perf(jacobian-action): major updates to jacobian action application by removing redudant quadrature work. ~5x increase in speed

CMakeLists.txt

@@ -37,6 +37,7 @@ find_package(PkgConfig REQUIRED)
 
 
 pkg_check_modules(stroid REQUIRED IMPORTED_TARGET stroid)
+pkg_check_modules(eigen3 REQUIRED IMPORTED_TARGET eigen3)
 
 add_library(mean_field)
 
@@ -92,7 +93,11 @@ target_sources(mean_field
         libmeanfield/impl/operators/prepared_rotation_displacement_force.cpp
         libmeanfield/impl/operators/prepared_displacement_operator.cpp
         libmeanfield/impl/models/polytropic.cpp
+        libmeanfield/impl/seed/lane_emden.cpp
+        libmeanfield/impl/seed/stellar_equilibrium_projection.cpp
+        libmeanfield/impl/solver/preconditioning_diagnostics.cpp
         libmeanfield/impl/operators/prepared_mass_normalization.cpp
+        libmeanfield/impl/operators/prepared_central_density_stellar_equilibrium.cpp
         libmeanfield/impl/operators/prepared_stellar_equilibrium.cpp
 )
 
@@ -125,6 +130,7 @@ target_sources(mean_field
         libmeanfield/interface/quadrature/policy.cppm
         libmeanfield/interface/quadrature/mfem.cppm
         libmeanfield/interface/solver/fields.cppm
+        libmeanfield/interface/solver/preconditioning_diagnostics.cppm
         libmeanfield/interface/operators/gravity_field.cppm
         libmeanfield/interface/operators/gravity_field_jacobian.cppm
         libmeanfield/interface/operators/kernels/gravity_kernels.cppm
@@ -151,6 +157,7 @@ target_sources(mean_field
         libmeanfield/interface/operators/kernels/rotation_displacement_force_kernels.cppm
         libmeanfield/interface/operators/prepared_rotation_displacement_force.cppm
         libmeanfield/interface/operators/prepared_displacement_operator.cppm
+        libmeanfield/interface/dimensions/quantities.cppm
         libmeanfield/interface/eos/quantities.cppm
         libmeanfield/interface/eos/relations.cppm
         libmeanfield/interface/eos/concepts.cppm
@@ -158,9 +165,14 @@ target_sources(mean_field
         libmeanfield/interface/eos/pressure_surface.cppm
         libmeanfield/interface/eos/runtime.cppm
         libmeanfield/interface/eos/polytropic.cppm
+        libmeanfield/interface/seed/lane_emden.cppm
         libmeanfield/interface/models/structure/structure_base.cppm
         libmeanfield/interface/models/structure/polytropic.cppm
         libmeanfield/interface/models/structure_profile.cppm
+        libmeanfield/interface/models/specifications.cppm
+        libmeanfield/interface/models/typed_stellar_model.cppm
+        libmeanfield/interface/models/compiled_fixed_mass.cppm
+        libmeanfield/interface/models/compiled_fixed_central_density.cppm
         libmeanfield/interface/surface/constant.cppm
         libmeanfield/interface/surface/dependencies.cppm
         libmeanfield/interface/surface/compiled.cppm
@@ -173,10 +185,18 @@ target_sources(mean_field
         libmeanfield/interface/deformation/radial_extensions.cppm
         libmeanfield/interface/deformation/domain_deformation.cppm
         libmeanfield/interface/models/stellar_model.cppm
+        libmeanfield/interface/operators/root_manifest.cppm
+        libmeanfield/interface/operators/prepared_constraint.cppm
         libmeanfield/interface/operators/prepared_mass_normalization.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/equilibrium/stellar_discretization.cppm
+        libmeanfield/interface/operators/stellar_equilibrium_problem.cppm
+        libmeanfield/interface/seed/stellar_equilibrium_projection.cppm
+        libmeanfield/interface/operators/stellar_equilibrium_system.cppm
 )
 
