECK(equilibrium::StellarEquilibriumModel<EarlierMultiplierModel>);
     STATIC_CHECK_FALSE(equilibrium::StellarEquilibriumModel<IncompleteModel>);
+    STATIC_CHECK_FALSE(operators::StellarEquilibriumRuntimeContribution<
+                       outer_manifest_report_test::EarlierMultiplier>::registered);
+    STATIC_CHECK_FALSE(operators::stellarEquilibriumBackendRuntimeAuthorized<
+                       outer_manifest_report_test::EarlierMultiplier,
+                       EarlierMultiplierModel>);
+    STATIC_CHECK(operators::StellarEquilibriumPhysicsAvailableFor<
+                 outer_manifest_report_test::EarlierMultiplier,
+                 EarlierMultiplierModel>);
     STATIC_CHECK_FALSE(std::same_as<BaseProblem, CentralDensityProblem>);
     STATIC_CHECK(BaseProblem::symbolicallySquare);
     STATIC_CHECK(CentralDensityProblem::symbolicallySquare);
     STATIC_CHECK_FALSE(BaseProblem::hasFixedCentralDensity);
     STATIC_CHECK(CentralDensityProblem::hasFixedCentralDensity);
+    STATIC_CHECK(AngularMomentumProblem::hasFixedAngularMomentum);
+    STATIC_CHECK_FALSE(AngularMomentumProblem::hasFixedCentralDensity);
+    STATIC_CHECK(AngularMomentumCentralDensityProblem::hasFixedAngularMomentum);
+    STATIC_CHECK(AngularMomentumCentralDensityProblem::hasFixedCentralDensity);
     STATIC_CHECK_FALSE(HasLegacyNumericalModelAdapter<BaseProblem>);
     STATIC_CHECK_FALSE(HasLegacyNumericalModelAdapter<CentralDensityProblem>);
     STATIC_CHECK(std::same_as<BaseProblem, equilibrium::StellarEquilibriumSystem<BaseModel>>);
     STATIC_CHECK(
-        std::same_as<typename BaseProblem::PreparedOperatorType, operators::PreparedStellarEquilibriumOperator>
+        std::same_as<
+            typename BaseProblem::PreparedOperatorType,
+            operators::PreparedVariadicStellarEquilibriumOperator<BaseModel>>
     );
     STATIC_CHECK(
         std::same_as<
             typename CentralDensityProblem::PreparedOperatorType,
-            operators::PreparedCentralDensityStellarEquilibriumOperator>
+            operators::PreparedVariadicStellarEquilibriumOperator<CentralDensityModel>>
     );
+    STATIC_CHECK(
+        std::same_as<
+            typename AngularMomentumProblem::PreparedOperatorType,
+            operators::PreparedVariadicStellarEquilibriumOperator<AngularMomentumModel>>
+    );
+    STATIC_CHECK_FALSE(std::same_as<
+                       typename BaseProblem::PreparedOperatorType,
+                       typename CentralDensityProblem::PreparedOperatorType>);
+    STATIC_CHECK_FALSE(std::same_as<
+                       typename AngularMomentumProblem::PreparedOperatorType,
+                       typename AngularMomentumCentralDensityProblem::PreparedOperatorType>);
+    STATIC_CHECK(AngularMomentumProblem::FormType::value_block_count == 7);
+    STATIC_CHECK(AngularMomentumCentralDensityProblem::FormType::value_block_count == 8);
+    STATIC_CHECK(std::same_as<
+                 typename BaseProblem::FormType,
+                 utils::blocks::surface_deformed_stellar_equilibrium_form>);
+    STATIC_CHECK(std::same_as<
+                 typename CentralDensityProblem::FormType,
+                 utils::blocks::central_density_bordered_stellar_equilibrium_form>);
     STATIC_CHECK(
         std::same_as<
             typename BaseProblem::CompiledSurfaceConstraintType,
@@ -97,6 +236,183 @@ TEST_CASE(
     STATIC_CHECK(material::CompiledThermodynamicEquations<typename BaseProblem::ThermodynamicEquationsType>);
 }
 