 
@@ -187,6 +207,7 @@ target_link_libraries(mean_field
         mfem
         PkgConfig::stroid
 )
+target_link_libraries(mean_field PRIVATE PkgConfig::eigen3)
 
 add_library(test_mod)
 target_sources(test_mod
@@ -209,6 +230,9 @@ pkg_check_modules(fourdst_config REQUIRED IMPORTED_TARGET fourdst_config)
 add_executable(tests
         tests/test_main.cpp
         tests/physics/gravity.cpp
+        tests/physics/dimensional_quantities.cpp
+        tests/seed/lane_emden.cpp
+        tests/seed/stellar_equilibrium_projection.cpp
         tests/geometry/volume.cpp
         tests/quadrature/policy.cpp
         tests/integrators/centrifugal.cpp
@@ -250,7 +274,13 @@ add_executable(tests
         tests/operators/prepared_rotation_displacement_force_analytic.cpp
         tests/operators/prepared_rotation_displacement_force_affine_deformation.cpp
         tests/operators/prepared_displacement_operator.cpp
+        tests/operators/root_manifest.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/stellar_model.cpp
+        tests/operators/stellar_equilibrium_system.cpp
         tests/deformation/contracts.cpp
         tests/deformation/surface_scalar_dof_map.cpp
         tests/deformation/nodal_radial_surface.cpp
@@ -263,6 +293,7 @@ add_executable(tests
         tests/field/field_registry.cpp
         tests/field/field_mfem.cpp
         tests/field/field_dof_map.cpp
+        tests/solver/preconditioning_diagnostics.cpp
 )
 
 target_link_libraries(tests PRIVATE mean_field test_mod Catch2::Catch2 Boost::boost)
@@ -283,6 +314,7 @@ target_link_libraries(experiment_mod
 add_executable(experiments
         experiments/experiment_main.cpp
         experiments/gravity_accuracy_budget.cpp
+        experiments/preconditioning_diagnostics.cpp
 )
 
 target_link_libraries(experiments PRIVATE mean_field test_mod experiment_mod Catch2::Catch2 Boost::boost)

experiments/README.md

@@ -147,3 +147,42 @@ The executable needs the same dependencies, generated module mapping, and
 configuration registration as the existing Catch2 test executable. Add
 `experiment_main.cpp` and `gravity_accuracy_budget.cpp` as a second executable
 next to that target; do not add them to the ordinary test executable.
+
+## P0 preconditioning baseline
+
+The P0 diagnostic establishes the unpreconditioned reference for the complete,
+central-density-closed `n = 3` stellar equilibrium Jacobian. It uses an identity
+inverse preconditioner with FGMRES, recomputes the true residual independently,
+records every residual block, and counts and times Jacobian and preconditioner
+applications. A separate fixed-operator Arnoldi measurement acts explicitly on
+the right-preconditioned product `J M^-1`. Its singular-value ratio is a
+projected Krylov-space condition proxy, not the condition number of the full
+Jacobian. The same output records Ritz values, clustering about one,
+nonnormality, and the real extent of the projected field of values.
+
+The extended baseline preserves the fixed 40-iteration FGMRES budget used by
+the original P0 run and increases the Arnoldi dimension from 12 to 48. It writes
+the complete reported FGMRES residual history, block-relative and
+manifest-scaled final residuals, the fraction of the squared residual in each
+physics block, timings for construction/projection/preparation/direct-residual
+measurement, and separate Arnoldi operator and orthogonalization timings. Live
+progress messages delimit every expensive phase and report every fourth
+Arnoldi application. The CSV records whether it came from a Debug or Release
+build.
+
+Run the focused synthetic verification tests with:
+
+```text
+./cmake-build-debug-homebrew/tests "[preconditioning][diagnostics][unit]"
+```
+
+Run the performance and spectral measurement separately with:
+
+```text
+mpirun -np 1 ./cmake-build-release-homebrew/experiments \
+    --experiment-output preconditioning_p0_identity_extended.csv \
+    --catch2 "[preconditioning][diagnostics][baseline]"
+```
+
+Set `MEANFIELD_SINGLE_JACOBIAN_BENCHMARK=1` to stop after the initial prepared
+Jacobian timing instead of running FGMRES and Arnoldi.