+TEST_CASE(
+    "Fixed Angular Momentum Root Uses Its Generated Angular Velocity In Every Physical Row",
+    "[fixed-angular-momentum][stellar-equilibrium][jacobian][integration]"
+) {
+    using namespace mean_field;
+
+    utils::Args arguments = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(arguments.mesh_file, arguments, 0);
+    REQUIRE(finiteElements.okay());
+
+    constexpr double radius = utils::RADIUS;
+    constexpr double mass = utils::MASS;
+    constexpr double targetAngularMomentum = 0.1;
+    const double polytropicConstant = 2.0 * utils::G * radius * radius / std::numbers::pi_v<double>;
+    const double seedCentralDensity =
+        std::numbers::pi_v<double> * mass / (4.0 * radius * radius * radius);
+    auto model = model::StellarModel(
+        eos::Polytrope({.n = 1.0, .K = polytropicConstant}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{mass}}),
+        integral::FixedAngularMomentum({
+            .Jtotal = dimensions::AngularMomentumValue{targetAngularMomentum},
+            .axis = {0.0, 0.0, 3.0}
+        })
+    );
+    auto problem = equilibrium::discretize(model, finiteElements);
+    auto projected = seed::makeProjectedEquilibriumState(
+        problem,
+        seed::LaneEmden({
+            .centralDensity = dimensions::DensityValue{seedCentralDensity},
+            .radialSampleCount = 1024
+        })
+    );
+    auto dependencies = make_dependencies();
+
+    const auto preparation = problem.Prepare(projected.values, dependencies);
+    CHECK(preparation.generatedPhysicalControl);
+    CHECK(preparation.physical.DidAnyWork());
+    CHECK(preparation.template specification<models::FixedAngularMomentum>().constraint.DidAnyWork());
+    CHECK(preparation.template specification<models::FixedAngularMomentum>().generatedRotation);
+    CHECK(problem.IsPrepared());
+    const auto angularReport = problem.GetPreparedOperator().GetAngularMomentumReport();
+    CHECK(angularReport.targetAngularMomentum == targetAngularMomentum);
+    CHECK(angularReport.angularVelocity > 0.0);
+    CHECK(angularReport.momentOfInertia > 0.0);
+    CHECK(std::abs(angularReport.scaledResidual) < 7.0e-4);
+
+    mfem::Vector direction(problem.StateSize());
+    direction = 0.0;
+    mfem::Vector angularVelocityDirection = problem.GetManifest().stateView(direction).block(
+        utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term
+    );
+    REQUIRE(angularVelocityDirection.Size() == 1);
+    angularVelocityDirection(0) = -0.37;
+    angularVelocityDirection.SyncAliasMemory(direction);
+
+    mfem::Vector analyticAction;
+    problem.ApplyLinearization(direction, analyticAction);
+
+    constexpr double step = 1.0e-5;
+    mfem::Vector plusState(projected.values);
+    plusState.Add(step, direction);
+    problem.Prepare(plusState, dependencies);
+    mfem::Vector plusResidual;
+    problem.BuildResidual(plusResidual);
+    mfem::Vector minusState(projected.values);
+    minusState.Add(-step, direction);
+    problem.Prepare(minusState, dependencies);
+    mfem::Vector minusResidual;
+    problem.BuildResidual(minusResidual);
+    plusResidual -= minusResidual;
+    plusResidual /= 2.0 * step;
+
+    auto analyticView = problem.GetManifest().residualView(analyticAction);
+    auto differenceView = problem.GetManifest().residualView(plusResidual);
+    const auto blockError = [&](const auto &term) {
+        const mfem::Vector analytic = analyticView.block(term);
+        const mfem::Vector difference = differenceView.block(term);
+        return relative_difference(analytic, difference);
+    };
+
+    const double surfaceError = blockError(utils::blocks::surface_deformation_field.shape_equilibrium_term);
+    const double enthalpyError = blockError(utils::blocks::enthalpy_field.specific_term);
+    const double angularMomentumError =
+        blockError(utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term);
+    INFO("Generated-Omega surface-row centered-difference error = " << surfaceError);
+    INFO("Generated-Omega hydrostatic-row centered-difference error = " << enthalpyError);
+    INFO("Generated-Omega invariant-row centered-difference error = " << angularMomentumError);
+    CHECK(surfaceError < 3.0e-7);
+    CHECK(enthalpyError < 3.0e-7);
+    CHECK(angularMomentumError < 3.0e-10);
+    CHECK(analyticView.block(utils::blocks::surface_deformation_field.shape_equilibrium_term).Norml2() > 0.0);
+    CHECK(analyticView.block(utils::blocks::enthalpy_field.specific_term).Norml2() > 0.0);
+    CHECK(analyticView.block(utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term).Norml2() > 0.0);
+    CHECK(analyticView.block(utils::blocks::gravity_field.gradient_term).Norml2() == 0.0);
+    CHECK(analyticView.block(utils::blocks::gravity_field.poisson_term).Norml2() == 0.0);
+    CHECK(analyticView.block(utils::blocks::density_field.mass_term).Norml2() == 0.0);
+    CHECK(analyticView.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term).Norml2() == 0.0);
+
+    auto zeroModel = model::StellarModel(
+        eos::Polytrope({.n = 1.0, .K = polytropicConstant}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{mass}}),
+        integral::FixedAngularMomentum({.Jtotal = dimensions::AngularMomentumValue{0.0}})
+    );
+    auto zeroProblem = equilibrium::discretize(zeroModel, finiteElements);
+    mfem::Vector zeroState(projected.values);
+    zeroProblem.GetManifest().stateView(zeroState).block(
+        utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term
+    ) = 0.0;
+    zeroProblem.Prepare(zeroState, dependencies);
+    mfem::Vector zeroAction;
+    zeroProblem.ApplyLinearization(direction, zeroAction);
+    auto zeroView = zeroProblem.GetManifest().residualView(zeroAction);
+    CHECK(zeroView.block(utils::blocks::surface_deformation_field.shape_equilibrium_term).Norml2() == 0.0);
+    CHECK(zeroView.block(utils::blocks::enthalpy_field.specific_term).Norml2() == 0.0);
+    CHECK(zeroView.block(utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term).Norml2() > 0.0);
+}
+
+TEST_CASE(
+    "Fixed Mass Reports Use The Inferred Outer Manifest Indices",
+    "[stellar-equilibrium][manifest][runtime][ordering]"
+) {
+    using namespace mean_field;
+    using Form = operators::CompiledStellarEquilibriumForm<EarlierMultiplierModel>;
+    using EarlierValue = utils::blocks::generated_value_block<
+        models::MultiplierFor<outer_manifest_report_test::EarlierMultiplier>>;
+    using MassValue = utils::blocks::fixed_total_mass::mass_normalization::value;
+
+    STATIC_CHECK(utils::blocks::type_index_v<
+                     EarlierValue,
+                     typename Form::value_blocks> == 5);
+    STATIC_CHECK(utils::blocks::type_index_v<
+                     MassValue,
+                     typename Form::value_blocks> == 6);
+
+    utils::Args arguments = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(arguments.mesh_file, arguments, 0);
+    REQUIRE(finiteElements.okay());
+
+    auto model = model::StellarModel(
+        eos::Polytrope({.n = 1.0, .K = 0.25}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        outer_manifest_report_test::EarlierMultiplier({
+            .target = dimensions::SpecificEnergyValue{0.75}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.25}})
+    );
+    auto problem = equilibrium::discretize(model, finiteElements);
+
+    mfem::Vector state(problem.StateSize());
+    state = 0.0;
+    const auto stateView = problem.GetManifest().stateView(state);
+    stateView.block(utils::blocks::density_field.mass_term) = 1.0;
+    stateView.block(utils::blocks::enthalpy_field.specific_term) = 1.0;
+    stateView.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term) = 0.25;
+
+    const auto preparation = problem.Prepare(
+        state,
+        make_dependencies(),
+        make_zero_rotation()
+    );
+    REQUIRE(preparation.physical.DidAnyWork());
+
+    const auto report = problem.GetPreparedOperator().GetFixedMassReport();
+    const auto &outerDescriptor =
+        problem.GetManifest().template specification<models::FixedTotalMass>();
+    CHECK(report.descriptor.stableId == outerDescriptor.stableId);
+    CHECK(report.descriptor.valueBlock == outerDescriptor.valueBlock);
+    CHECK(report.descriptor.residualBlock == outerDescriptor.residualBlock);
+    CHECK(report.descriptor.valueBlock == 6);
+    CHECK(report.descriptor.residualBlock == 6);
+    CHECK(report.descriptor.target == 1.25);
+    CHECK(report.dimensionalResidual == report.achieved - report.descriptor.target);
+    CHECK(report.scaledResidual ==
+          report.dimensionalResidual / report.descriptor.residualScale);
+}
+
 TEST_CASE(
     "Discretized Stellar Equilibrium Problem Is Exactly Equivalent To The Legacy Construction Path",
     tags::stellar_equilibrium_problem_integration
@@ -122,6 +438,7 @@ TEST_CASE(
         discretization
     );
     auto &modelDrivenOperator = equilibriumProblem.GetPreparedOperator();
+    const auto &physicalOperator = equilibriumProblem.GetPhysicalOperator();
 
     CHECK(equilibriumProblem.StateSize() == legacyOperator.Width());
     CHECK(equilibriumProblem.EquationSize() == legacyOperator.Height());
@@ -129,16 +446,16 @@ TEST_CASE(
     CHECK(&equilibriumProblem.GetDiscretization().finiteElementModel() == &f);
     CHECK(&equilibriumProblem.GetDiscretization().domainMapper() == f.domainMapperStateless.get());
     CHECK(equilibriumProblem.GetDiscretization().isCurrent());
-    CHECK(modelDrivenOperator.GetTargetMass() == 1.25);
+    CHECK(physicalOperator.GetTargetMass() == 1.25);
-    CHECK(modelDrivenOperator.GetSurfaceConstraintOperator().GetPhysicalCondition().targetPressure == 0.0);
+    CHECK(physicalOperator.GetSurfaceConstraintOperator().GetPhysicalCondition().targetPressure == 0.0);
     CHECK(equilibriumProblem.GetCompiledSurfaceConstraint().targetPressure() == dimensions::PressureValue{0.0});
-    CHECK(modelDrivenOperator.GetDomainDeformation().matchesCurrentDiscretization());
+    CHECK(physicalOperator.GetDomainDeformation().matchesCurrentDiscretization());
     CHECK(&equilibriumProblem.GetLinearizationOperator() == &modelDrivenOperator);
-    CHECK(equilibriumProblem.GetManifest().constraints()[0].target == 1.25);
+    CHECK(equilibriumProblem.GetManifest().template specification<models::FixedTotalMass>().target == 1.25);
 
     mfem::Vector state(legacyOperator.Width());
     state                                                        = 0.0;
-    const auto stateView                                         = legacyOperator.GetRootStateView(state);
+    const auto stateView = legacyOperator.GetRootManifest().stateView(state);
     stateView.block(utils::blocks::density_field.mass_term)      = 1.0;
     stateView.block(utils::blocks::enthalpy_field.specific_term) = 1.0;
 