experiments/coupled_gauge_modes.cpp

@@ -679,6 +679,93 @@ TEST_CASE(
     null_space::report_progress(communicator, "coupled reduced surface-mode probe complete; writing CSV output");
 }
 
+TEST_CASE(
+    "Fixed Central Density Phase Couples To The N3 Homology Tangent",
+    "[null_space][homology][central_density][phase]"
+) {
+    mean_field::utils::Args args = test_utils::setup_args();
+    null_space::N3Equilibrium fixture(std::move(args));
+    const MPI_Comm communicator        = fixture.fem().mesh->GetComm();
+
+    const std::vector<GaugeMode> modes = make_gauge_modes(fixture);
+    const auto homology = std::ranges::find_if(modes, [](const GaugeMode &mode) { return mode.family == "homology"; });
+    REQUIRE(homology != modes.end());
+
+    const auto &layout          = fixture.stellar_operator().GetLayout();
+    const mfem::Vector enthalpy = null_space::const_value_view(fixture.state(), layout, null_space::enthalpyValue);
+    const mfem::Vector enthalpyDirection =
+        null_space::const_value_view(homology->direction, layout, null_space::enthalpyValue);
+
+    const mean_field::field::FieldDofMap enthalpyMap =
+        mean_field::field::make_field_dof_map<mean_field::field::Enthalpy, null_space::DomainSchema>(
+            *fixture.fem().enthalpyFes
+        );
+    mfem::Vector origin(fixture.fem().mesh->SpaceDimension());
+    origin = 0.0;
+    mean_field::field::FieldPointDofMap centerDof =
+        mean_field::field::make_field_point_dof_map<mean_field::field::Enthalpy>(
+            *fixture.fem().enthalpyFes, enthalpyMap, origin, 1.0e-12
+        );
+
+    double localCentralEnthalpy = 0.0;
+    for (const int reducedDof : centerDof.reduced_dofs()) {
+        localCentralEnthalpy += enthalpy(reducedDof);
+    }
+    double centralEnthalpy = 0.0;
+    MPI_Allreduce(&localCentralEnthalpy, &centralEnthalpy, 1, MPI_DOUBLE, MPI_SUM, communicator);
+    REQUIRE(std::isfinite(centralEnthalpy));
+    REQUIRE(centralEnthalpy > 0.0);
+
+    const auto &equationOfState                       = fixture.model().equationOfState();
+    const mean_field::eos::DensityValue targetDensity = mean_field::eos::evaluate<mean_field::eos::quantity::Density>(
+        equationOfState, mean_field::eos::SpecificEnthalpyValue{centralEnthalpy}
+    );
+    const mean_field::models::CompiledFixedCentralDensity compiled =
+        mean_field::models::compileConstraint(mean_field::models::FixedCentralDensity{targetDensity}, equationOfState);