@@ -163,3 +480,65 @@ TEST_CASE(
     equilibriumProblem.ApplyLinearization(direction, modelDrivenAction);
     CHECK(relative_difference(modelDrivenAction, legacyAction) < 2.0e-15);
 }
+
+TEST_CASE(
+    "Fixed Angular Momentum Composes With The Optional Central Density Phase At Runtime",
+    "[fixed-angular-momentum][central-density][stellar-equilibrium][integration]"
+) {
+    using namespace mean_field;
+
+    utils::Args arguments = test_utils::setup_args();
+    fem::FEM finiteElements = fem::setup_fem(arguments.mesh_file, arguments, 0);
+    REQUIRE(finiteElements.okay());
+    auto model = model::StellarModel(
+        eos::Polytrope({.n = 1.0, .K = 0.25}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}}),
+        integral::FixedAngularMomentum({.Jtotal = dimensions::AngularMomentumValue{0.2}}),
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{1.0}})
+    );
+    auto problem = equilibrium::discretize(model, finiteElements);
+    using Problem = std::remove_cvref_t<decltype(problem)>;
+    STATIC_CHECK(Problem::FormType::value_block_count == 8);
+    STATIC_CHECK(Problem::FormType::residual_block_count == 8);
+
+    mfem::Vector state(problem.StateSize());
+    state = 0.0;
+    const auto stateView = problem.GetManifest().stateView(state);
+    stateView.block(utils::blocks::density_field.mass_term) = 1.0;
+    stateView.block(utils::blocks::enthalpy_field.specific_term) = 1.0;
+    stateView.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term) = 0.25;
+    stateView.block(utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term) = 0.4;
+    stateView.block(utils::blocks::fixed_central_density_phase.central_value_term) = 0.03;
+
+    const auto report = problem.Prepare(state, make_dependencies());
+    CHECK(report.template specification<models::FixedCentralDensity>().constraint.DidAnyWork());
+    CHECK(report.template specification<models::FixedAngularMomentum>().constraint.DidAnyWork());
+    CHECK(problem.IsPrepared());
+    CHECK(problem.StateSize() == problem.GetPhysicalOperator().Width() + 2);
+    REQUIRE(problem.GetManifest().constraints().size() == 4);
+    CHECK(problem.GetManifest().template specification<models::FixedAngularMomentum>().stableId ==
+          "FixedAngularMomentum");
+    CHECK(problem.GetManifest().template specification<models::FixedCentralDensity>().stableId ==
+          "FixedCentralDensity");
+
+    mfem::Vector residual;
+    problem.BuildResidual(residual);
+    REQUIRE(residual.Size() == problem.EquationSize());
+    const auto residualView = problem.GetManifest().residualView(residual);
+    CHECK(std::isfinite(
+        residualView.block(utils::blocks::fixed_angular_momentum_constraint.angular_velocity_term)(0)
+    ));
+    CHECK(std::isfinite(residualView.block(utils::blocks::fixed_central_density_phase.central_value_term)(0)));
+
+    mfem::Vector direction(problem.StateSize());
+    for (int index = 0; index < direction.Size(); ++index) {
+        direction(index) = 0.01 * std::sin(0.17 * static_cast<double>(index + 1));
+    }
+    mfem::Vector action;
+    problem.ApplyLinearization(direction, action);
+    REQUIRE(action.Size() == problem.EquationSize());
+    for (int index = 0; index < action.Size(); ++index) {
+        CHECK(std::isfinite(action(index)));
+    }
+}

tests/preconditioning/equilibrium_coordinates.cpp

@@ -6,6 +6,7 @@
 #include <numbers>
 #include <stdexcept>
 #include <type_traits>
+#include <utility>
 
 #include <catch2/catch_test_macros.hpp>
 #include <mfem.hpp>
@@ -56,6 +57,33 @@ namespace {
         blocks::type_list<>,
         preconditioning::IdentityOperatorCharacteristics,
         preconditioning::backend::Identity>;
+    using LifetimeModel = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::models::FixedTotalMass,
+        mean_field::models::FixedCentralDensity>>;
+    using LifetimeProblem = mean_field::equilibrium::StellarEquilibriumProblem<LifetimeModel>;
+    using LifetimeBlock = decltype(preconditioning::makePreconditioner(
+        std::declval<const LifetimeProblem &>()
+    ));
+    using LifetimePrepared = preconditioning::PreparedStellarPreconditioner<
+        LifetimeProblem,
+        LifetimeBlock>;
+
+    template <typename Problem, typename Block>
+    concept CanPrepareStellarPreconditioner = requires(const Problem &problem, Block block) {
+        preconditioning::prepare(problem, std::move(block));
+    };
+
+    template <typename Problem, typename Block>
+    concept CanPrepareStellarPreconditionerFromTemporary = requires(Block block) {
+        preconditioning::prepare(std::declval<Problem &&>(), std::move(block));
+    };
+
+    template <typename Problem, typename Block>
+    concept CanPrepareStellarPreconditionerFromConstTemporary = requires(Block block) {
+        preconditioning::prepare(std::declval<const Problem &&>(), std::move(block));
+    };
 
     [[nodiscard]] blocks::form_layout<Form> makeUnevenLayout() {
         return {
@@ -102,6 +130,12 @@ TEST_CASE(
 ) {
     STATIC_CHECK(preconditioning::EquilibriumCoordinateComponentFor<GroupedComponent, Form>);
     STATIC_CHECK_FALSE(preconditioning::EquilibriumCoordinateComponentFor<IncompleteComponent, Form>);
+    STATIC_CHECK(CanPrepareStellarPreconditioner<LifetimeProblem, LifetimeBlock>);
+    STATIC_CHECK_FALSE(CanPrepareStellarPreconditionerFromTemporary<LifetimeProblem, LifetimeBlock>);
+    STATIC_CHECK_FALSE(CanPrepareStellarPreconditionerFromConstTemporary<LifetimeProblem, LifetimeBlock>);
+    STATIC_CHECK(std::constructible_from<LifetimePrepared, const LifetimeProblem &, LifetimeBlock>);
+    STATIC_CHECK_FALSE(std::constructible_from<LifetimePrepared, LifetimeProblem &&, LifetimeBlock>);
+    STATIC_CHECK_FALSE(std::constructible_from<LifetimePrepared, const LifetimeProblem &&, LifetimeBlock>);
 
     const auto layout = makeUnevenLayout();
     preconditioning::EquilibriumPreconditionerCoordinateMap<Form, GroupedComponent> coordinates(layout);

tests/preconditioning/material_surface.cpp

@@ -2,6 +2,7 @@
 #include <array>
 #include <cmath>
 #include <concepts>
+#include <memory>
 #include <numbers>
 #include <stdexcept>
 #include <type_traits>
@@ -13,6 +14,130 @@
 import mean_field;
 import test_helpers;
 
+namespace material_surface_runtime_contract_test {
+    struct RegisteredAlternateEquationOfState final {
+        struct Parameters final { };
+
+        using ModelDefinition = mean_field::models::ConstitutiveLaw<
+            RegisteredAlternateEquationOfState,
+            "RegisteredAlternateMaterialSurfaceEquationOfState">;
+        using Relations = mean_field::eos::RelationCatalog<mean_field::eos::SpecificEnthalpyFromPressure>;
+
+        explicit RegisteredAlternateEquationOfState(Parameters) noexcept { }
+
+        [[nodiscard]] mean_field::dimensions::SpecificEnthalpyValue evaluate(
+            mean_field::eos::SpecificEnthalpyFromPressure,
+            mean_field::dimensions::PressureValue
+        ) const;
+    };
+
+    class AlternatePhysicalCore final : public mfem::Operator {
+    public:
+        using BackendSpecifications = mean_field::models::ModelTypeList<
+            RegisteredAlternateEquationOfState,
+            mean_field::surface::Isobaric,
+            mean_field::models::FixedTotalMass>;
+
+        using mfem::Operator::Operator;
+
+        void Mult(const mfem::Vector &, mfem::Vector &) const override;
+
+        [[nodiscard]] const mean_field::operators::StellarEquilibriumLayout &GetLayout() const noexcept;
+        [[nodiscard]] mean_field::operators::PreparedStellarEquilibriumReport Prepare(
+            const mfem::Vector &,
+            const mean_field::operators::StellarEquilibriumDependencies &,
+            const mean_field::physics::RigidRotation &
+        );
+        void BuildResidual(mfem::Vector &) const;
+        [[nodiscard]] bool IsPrepared() const noexcept;
+        [[nodiscard]] mean_field::operators::RootConstraintReport GetFixedMassReport() const;
+        [[nodiscard]] const mean_field::operators::StellarEquilibriumDependencies &GetDependencies() const;
+        [[nodiscard]] const mean_field::operators::StellarEquilibriumDependencyStamp &
+        GetGeneratedDisplacementDependency() const;
+        [[nodiscard]] const mean_field::operators::PreparedPressureSurfaceConstraint &
+        GetSurfaceConstraintOperator() const;
+    };
+
+    template <mean_field::model::StellarModelType Model>
+    class AlternateEquationOfStateRuntime final {
+    public:
+        using Report = mean_field::operators::EmptySpecificationPreparationReport;
+
+        AlternateEquationOfStateRuntime(
+            mean_field::fem::FEM &,
+            const mean_field::mapping::DomainMapper &,
+            AlternatePhysicalCore &,
+            const Model &
+        ) noexcept { }
+
+        template <typename StateView, typename Controls>
+        void ReadPhysicalControls(const StateView &, Controls &) noexcept { }
+
+        template <typename StateView>
+        [[nodiscard]] Report PrepareAfterPhysical(
+            const StateView &,
+            const mean_field::operators::StellarEquilibriumDependencies &,
+            const AlternatePhysicalCore &
+        ) noexcept {
+            return {};
+        }
+
+        template <typename ResidualView>
+        void AddResidual(const ResidualView &) const noexcept { }
+
+        template <typename DirectionView, typename ActionView>
+        void AddJacobianAction(
+            const DirectionView &,
+            const ActionView &,
+            const AlternatePhysicalCore &
+        ) const noexcept { }
+
+        [[nodiscard]] constexpr bool IsPrepared() const noexcept {
+            return true;
+        }
+    };
+} // namespace material_surface_runtime_contract_test
+
+namespace mean_field::operators {
+    template <>
+    struct StellarEquilibriumCoreRuntime<
+        material_surface_runtime_contract_test::RegisteredAlternateEquationOfState> {
+        static constexpr bool registered = true;
+        using CoreType = material_surface_runtime_contract_test::AlternatePhysicalCore;
+
+        [[nodiscard]] static std::unique_ptr<CoreType> Make(
+            fem::FEM &,
+            const mapping::DomainMapper &,
+            const material_surface_runtime_contract_test::RegisteredAlternateEquationOfState &,
+            const models::CompiledFixedMass &,
+            PressureSurfaceConstraintView,
+            deformation::PreparedDomainDeformationRuntime
+        );
+
+        [[nodiscard]] static int SurfaceEquationCount(const CoreType &) noexcept;
+    };
+
+    template <>
+    struct StellarEquilibriumRuntimeContribution<
+        material_surface_runtime_contract_test::RegisteredAlternateEquationOfState>
+        : PreparedStellarEquilibriumContribution<
+              material_surface_runtime_contract_test::AlternateEquationOfStateRuntime> { };
+} // namespace mean_field::operators
+
+namespace mean_field::preconditioning {
+    /*
+     * This registration deliberately advertises a distinct core without an
+     * executable material/surface implementation.  Registration alone must
+     * therefore remain insufficient for MaterialSurfaceRuntimeFor.
+     */
+    template <>
+    struct MaterialSurfaceEquationOfStateBackend<
+        material_surface_runtime_contract_test::RegisteredAlternateEquationOfState> {
+        static constexpr bool registered = true;
+        using CoreType = material_surface_runtime_contract_test::AlternatePhysicalCore;
+    };
+} // namespace mean_field::preconditioning
+
 namespace {
     namespace backend                         = mean_field::preconditioning::backend;
     namespace blocks                          = mean_field::utils::blocks;
@@ -25,6 +150,11 @@ namespace {
         mean_field::constraint::FixedCentralDensity>>;
     using PolytropicProblem                   = mean_field::equilibrium::StellarEquilibriumProblem<PolytropicModel>;
     using PolytropicMaterialSurfaceDescriptor = preconditioning::MaterialSurfaceDescriptorFor<PolytropicProblem>;
+    using RegisteredAlternateModel = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        material_surface_runtime_contract_test::RegisteredAlternateEquationOfState,
+        mean_field::surface::Isobaric,
+        mean_field::integral::FixedTotalMass>>;
+    using RegisteredAlternateProblem = mean_field::equilibrium::StellarEquilibriumProblem<RegisteredAlternateModel>;
     using MaterialSurfaceDiagonal             = preconditioning::MaterialSurfaceBlock<
         PolytropicMaterialSurfaceDescriptor,
         backend::Diagonal,
@@ -44,6 +174,30 @@ namespace {
         PolytropicMaterialSurfaceDescriptor,
         preconditioning::ApproximateMaterialSurfaceLDU,
         backend::FixedCycles>;
+    using RegisteredAlternateGravityComponent = preconditioning::GravityFieldBlock<
+        backend::Diagonal,
+        FixedCycleAMG,
+        preconditioning::GravityApproximateLDU>;
+
+    using RegisteredAlternateMaterialSurfaceDescriptor =
+        preconditioning::MaterialSurfaceDescriptorFor<RegisteredAlternateProblem>;
+
+    struct DistinctPhysicalCore final { };
+
+    template <typename Problem>
+    concept CanMakeDefaultMaterialSurface = requires(const Problem &problem) {
+        preconditioning::materialSurfaceBlock(problem);
+    };
+
+    template <typename Problem>
+    concept CanPrepareDefaultMaterialSurface = requires(const Problem &problem) {
+        preconditioning::prepare(problem, preconditioning::materialSurfaceBlock(problem));
+    };
+
+    template <typename Problem>
+    concept CanMakeDefaultStellarStructure = requires(const Problem &problem) {
+        preconditioning::stellarStructureBlock(problem);
+    };
 
     class KnownCouplings final {
     public:
@@ -161,6 +315,53 @@ TEST_CASE(
     STATIC_CHECK(preconditioning::PreconditionerComponent<MaterialSurfaceDiagonal>);
     STATIC_CHECK(preconditioning::PreconditionerComponent<MaterialSurfaceH1>);
     STATIC_CHECK(preconditioning::MaterialSurfaceDescriptor<PolytropicMaterialSurfaceDescriptor>);
+    STATIC_CHECK(preconditioning::ImplementedMaterialSurfaceEquationOfState<
+                 mean_field::eos::Polytrope>);
+    STATIC_CHECK_FALSE(
+        preconditioning::ImplementedMaterialSurfaceEquationOfState<int>);
+    STATIC_CHECK(preconditioning::ImplementedMaterialSurfaceDescriptor<
+                 PolytropicMaterialSurfaceDescriptor>);
+    STATIC_CHECK(preconditioning::ExecutableMaterialSurfaceRuntimeFor<
+                 mean_field::eos::Polytrope,
+                 mean_field::operators::PreparedStellarEquilibriumOperator>);
+    STATIC_CHECK(std::same_as<
+                 typename preconditioning::MaterialSurfaceEquationOfStateBackend<
+                     mean_field::eos::Polytrope>::CoreType,
+                 mean_field::operators::PreparedStellarEquilibriumOperator>);
+    STATIC_CHECK(preconditioning::MaterialSurfaceRuntimeFor<
+                 PolytropicMaterialSurfaceDescriptor,
+                 mean_field::operators::PreparedStellarEquilibriumOperator>);
+    STATIC_CHECK_FALSE(preconditioning::MaterialSurfaceRuntimeFor<
+                       PolytropicMaterialSurfaceDescriptor,
+                       DistinctPhysicalCore>);
+    STATIC_CHECK(preconditioning::MaterialSurfaceDescriptor<
+                 RegisteredAlternateMaterialSurfaceDescriptor>);
+    STATIC_CHECK(mean_field::equilibrium::StellarEquilibriumModel<RegisteredAlternateModel>);
+    STATIC_CHECK(std::same_as<
+                 typename RegisteredAlternateProblem::PhysicalCoreType,
+                 material_surface_runtime_contract_test::AlternatePhysicalCore>);
+    STATIC_CHECK(preconditioning::ImplementedMaterialSurfaceEquationOfState<
+                 material_surface_runtime_contract_test::RegisteredAlternateEquationOfState>);
+    STATIC_CHECK(preconditioning::ImplementedMaterialSurfaceDescriptor<
+                 RegisteredAlternateMaterialSurfaceDescriptor>);
+    STATIC_CHECK_FALSE(preconditioning::ExecutableMaterialSurfaceRuntimeFor<
+                       material_surface_runtime_contract_test::RegisteredAlternateEquationOfState,
+                       material_surface_runtime_contract_test::AlternatePhysicalCore>);
+    STATIC_CHECK_FALSE(preconditioning::MaterialSurfaceRuntimeFor<
+                       RegisteredAlternateMaterialSurfaceDescriptor,
+                       material_surface_runtime_contract_test::AlternatePhysicalCore>);
+    STATIC_CHECK(preconditioning::MaterialSurfacePreconditionerProblem<PolytropicProblem>);
+    STATIC_CHECK_FALSE(preconditioning::MaterialSurfacePreconditionerProblem<RegisteredAlternateProblem>);
+    STATIC_CHECK(CanMakeDefaultMaterialSurface<PolytropicProblem>);
+    STATIC_CHECK_FALSE(CanMakeDefaultMaterialSurface<RegisteredAlternateProblem>);
+    STATIC_CHECK(CanPrepareDefaultMaterialSurface<PolytropicProblem>);
+    STATIC_CHECK_FALSE(CanPrepareDefaultMaterialSurface<RegisteredAlternateProblem>);
+    STATIC_CHECK_FALSE(preconditioning::StellarStructurePreconditionerProblem<RegisteredAlternateProblem>);
+    STATIC_CHECK_FALSE(preconditioning::StellarStructurePreparableFor<
+                       RegisteredAlternateProblem,
+                       MaterialSurfaceDiagonal,
+                       RegisteredAlternateGravityComponent>);
+    STATIC_CHECK_FALSE(CanMakeDefaultStellarStructure<RegisteredAlternateProblem>);
     STATIC_CHECK(
         mean_field::material::CompiledThermodynamicEquations<typename PolytropicProblem::ThermodynamicEquationsType>
     );
@@ -171,7 +372,7 @@ TEST_CASE(
     );
     STATIC_CHECK(MaterialSurfaceDiagonal::CorrectionBlocks::size == 3);
     STATIC_CHECK(MaterialSurfaceDiagonal::ResidualBlocks::size == 3);
-    STATIC_CHECK(MaterialSurfaceDiagonal::RequiredCouplings::size == 8);
+    STATIC_CHECK(MaterialSurfaceDiagonal::RequiredCouplings::size == 9);
     STATIC_CHECK(preconditioning::CompletePreconditionerFor<Plan, Form>);
     STATIC_CHECK(preconditioning::CompatiblePreconditionerFor<Plan, Form, JacobianForm>);
     STATIC_CHECK(preconditioning::backend::ArnoldiAdmissible<typename MaterialSurfaceDiagonal::BackendType>);
@@ -409,7 +610,7 @@ TEST_CASE(
 