+    mean_field::operators::PreparedCentralDensityConstraint phase(std::move(centerDof), communicator);
+    phase.Prepare(compiled, enthalpy, 0.0, {.enthalpy = {.identity = 3251, .revision = 1}});
+
+    mfem::Vector enthalpyAction(enthalpy.Size());
+    mfem::Vector phaseAction(1);
+    enthalpyAction = 0.0;
+    phaseAction    = 0.0;
+    phase.ApplyJacobian(
+        {.enthalpyVariation = enthalpyDirection, .borderVariation = 0.0},
+        {.enthalpyAction = enthalpyAction, .phaseAction = phaseAction}
+    );
+
+    const double couplingScale         = std::max(1.0, std::abs(compiled.targetEnthalpy().value()));
+    const double enthalpyDirectionNorm = null_space::global_norm(enthalpyDirection, communicator);
+    const double homologyDirectionNorm = null_space::global_norm(homology->direction, communicator);
+    const double absolutePhaseCoupling = std::abs(phaseAction(0));
+    INFO("Central enthalpy = " << centralEnthalpy);
+    INFO("N3 homology phase coupling = " << phaseAction(0));
+    REQUIRE(std::isfinite(phaseAction(0)));
+    REQUIRE(enthalpyDirectionNorm > 0.0);
+    REQUIRE(homologyDirectionNorm > 0.0);
+    CHECK(absolutePhaseCoupling > 100.0 * std::numeric_limits<double>::epsilon() * couplingScale);
+
+    int rank = 0;
+    MPI_Comm_rank(communicator, &rank);
+    if (rank == 0) {
+        experiment::record_experiment_result(
+            "fixed_central_density_homology_coupling", "n3_homology",
+            {{"mesh_file", test_utils::setup_args().mesh_file},
+             {"local_state_dofs", std::to_string(fixture.stellar_operator().Width())}},
+            {{"target_density", compiled.targetDensity().value()},
+             {"target_enthalpy", compiled.targetEnthalpy().value()},
+             {"central_enthalpy", centralEnthalpy},
+             {"homology_phase_action", phaseAction(0)},
+             {"absolute_phase_coupling", absolutePhaseCoupling},
+             {"target_scaled_phase_coupling", absolutePhaseCoupling / couplingScale},
+             {"enthalpy_direction_norm", enthalpyDirectionNorm},
+             {"enthalpy_normalized_phase_coupling", absolutePhaseCoupling / enthalpyDirectionNorm},
+             {"homology_direction_norm", homologyDirectionNorm},
+             {"state_normalized_phase_coupling", absolutePhaseCoupling / homologyDirectionNorm}}
+        );
+    }
+}
+
 TEST_CASE(
     "N3 Homology Mass Cancellation At The Registered Polynomial Order",
     "[null_space][homology][mass_normalization][convergence][p_refinement]"