     mfem::Vector fullDirection(physical.Width());
     fullDirection                = 0.0;
-    const auto fullDirectionView = physical.GetRootManifest().directionView(fullDirection);
+    const auto fullDirectionView = physical.GetRootManifest().stateView(fullDirection);
     const auto &offsets          = restricted.GetOffsets();
     const mfem::Vector densityDirection(restrictedDirection.GetData(), offsets[1]);
     const mfem::Vector surfaceDirection(restrictedDirection.GetData() + offsets[1], offsets[2] - offsets[1]);

tests/preconditioning/stellar_equilibrium.cpp

@@ -71,6 +71,10 @@ namespace preconditioning_runtime_test {
             ++m_snapshot.geometry.revision;
         }
 
+        void AdvancePreparation() noexcept {
+            ++m_snapshot.preparedOperatorGeneration;
+        }
+
         void SetPrepared(const bool prepared) noexcept {
             m_prepared = prepared;
         }
@@ -108,6 +112,136 @@ namespace preconditioning_runtime_test {
     };
 } // namespace preconditioning_runtime_test
 
+namespace unsupported_physical_preconditioner_test {
+    class Constraint;
+    class PreparedConstraint;
+
+    class Constraint final {
+    public:
+        struct Parameters final {
+            mean_field::dimensions::SpecificEnthalpyValue target;
+        };
+
+        using TargetValue = mean_field::dimensions::SpecificEnthalpyValue;
+        using ScalarDescription = mean_field::stellar::ScalarConstraint<
+            mean_field::dimensions::quantity::SpecificEnthalpy,
+            mean_field::dimensions::quantity::Dimensionless,
+            mean_field::dimensions::quantity::SpecificEnthalpy,
+            "test.unsupported_physical_edge.coordinate",
+            "q_u",
+            "test.unsupported_physical_edge.residual",
+            "R_u">;
+        using ModelDefinition = mean_field::constraint::ScalarPhaseCondition<
+            Constraint,
+            "UnsupportedPhysicalPreconditionerEdge",
+            mean_field::stellar::Reads<
+                mean_field::stellar::state::SpecificEnthalpy,
+                mean_field::stellar::state::OwnGeneratedCoordinate>,
+            mean_field::stellar::Changes<
+                mean_field::stellar::equation::PoissonEquation>,
+            ScalarDescription>;
+        using EquilibriumPhysics =
+            mean_field::operators::LocalSpecificationEquilibriumPhysics<
+                PreparedConstraint>;
+
+        explicit constexpr Constraint(const Parameters parameters) noexcept
+            : m_target(parameters.target) {
+        }
+
+        [[nodiscard]] constexpr TargetValue target() const noexcept {
+            return m_target;
+        }
+
+    private:
+        TargetValue m_target;
+    };
+
+    struct PreparationReport final {
+        bool stateChanged{true};
+    };
+
+    class PreparedConstraint final {
+    public:
+        using Report = PreparationReport;
+
+        explicit PreparedConstraint(const Constraint &) noexcept {
+        }
+
+        template <typename StateView>
+        [[nodiscard]] Report PrepareAfterPhysical(const StateView &) noexcept {
+            m_isPrepared = true;
+            return {};
+        }
+
+        template <typename Row>
+        [[nodiscard]] mean_field::stellar::StructuralZero AddResidual(
+            mean_field::stellar::equation::OwnConstraint,
+            Row &
+        ) const noexcept {
+            return mean_field::stellar::structuralZero;
+        }
+
+        template <typename Row>
+        [[nodiscard]] mean_field::stellar::StructuralZero AddResidual(
+            mean_field::stellar::equation::PoissonEquation,
+            Row &
+        ) const noexcept {
+            return mean_field::stellar::structuralZero;
+        }
+
+        template <typename Direction, typename Row>
+        [[nodiscard]] mean_field::stellar::StructuralZero AddJacobianAction(
+            mean_field::stellar::Derivative<
+                mean_field::stellar::equation::OwnConstraint,
+                mean_field::stellar::state::SpecificEnthalpy>,
+            const Direction &,
+            Row &
+        ) const noexcept {
+            return mean_field::stellar::zeroDerivative;
+        }
+
+        template <typename Direction, typename Row>
+        [[nodiscard]] mean_field::stellar::StructuralZero AddJacobianAction(
+            mean_field::stellar::Derivative<
+                mean_field::stellar::equation::OwnConstraint,
+                mean_field::stellar::state::OwnGeneratedCoordinate>,
+            const Direction &,
+            Row &
+        ) const noexcept {
+            return mean_field::stellar::zeroDerivative;
+        }
+
+        template <typename Direction, typename Row>
+        [[nodiscard]] mean_field::stellar::StructuralZero AddJacobianAction(
+            mean_field::stellar::Derivative<
+                mean_field::stellar::equation::PoissonEquation,
+                mean_field::stellar::state::SpecificEnthalpy>,
+            const Direction &,
+            Row &
+        ) const noexcept {
+            return mean_field::stellar::zeroDerivative;
+        }
+
+        template <typename Direction, typename Row>
+        [[nodiscard]] mean_field::stellar::StructuralZero AddJacobianAction(
+            mean_field::stellar::Derivative<
+                mean_field::stellar::equation::PoissonEquation,
+                mean_field::stellar::state::OwnGeneratedCoordinate>,
+            const Direction &,
+            Row &
+        ) const noexcept {
+            return mean_field::stellar::zeroDerivative;
+        }
+
+        [[nodiscard]] bool IsPrepared() const noexcept {
+            return m_isPrepared;
+        }
+
+    private:
+        bool m_isPrepared{false};
+    };
+} // namespace unsupported_physical_preconditioner_test
+
 template <> struct mean_field::preconditioning::StellarEquilibriumProblemTraits<preconditioning_runtime_test::Problem> {
     using Problem                    = preconditioning_runtime_test::Problem;
     using Form                       = preconditioning_runtime_test::Form;
@@ -146,14 +280,40 @@ namespace {
     namespace blocks                = mean_field::utils::blocks;
     namespace preconditioning       = mean_field::preconditioning;
 
-    using ModelWithoutPhase         = mean_field::operators::StellarEquilibriumSpecificationModel;
+    using ModelWithoutPhase = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
-    using CentralDensityModel       = mean_field::operators::CentralDensityStellarEquilibriumSpecificationModel;
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::models::FixedTotalMass>>;
+    using CentralDensityModel = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::models::FixedTotalMass,
+        mean_field::models::FixedCentralDensity>>;
+    using AngularMomentumModel = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::models::FixedTotalMass,
+        mean_field::models::FixedAngularMomentum>>;
+    using ProvenZeroPhysicalEdgeModel = mean_field::model::StellarModel<
+        mean_field::models::SpecificationSet<
+            mean_field::eos::Polytrope,
+            mean_field::surface::Isobaric,
+            mean_field::models::FixedTotalMass,
+            unsupported_physical_preconditioner_test::Constraint>>;
 
     using ProblemWithoutPhase       = mean_field::equilibrium::StellarEquilibriumProblem<ModelWithoutPhase>;
     using CentralDensityProblem     = mean_field::equilibrium::StellarEquilibriumProblem<CentralDensityModel>;
+    using AngularMomentumProblem    = mean_field::equilibrium::StellarEquilibriumProblem<AngularMomentumModel>;
+    using ProvenZeroPhysicalEdgeProblem =
+        mean_field::equilibrium::StellarEquilibriumProblem<ProvenZeroPhysicalEdgeModel>;
     using PlanWithoutPhase          = preconditioning::IdentityPreconditionerPlanFor<ProblemWithoutPhase>;
     using CentralDensityPlan        = preconditioning::IdentityPreconditionerPlanFor<CentralDensityProblem>;
 
+    template <typename Problem>
+    concept CanMakeDefaultStellarStructureBlock = requires(const Problem &problem) {
+        preconditioning::stellarStructureBlock(problem);
+    };
+
     using RefreshingDensityIdentity = preconditioning::ComponentDeclaration<
         blocks::type_list<blocks::density::mass::value>,
         blocks::type_list<blocks::density::mass::residual>,
@@ -203,6 +363,73 @@ TEST_CASE(
     STATIC_CHECK(CentralDensityPlan::ComponentTypes::size == 7);
 }
 
+TEST_CASE(
+    "Default Stellar Structure Availability Distinguishes Proven Zeros From Unhandled Physical Edges",
+    "[preconditioning][stellar_structure][type_contract][compiler]"
+) {
+    using BaseCompilation =
+        mean_field::operators::CompiledStellarEquilibriumSystem<ModelWithoutPhase>;
+    using BaseSupport =
+        preconditioning::DefaultStellarStructurePhysicalTopologySupport<
+            ModelWithoutPhase>;
+    using ProvenZeroSupport =
+        preconditioning::DefaultStellarStructurePhysicalTopologySupport<
+            ProvenZeroPhysicalEdgeModel>;
+    using TrustedFixedMassEdge =
+        mean_field::operators::StellarEquilibriumJacobianCoupling<
+            blocks::enthalpy::specific::residual,
+            blocks::density::mass::value>;
+    using ProvenZeroPoissonEnthalpyEdge =
+        mean_field::operators::StellarEquilibriumJacobianCoupling<
+            blocks::gravity::poisson::residual,
+            blocks::enthalpy::specific::value>;
+
+    // FixedTotalMass contributes h <- rho outside the generic five-field base
+    // graph. It remains supported because that specification is explicitly
+    // embedded in the trusted numerical core, not because of a model-pack
+    // special case.
+    STATIC_CHECK_FALSE(mean_field::utils::blocks::contains_type_v<
+                       TrustedFixedMassEdge,
+                       typename BaseCompilation::BaseJacobianCouplings>);
+    STATIC_CHECK(mean_field::utils::blocks::contains_type_v<
+                 TrustedFixedMassEdge,
+                 typename BaseCompilation::ContributionJacobianCouplings>);
+    STATIC_CHECK(BaseSupport::UnsupportedCouplings::size == 0);
+    STATIC_CHECK(preconditioning::DefaultStellarStructurePhysicalTopologySupportedFor<
+                 ModelWithoutPhase>);
+    STATIC_CHECK(preconditioning::DefaultStellarStructurePhysicalTopologySupportedFor<
+                 CentralDensityModel>);
+    STATIC_CHECK(preconditioning::DefaultStellarStructurePhysicalTopologySupportedFor<
+                 AngularMomentumModel>);
+    STATIC_CHECK(preconditioning::StellarStructurePreconditionerProblem<
+                 ProblemWithoutPhase>);
+    STATIC_CHECK(CanMakeDefaultStellarStructureBlock<ProblemWithoutPhase>);
+    STATIC_CHECK(preconditioning::DefaultStellarPreconditionerAvailableFor<
+                 ProblemWithoutPhase>);
+
+    // The mock's novel Poisson <- enthalpy edge is absent from the structure
+    // backend, but its exact nested provider returns StructuralZero.  That is
+    // a compile-time proof that no preconditioner term is missing; generated-
+    // coordinate edges are handled independently by the inferred border.
+    STATIC_CHECK(mean_field::equilibrium::StellarEquilibriumModel<
+                 ProvenZeroPhysicalEdgeModel>);
+    STATIC_CHECK(mean_field::equilibrium::DiscretizedStellarEquilibriumProblem<
+                 ProvenZeroPhysicalEdgeProblem>);
+    STATIC_CHECK(ProvenZeroSupport::UnsupportedCouplings::size == 0);
+    STATIC_CHECK(mean_field::utils::blocks::contains_type_v<
+                 ProvenZeroPoissonEnthalpyEdge,
+                 typename mean_field::operators::CompiledStellarEquilibriumSystem<
+                     ProvenZeroPhysicalEdgeModel>::ContributionJacobianCouplings>);
+    STATIC_CHECK(preconditioning::DefaultStellarStructurePhysicalTopologySupportedFor<
+                 ProvenZeroPhysicalEdgeModel>);
+    STATIC_CHECK(preconditioning::StellarStructurePreconditionerProblem<
+                 ProvenZeroPhysicalEdgeProblem>);
+    STATIC_CHECK(CanMakeDefaultStellarStructureBlock<
+                 ProvenZeroPhysicalEdgeProblem>);
+    STATIC_CHECK(preconditioning::DefaultStellarPreconditionerAvailableFor<
+                 ProvenZeroPhysicalEdgeProblem>);
+}
+
 TEST_CASE(
     "Prepared Stellar Identity Preconditioning Is Bitwise Equivalent To The P0 Baseline",
     tags::preconditioning_runtime_unit
@@ -286,6 +513,14 @@ TEST_CASE(
     CHECK_FALSE(geometryRefresh.changes.linearization);
     CHECK(preconditioner.GetStatistics().refreshes == 2);
 
+    problem.AdvancePreparation();
+    CHECK_FALSE(preconditioner.IsCurrent());
+    const auto preparationRefresh = preconditioner.Refresh();
+    CHECK(preparationRefresh.changes.linearization);
+    CHECK_FALSE(preparationRefresh.changes.geometry);
+    CHECK(preconditioner.IsCurrent());
+    CHECK(preconditioner.GetStatistics().refreshes == 3);
+
     problem.SetPrepared(false);
     CHECK_FALSE(preconditioner.IsCurrent());
     CHECK_THROWS_AS(preconditioner.Refresh(), std::logic_error);

tests/preconditioning/stellar_structure.cpp

@@ -114,6 +114,7 @@ namespace {
         mean_field::integral::FixedTotalMass,
         mean_field::constraint::FixedCentralDensity>>;
     using PolytropicProblem = mean_field::equilibrium::StellarEquilibriumProblem<PolytropicModel>;
+    using PolytropicMaterialSurfaceDescriptor = preconditioning::MaterialSurfaceDescriptorFor<PolytropicProblem>;
     using MaterialComponent =
         decltype(preconditioning::materialSurfaceBlock(std::declval<const PolytropicProblem &>()));
     using FixedAMG = backend::HypreBoomerAMG<backend::FixedCycles>;
@@ -126,6 +127,13 @@ namespace {
         preconditioning::IndependentStellarSubsystems{}
     ));
 
+    template <typename Problem>
+    concept CanPrepareDefaultStellarStructure = requires(const Problem &problem) {
+        preconditioning::prepare(problem, preconditioning::stellarStructureBlock(problem));
+    };
+
+    struct DistinctPhysicalCore final { };
+
     [[nodiscard]] mean_field::operators::StellarEquilibriumDependencies
     makeDependencies(const std::uint64_t revision = 1) {
         return {
@@ -173,11 +181,26 @@ TEST_CASE(
         preconditioning::Coupling<blocks::enthalpy::specific::residual, blocks::gravity::poisson::value>>;
 