experiments/preconditioning_diagnostics.cpp

@@ -0,0 +1,455 @@
+#include <algorithm>
+#include <chrono>
+#include <cmath>
+#include <cstdlib>
+#include <iostream>
+#include <map>
+#include <numbers>
+#include <ranges>
+#include <span>
+#include <string>
+#include <utility>
+#include <vector>
+
+#include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
+#include <mpi.h>
+
+import mean_field;
+import test_helpers;
+import experiment;
+
+namespace {
+    using Clock = std::chrono::steady_clock;
+
+    [[nodiscard]] const char *build_configuration() noexcept {
+#ifdef NDEBUG
+        return "release";
+#else
+        return "debug";
+#endif
+    }
+
+    [[nodiscard]] double maximum_rank_seconds(
+        const Clock::time_point start,
+        const MPI_Comm communicator
+    ) {
+        const double localSeconds = std::chrono::duration<double>(Clock::now() - start).count();
+        double maximumSeconds{0.0};
+        MPI_Allreduce(&localSeconds, &maximumSeconds, 1, MPI_DOUBLE, MPI_MAX, communicator);
+        return maximumSeconds;
+    }
+
+    void announce(
+        const MPI_Comm communicator,
+        const std::string &message
+    ) {
+        int rank{0};
+        MPI_Comm_rank(communicator, &rank);
+        if (rank == 0) {
+            std::cout << message << std::endl;
+        }
+    }
+
+    class ArnoldiProgressOperator final : public mfem::Operator {
+    public:
+        ArnoldiProgressOperator(
+            const mfem::Operator &operation,
+            const MPI_Comm communicator,
+            const int expectedApplications,
+            const int reportingInterval
+        )
+            : mfem::Operator(
+                  operation.Height(),
+                  operation.Width()
+              ),
+              m_operation(&operation),
+              m_communicator(communicator),
+              m_expectedApplications(expectedApplications),
+              m_reportingInterval(reportingInterval) {
+        }
+
+        void Mult(
+            const mfem::Vector &input,
+            mfem::Vector &output
+        ) const override {
+            m_operation->Mult(input, output);
+            ++m_completedApplications;
+            if (m_completedApplications == 1 || m_completedApplications == m_expectedApplications ||
+                m_completedApplications % m_reportingInterval == 0) {
+                announce(
+                    m_communicator, "Arnoldi progress: " + std::to_string(m_completedApplications) + "/" +
+                                        std::to_string(m_expectedApplications) + " Jacobian applications"
+                );
+            }
+        }
+
+    private:
+        const mfem::Operator *m_operation;
+        MPI_Comm m_communicator;
+        int m_expectedApplications;
+        int m_reportingInterval;
+        mutable int m_completedApplications{0};
+    };
+
+    [[nodiscard]] mean_field::operators::StellarEquilibriumDependencies make_dependencies() {
+        return {
+            .discretization     = {.identity = 8101, .revision = 1},
+            .density            = {.identity = 8103, .revision = 1},
+            .surfaceDeformation = {.identity = 8107, .revision = 1},
+            .gravityGradient    = {.identity = 8111, .revision = 1},
+            .gravityPotential   = {.identity = 8117, .revision = 1},
+            .enthalpy           = {.identity = 8123, .revision = 1},
+            .bernoulliConstant  = {.identity = 8129, .revision = 1},
+            .rotation           = {.identity = 8131, .revision = 1},
+            .targetMass         = {.identity = 8137, .revision = 1}
+        };
+    }
+
+    [[nodiscard]] mean_field::physics::RigidRotation make_zero_rotation() {
+        mfem::Vector angularVelocity(3);
+        mfem::Vector center(3);
+        angularVelocity = 0.0;
+        center          = 0.0;
+        return {angularVelocity, center};
+    }
+
+    [[nodiscard]] double global_norm(
+        const mfem::Vector &vector,
+        const MPI_Comm communicator
+    ) {
+        const double localSquaredNorm = vector * vector;
+        double globalSquaredNorm{0.0};
+        MPI_Allreduce(&localSquaredNorm, &globalSquaredNorm, 1, MPI_DOUBLE, MPI_SUM, communicator);