     STATIC_CHECK(preconditioning::PreconditionerComponent<PolytropicStructure>);
+    STATIC_CHECK(preconditioning::ExecutableStellarStructureRuntimeFor<
+                 mean_field::operators::PreparedStellarEquilibriumOperator>);
+    STATIC_CHECK_FALSE(preconditioning::ExecutableStellarStructureRuntimeFor<DistinctPhysicalCore>);
+    STATIC_CHECK(preconditioning::StellarStructureRuntimeFor<
+                 PolytropicMaterialSurfaceDescriptor,
+                 mean_field::operators::PreparedStellarEquilibriumOperator>);
+    STATIC_CHECK_FALSE(preconditioning::StellarStructureRuntimeFor<
+                       PolytropicMaterialSurfaceDescriptor,
+                       DistinctPhysicalCore>);
+    STATIC_CHECK(preconditioning::StellarStructurePreconditionerProblem<PolytropicProblem>);
+    STATIC_CHECK(preconditioning::StellarStructurePreparableFor<
+                 PolytropicProblem,
+                 MaterialComponent,
+                 GravityComponent>);
+    STATIC_CHECK(CanPrepareDefaultStellarStructure<PolytropicProblem>);
     STATIC_CHECK(std::same_as<typename PolytropicStructure::MaterialToGravityCouplings, ExpectedMaterialToGravity>);
     STATIC_CHECK(std::same_as<typename PolytropicStructure::GravityToMaterialCouplings, ExpectedGravityToMaterial>);
     STATIC_CHECK(PolytropicStructure::MaterialToGravityCouplings::size == 3);
     STATIC_CHECK(PolytropicStructure::GravityToMaterialCouplings::size == 2);
-    STATIC_CHECK(PolytropicStructure::RequiredCouplings::size == 16);
+    STATIC_CHECK(PolytropicStructure::RequiredCouplings::size == 17);
     STATIC_CHECK(preconditioning::backend::ArnoldiAdmissible<typename PolytropicStructure::BackendType>);
 
     using FixedMassIdentity = preconditioning::IdentityBlock<
@@ -265,7 +288,7 @@ TEST_CASE(
 
     mfem::Vector materialOnlyDirection(physical.Width());
     materialOnlyDirection         = 0.0;
-    auto materialOnlyView         = physical.GetRootManifest().directionView(materialOnlyDirection);
+    auto materialOnlyView         = physical.GetRootManifest().stateView(materialOnlyDirection);
     mfem::Vector materialDensity  = materialOnlyView.block(blocks::density_field.mass_term);
     mfem::Vector materialSurface  = materialOnlyView.block(blocks::surface_deformation_field.parameters_term);
     mfem::Vector materialEnthalpy = materialOnlyView.block(blocks::enthalpy_field.specific_term);
@@ -290,7 +313,7 @@ TEST_CASE(
 
     mfem::Vector gravityOnlyDirection(physical.Width());
     gravityOnlyDirection          = 0.0;
-    auto gravityOnlyView          = physical.GetRootManifest().directionView(gravityOnlyDirection);
+    auto gravityOnlyView          = physical.GetRootManifest().stateView(gravityOnlyDirection);
     mfem::Vector gravityGradient  = gravityOnlyView.block(blocks::gravity_field.gradient_term);
     mfem::Vector gravityPotential = gravityOnlyView.block(blocks::gravity_field.poisson_term);
     const mfem::Vector sourceGravityGradient(

tests/seed/stellar_equilibrium_projection.cpp

@@ -13,6 +13,137 @@ import mean_field;
 import test_helpers;
 
 namespace {
+    template <typename... Specifications>
+    using ProjectionModelWith = mean_field::model::StellarModel<
+        mean_field::models::SpecificationSet<Specifications...>>;
+
+    class UnregisteredProjectionConstraint final {
+    public:
+        struct Parameters final { };
+        using ModelDefinition = mean_field::constraint::PhaseCondition<
+            UnregisteredProjectionConstraint,
+            "UnregisteredProjectionConstraint",
+            mean_field::models::DependsOn<mean_field::models::stellar::state::SpecificEnthalpy>,
+            mean_field::models::Affects<mean_field::models::stellar::equation::HydrostaticBalance>>;
+
+        explicit constexpr UnregisteredProjectionConstraint(Parameters) noexcept {
+        }
+    };
+
+    class ExplicitNoChangeProjectionConstraint final {
+    public:
+        struct Parameters final { };
+        using ModelDefinition = mean_field::constraint::PhaseCondition<
+            ExplicitNoChangeProjectionConstraint,
+            "ExplicitNoChangeProjectionConstraint",
+            mean_field::models::DependsOn<mean_field::models::stellar::state::SpecificEnthalpy>,
+            mean_field::models::Affects<mean_field::models::stellar::equation::HydrostaticBalance>>;
+        using RadialProjection = mean_field::seed::projection::Use<mean_field::seed::projection::NoStateChange>;
+
+        explicit constexpr ExplicitNoChangeProjectionConstraint(Parameters) noexcept {
+        }
+    };
+
+    struct IncompleteProjectionPhysics final {
+        static constexpr bool registered         = true;
+        static constexpr bool providesRadialMass = false;
+
+        template <typename Model>
+        static constexpr bool supports = true;
+    };
+
+    class IncompleteProjectionConstraint final {
+    public:
+        struct Parameters final { };
+        using ModelDefinition = mean_field::constraint::PhaseCondition<
+            IncompleteProjectionConstraint,
+            "IncompleteProjectionConstraint",
+            mean_field::models::DependsOn<mean_field::models::stellar::state::SpecificEnthalpy>,
+            mean_field::models::Affects<mean_field::models::stellar::equation::HydrostaticBalance>>;
+        using RadialProjection = mean_field::seed::projection::Use<IncompleteProjectionPhysics>;
+
+        explicit constexpr IncompleteProjectionConstraint(Parameters) noexcept {
+        }
+    };
+
+    class UnregisteredProjectionEquationOfState final {
+    public:
+        struct Parameters final { };
+        using ModelDefinition = mean_field::eos::ConstitutiveLaw<
+            UnregisteredProjectionEquationOfState,
+            "UnregisteredProjectionEquationOfState">;
+
+        explicit constexpr UnregisteredProjectionEquationOfState(Parameters) noexcept {
+        }
+    };
+
+    class UnregisteredProjectionSurface final {
+    public:
+        struct Parameters final { };
+        using ModelDefinition = mean_field::surface::BoundaryCondition<
+            UnregisteredProjectionSurface,
+            "UnregisteredProjectionSurface">;
+
+        explicit constexpr UnregisteredProjectionSurface(Parameters) noexcept {
+        }
+    };
+
+    struct SecondRadialMassProjectionPhysics final {
+        static constexpr bool registered         = true;
+        static constexpr bool providesRadialMass = true;
+
+        template <typename Model>
+        static constexpr bool supports = true;
+
+        template <typename Specification>
+        [[nodiscard]] static mean_field::dimensions::MassValue targetMass(const Specification &specification) {
+            return specification.targetMass();
+        }
+
+        template <typename Specification, typename Model>
+        static void validate(
+            const Specification &,
+            const Model &,
+            const mean_field::seed::RadialProfile &,
+            const mean_field::seed::StellarEquilibriumProjectionOptions &
+        ) noexcept {
+        }
+
+        template <typename Specification, typename Model>
+        static void initialize(
+            const Specification &,
+            const Model &,
+            const mean_field::seed::RadialProjectionScales &,
+            mean_field::seed::RadialProjectionState &,
+            mfem::Vector coordinate
+        ) {
+            if (coordinate.Size() == 1) {
+                coordinate(0) = 0.0;
+            }
+        }
+    };
+
+    class SecondRadialMassConstraint final {
+    public:
+        struct Parameters final {
+            mean_field::dimensions::MassValue mass;
+        };
+        using ModelDefinition = mean_field::integral::FixedWithMultiplier<
+            SecondRadialMassConstraint,
+            "SecondRadialMassConstraint">;
+        using RadialProjection = mean_field::seed::projection::Use<SecondRadialMassProjectionPhysics>;
+
+        explicit constexpr SecondRadialMassConstraint(Parameters parameters) noexcept : m_mass(parameters.mass) {
+        }
+
+        [[nodiscard]] constexpr mean_field::dimensions::MassValue targetMass() const noexcept {
+            return m_mass;
+        }
+
+    private:
+        mean_field::dimensions::MassValue m_mass;
+    };
+
     [[nodiscard]] mean_field::operators::StellarEquilibriumDependencies make_dependencies() {
         return {
             .discretization     = {.identity = 7001, .revision = 1},
@@ -42,6 +173,65 @@ namespace {
     }
 } // namespace
 
+TEST_CASE(
+    "Radial Projection Capabilities Are Inferred From Every Model Specification",
+    tags::stellar_seed_projection_type_contract
+) {
+    using namespace mean_field;
+    using BaseModel = ProjectionModelWith<eos::Polytrope, surface::Isobaric, integral::FixedTotalMass>;
+    using CentralModel = ProjectionModelWith<
+        eos::Polytrope,
+        surface::Isobaric,
+        integral::FixedTotalMass,
+        constraint::FixedCentralDensity>;
+    using AngularModel = ProjectionModelWith<
+        eos::Polytrope,
+        surface::Isobaric,
+        integral::FixedTotalMass,
+        integral::FixedAngularMomentum>;
+    using ExplicitExtensionModel = ProjectionModelWith<
+        eos::Polytrope,
+        surface::Isobaric,
+        integral::FixedTotalMass,
+        ExplicitNoChangeProjectionConstraint>;
+    using MissingConstraintRuleModel = ProjectionModelWith<
+        eos::Polytrope,
+        surface::Isobaric,
+        integral::FixedTotalMass,
+        UnregisteredProjectionConstraint>;
+    using IncompleteConstraintRuleModel = ProjectionModelWith<
+        eos::Polytrope,
+        surface::Isobaric,
+        integral::FixedTotalMass,
+        IncompleteProjectionConstraint>;
+    using MissingEquationOfStateRuleModel = ProjectionModelWith<
+        UnregisteredProjectionEquationOfState,
+        surface::Isobaric,
+        integral::FixedTotalMass>;
+    using MissingSurfaceRuleModel = ProjectionModelWith<
+        eos::Polytrope,
+        UnregisteredProjectionSurface,
+        integral::FixedTotalMass>;
+    using MissingMassProviderModel = ProjectionModelWith<eos::Polytrope, surface::Isobaric>;
+    using AmbiguousMassProviderModel = ProjectionModelWith<
+        eos::Polytrope,
+        surface::Isobaric,
+        integral::FixedTotalMass,
+        SecondRadialMassConstraint>;
+
+    STATIC_CHECK(seed::RadialProfileProjectableModel<BaseModel>);
+    STATIC_CHECK(seed::RadialProfileProjectableModel<CentralModel>);
+    STATIC_CHECK(seed::RadialProfileProjectableModel<AngularModel>);
+    STATIC_CHECK(seed::RadialProfileProjectableModel<ExplicitExtensionModel>);
+    STATIC_CHECK_FALSE(seed::RadialProfileProjectableModel<MissingConstraintRuleModel>);
+    STATIC_CHECK_FALSE(seed::RadialProfileProjectableModel<IncompleteConstraintRuleModel>);
+    STATIC_CHECK_FALSE(seed::RadialProfileProjectableModel<MissingEquationOfStateRuleModel>);
+    STATIC_CHECK_FALSE(seed::RadialProfileProjectableModel<MissingSurfaceRuleModel>);
+    STATIC_CHECK_FALSE(seed::RadialProfileProjectableModel<MissingMassProviderModel>);
+    STATIC_CHECK_FALSE(seed::RadialProfileProjectableModel<AmbiguousMassProviderModel>);
+    STATIC_CHECK_FALSE(seed::RadialProfileProjectableModel<int>);
+}
+
 TEST_CASE(
     "Projected Equilibrium States Preserve Their Compiled Stellar Model Type",
     tags::stellar_seed_projection_type_contract
@@ -116,9 +306,9 @@ TEST_CASE(
     REQUIRE(centralDensityBorder.Size() == 1);
     CHECK(centralDensityBorder(0) == 0.0);
 
-    const operators::PreparedCentralDensityStellarEquilibriumReport preparation =
+    const auto preparation = problem.Prepare(projected.values, make_dependencies(), make_zero_rotation());
-        problem.Prepare(projected.values, make_dependencies(), make_zero_rotation());
     CHECK(preparation.assembledResidual);
+    CHECK(preparation.template specification<models::FixedCentralDensity>().constraint.DidAnyWork());
 
     mfem::Vector residual;
     problem.BuildResidual(residual);

tests/user-api/stellar_equilibrium.cpp

@@ -2,6 +2,7 @@
 #include <type_traits>
 
 #include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
 
 import mean_field;
 import test_helpers;
@@ -11,6 +12,25 @@ namespace {
         const mean_field::utils::Args arguments = test_utils::setup_args();
         return mean_field::fem::setup_fem(arguments.mesh_file, arguments, 0);
     }
+
+    template <typename Problem>
+    concept PreparesWithGeneratedRotation = requires(
+        Problem &problem,
+        const mfem::Vector &state,
+        const mean_field::operators::StellarEquilibriumDependencies &dependencies
+    ) {
+        problem.Prepare(state, dependencies);
+    };
+
+    template <typename Problem>
+    concept PreparesWithPrescribedRotation = requires(
+        Problem &problem,
+        const mfem::Vector &state,
+        const mean_field::operators::StellarEquilibriumDependencies &dependencies,
+        const mean_field::physics::RigidRotation &rotation
+    ) {
+        problem.Prepare(state, dependencies, rotation);
+    };
 } // namespace
 
 TEST_CASE(
@@ -42,6 +62,49 @@ TEST_CASE(
     CHECK(gravity.potentialSchurBackend().application.cycles == 3);
 }
 
+TEST_CASE(
+    "Fixed Angular Momentum User API Generates Rotation And Its Composable Solver Border",
+    "[user-api][fixed-angular-momentum][type]"
+) {
+    using namespace mean_field;
+
+    auto finiteElements = makeFiniteElements();
+    REQUIRE(finiteElements.okay());
+
+    auto model = model::StellarModel(
+        eos::Polytrope({.n = 1.0, .K = 0.25}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}}),
+        integral::FixedAngularMomentum({
+            .Jtotal = dimensions::AngularMomentumValue{0.2},
+            .axis = {0.0, 0.0, 2.0}
+        })
+    );
+
+    auto problem = equilibrium::discretize(model, finiteElements);
+    auto preconditioner = preconditioning::makePreconditioner(problem);
+    using Problem = std::remove_cvref_t<decltype(problem)>;
+    using Preconditioner = std::remove_cvref_t<decltype(preconditioner)>;
+
+    STATIC_CHECK(Problem::hasFixedAngularMomentum);
+    STATIC_CHECK_FALSE(Problem::hasFixedCentralDensity);
+    STATIC_CHECK(PreparesWithGeneratedRotation<Problem>);
+    STATIC_CHECK_FALSE(PreparesWithPrescribedRotation<Problem>);
+    STATIC_CHECK(Problem::FormType::value_block_count == 7);
+    STATIC_CHECK(Problem::FormType::residual_block_count == 7);
+    STATIC_CHECK(preconditioning::PreconditionerComponent<Preconditioner>);
+    STATIC_CHECK(Preconditioner::borderValueArity == 2);
+    STATIC_CHECK(Preconditioner::borderResidualArity == 2);
+
+    CHECK(problem.StateSize() == problem.EquationSize());
+    CHECK(problem.StateSize() == problem.GetPhysicalOperator().Width() + 1);
+    REQUIRE(problem.GetManifest().constraints().size() == 3);
+    CHECK(problem.GetManifest().template specification<models::FixedAngularMomentum>().stableId ==
+          "FixedAngularMomentum");
+    CHECK(problem.GetManifest().valueBlocks().back().symbol == "Omega");
+    CHECK(problem.GetManifest().residualBlocks().back().symbol == "R_J");
+}
+
 TEST_CASE(
     "Intermediate User API Selects A Coupled Stellar Factorization",
     "[user-api][intermediate]"