+        return std::sqrt(std::max(globalSquaredNorm, 0.0));
+    }
+
+    [[nodiscard]] mfem::Vector make_block_balanced_direction(
+        const int stateSize,
+        const std::span<const mean_field::operators::RootBlockDescriptor> valueBlocks,
+        const MPI_Comm communicator
+    ) {
+        mfem::Vector direction(stateSize);
+        direction = 0.0;
+
+        for (const mean_field::operators::RootBlockDescriptor &block : valueBlocks) {
+            mfem::Vector values(direction.GetData() + block.offset, block.size);
+            for (int index = 0; index < values.Size(); ++index) {
+                const double ordinal = static_cast<double>(block.canonicalIndex + 1);
+                values(index)        = std::sin(0.6180339887498948 * static_cast<double>(index + 1) + ordinal);
+            }
+            const double norm = global_norm(values, communicator);
+            if (norm > 0.0) {
+                values /= norm;
+            }
+        }
+        return direction;
+    }
+
+    void require_finite(const double value) {
+        REQUIRE(std::isfinite(value));
+    }
+
+    [[nodiscard]] std::map<
+        std::string,
+        std::string>
+    common_parameters(
+        const std::string &measurement,
+        const int stateSize
+    ) {
+        return {
+            {"build_configuration", build_configuration()},
+            {"equation_of_state", "Polytrope(n=3)"},
+            {"experiment_schema", "p0_extended_v2"},
+            {"linearization_state", "projected_lane_emden"},
+            {"measurement", measurement},
+            {"mesh_file", test_utils::setup_args().mesh_file},
+            {"preconditioner", "identity"},
+            {"preconditioned_product", "J M^-1"},
+            {"root_dimension", std::to_string(stateSize)}
+        };
+    }
+} // namespace
+
+TEST_CASE(
+    "Stellar Equilibrium P0 Identity Preconditioning Baseline",
+    "[preconditioning][diagnostics][baseline][spectrum]"
+) {
+    using namespace mean_field;
+
+    constexpr int arnoldiDimension          = 48;
+    const MPI_Comm world                    = MPI_COMM_WORLD;
+    const Clock::time_point experimentStart = Clock::now();
+    announce(world, "P0 extended baseline: constructing the finite-element discretization");
+
+    const Clock::time_point finiteElementSetupStart = Clock::now();
+    utils::Args args                                = test_utils::setup_args();
+    fem::FEM finiteElementModel                     = fem::setup_fem(args.mesh_file, args, 0);
+    REQUIRE(finiteElementModel.okay());
+    const MPI_Comm communicator            = finiteElementModel.mesh->GetComm();
+    const double finiteElementSetupSeconds = maximum_rank_seconds(finiteElementSetupStart, communicator);
+    announce(
+        communicator, "P0 extended baseline: finite-element setup completed in " +
+                          std::to_string(finiteElementSetupSeconds) + " seconds"
+    );
+
+    constexpr double stellarRadius                                  = utils::RADIUS;
+    constexpr double targetMass                                     = utils::MASS;
+    const Clock::time_point calibrationStart                        = Clock::now();
+    const seed::DimensionlessLaneEmdenSolution dimensionlessProfile = seed::integrateLaneEmden(3.0, 10.0);
+    REQUIRE(dimensionlessProfile.firstZeroCoordinate.has_value());
+    const double surfaceCoordinate = *dimensionlessProfile.firstZeroCoordinate;
+    const double surfaceDerivative =
+        dimensionlessProfile.thetaDerivative(dimensionlessProfile.thetaDerivative.Size() - 1);
+    const double dimensionlessMass = -surfaceCoordinate * surfaceCoordinate * surfaceDerivative;
+    REQUIRE(dimensionlessMass > 0.0);
+
+    const double massScale          = targetMass / (4.0 * std::numbers::pi_v<double> * dimensionlessMass);
+    const double polytropicConstant = std::numbers::pi_v<double> * utils::G * std::pow(massScale, 2.0 / 3.0);
+    const double radialScale        = stellarRadius / surfaceCoordinate;
+    const double centralDensity =
+        std::pow(polytropicConstant / (std::numbers::pi_v<double> * utils::G * radialScale * radialScale), 1.5);
+    const double calibrationSeconds                  = maximum_rank_seconds(calibrationStart, communicator);
+
+    const Clock::time_point problemConstructionStart = Clock::now();
+    const auto stellarModel                          = model::StellarModel(
+        eos::Polytrope({.n = 3.0, .K = polytropicConstant}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{targetMass}}),
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{centralDensity}})
+    );
+    auto problem                            = equilibrium::discretize(stellarModel, finiteElementModel);
+    const double problemConstructionSeconds = maximum_rank_seconds(problemConstructionStart, communicator);
+    announce(communicator, "P0 extended baseline: projecting the Lane-Emden seed");
+
+    const Clock::time_point seedProjectionStart = Clock::now();
+    const auto projected = seed::makeProjectedEquilibriumState(problem, seed::LaneEmden({.radialSampleCount = 4096}));
+    const double seedProjectionSeconds = maximum_rank_seconds(seedProjectionStart, communicator);
+    announce(communicator, "P0 extended baseline: preparing the complete equilibrium operator");
+
+    const Clock::time_point operatorPreparationStart = Clock::now();
+    const operators::PreparedCentralDensityStellarEquilibriumReport preparation =
+        problem.Prepare(projected.values, make_dependencies(), make_zero_rotation());
+    REQUIRE(preparation.assembledResidual);
+    const double operatorPreparationSeconds = maximum_rank_seconds(operatorPreparationStart, communicator);
+
+    const mfem::Operator &rawJacobian       = problem.GetLinearizationOperator();
+    mfem::Vector knownDirection =
+        make_block_balanced_direction(problem.StateSize(), problem.GetManifest().valueBlocks(), communicator);
+    mfem::Vector rightHandSide(problem.EquationSize());
+    const Clock::time_point applicationStart = Clock::now();
+    rawJacobian.Mult(knownDirection, rightHandSide);
+    const double applicationSeconds = maximum_rank_seconds(applicationStart, communicator);
+    REQUIRE(rightHandSide.Size() == problem.EquationSize());
+    require_finite(global_norm(rightHandSide, communicator));
+    announce(
+        communicator, "P0 extended baseline: first prepared Jacobian application completed in " +
+                          std::to_string(applicationSeconds) + " seconds"
+    );
+
+    if (std::getenv("MEANFIELD_SINGLE_JACOBIAN_BENCHMARK") != nullptr) {
+        int rank{0};
+        MPI_Comm_rank(communicator, &rank);
+        if (rank == 0) {
+            std::cout << "Single prepared Jacobian application: " << applicationSeconds << " seconds\n";
+        }
+        return;
+    }
+
+    solver::IdentityPreconditioner identity(problem.StateSize());
+    solver::InstrumentedOperator instrumentedJacobian(rawJacobian);
+    solver::InstrumentedPreconditioner instrumentedPreconditioner(identity);
+    solver::ResidualHistoryMonitor monitor;
+    mfem::FGMRESSolver krylov(communicator);
+    krylov.SetPreconditioner(instrumentedPreconditioner);
+    krylov.SetOperator(instrumentedJacobian);
+    krylov.SetMonitor(monitor);
+    krylov.SetRelTol(1.0e-8);
+    krylov.SetAbsTol(1.0e-12);
+    krylov.SetMaxIter(40);
+    krylov.SetKDim(20);
+    krylov.SetPrintLevel(1);
+
+    mfem::Vector solution(problem.StateSize());
+    solution = 0.0;
+    const operators::PreparedStellarEquilibriumStatistics statisticsBeforeSolve =
+        problem.GetPreparedOperator().GetPhysicalOperator().GetStatistics();
+    announce(communicator, "P0 extended baseline: starting the 40-iteration identity-preconditioned FGMRES solve");
+    const Clock::time_point solveStart = Clock::now();
+    krylov.Mult(rightHandSide, solution);
+    const double localSolveSeconds = std::chrono::duration<double>(Clock::now() - solveStart).count();
+    const operators::PreparedStellarEquilibriumStatistics statisticsAfterSolve =
+        problem.GetPreparedOperator().GetPhysicalOperator().GetStatistics();
+
+    announce(communicator, "P0 extended baseline: independently reconstructing the true residual");
+    const Clock::time_point directResidualStart           = Clock::now();
+    const solver::LinearSolveMeasurement solveMeasurement = solver::measureLinearSolve(
+        krylov, rawJacobian, rightHandSide, solution, problem.GetManifest().residualBlocks(),
+        instrumentedJacobian.GetStatistics(), instrumentedPreconditioner.GetStatistics(),
+        instrumentedPreconditioner.GetLifecycleStatistics(), monitor, localSolveSeconds, communicator
+    );
+    const double directResidualMeasurementSeconds = maximum_rank_seconds(directResidualStart, communicator);
+    require_finite(solveMeasurement.directResidual.relativeResidual);
+    require_finite(solveMeasurement.solveSecondsMaximumRank);
+
+    std::map<std::string, double> solveMetrics{
+        {"solver_converged", solveMeasurement.solverConverged ? 1.0 : 0.0},
+        {"outer_iterations", static_cast<double>(solveMeasurement.outerIterations)},
+        {"reported_initial_residual_norm", solveMeasurement.solverReportedInitialNorm},
+        {"reported_final_residual_norm", solveMeasurement.solverReportedFinalNorm},
+        {"reported_residual_reduction", solveMeasurement.solverReportedResidualReduction},
+        {"true_residual_norm", solveMeasurement.directResidual.trueResidualNorm},
+        {"true_relative_residual", solveMeasurement.directResidual.relativeResidual},
+        {"rhs_norm", solveMeasurement.directResidual.rightHandSideNorm},
+        {"true_residual_digits_per_jacobian_application",
+         solveMeasurement.trueResidualDigitsReducedPerJacobianApplication},
+        {"finite_element_setup_seconds", finiteElementSetupSeconds},
+        {"lane_emden_calibration_seconds", calibrationSeconds},
+        {"equilibrium_problem_construction_seconds", problemConstructionSeconds},
+        {"seed_projection_seconds", seedProjectionSeconds},
+        {"operator_preparation_seconds", operatorPreparationSeconds},
+        {"initial_jacobian_application_seconds", applicationSeconds},
+        {"direct_residual_measurement_seconds", directResidualMeasurementSeconds},
+        {"solve_seconds_maximum_rank", solveMeasurement.solveSecondsMaximumRank},
+        {"jacobian_applications", static_cast<double>(solveMeasurement.jacobian.applications)},
+        {"jacobian_application_seconds", solveMeasurement.jacobian.totalSeconds},
+        {"jacobian_maximum_application_seconds", solveMeasurement.jacobian.maximumSeconds},
+        {"inverse_preconditioner_applications",
+         static_cast<double>(solveMeasurement.inversePreconditioner.applications)},
+        {"inverse_preconditioner_application_seconds", solveMeasurement.inversePreconditioner.totalSeconds},
+        {"inverse_preconditioner_maximum_application_seconds", solveMeasurement.inversePreconditioner.maximumSeconds},
+        {"inverse_preconditioner_setups", static_cast<double>(solveMeasurement.inversePreconditionerLifecycle.setups)},
+        {"inverse_preconditioner_refreshes",
+         static_cast<double>(solveMeasurement.inversePreconditionerLifecycle.refreshes)},
+        {"inverse_preconditioner_setup_seconds", solveMeasurement.inversePreconditionerLifecycle.setupSeconds},
+        {"inverse_preconditioner_refresh_seconds", solveMeasurement.inversePreconditionerLifecycle.refreshSeconds},
+        {"prepared_residual_assemblies_during_solve",
+         static_cast<double>(statisticsAfterSolve.residualAssemblies - statisticsBeforeSolve.residualAssemblies)},
+        {"prepared_geometry_builds_during_solve",
+         static_cast<double>(
+             statisticsAfterSolve.generatedGeometryBuilds - statisticsBeforeSolve.generatedGeometryBuilds
+         )},
+        {"prepared_jacobian_applications_during_solve",
+         static_cast<double>(statisticsAfterSolve.jacobianApplications - statisticsBeforeSolve.jacobianApplications)}
+    };
+    for (const solver::ResidualBlockMeasurement &block : solveMeasurement.directResidual.blocks) {
+        const std::string prefix                                   = "residual_block." + block.stableId;
+        solveMetrics[prefix + ".descriptor_scale"]                 = block.descriptorScale;
+        solveMetrics[prefix + ".rhs_norm"]                         = block.rightHandSideNorm;
+        solveMetrics[prefix + ".true_norm"]                        = block.trueResidualNorm;
+        solveMetrics[prefix + ".block_relative_residual"]          = block.blockRelativeResidual;
+        solveMetrics[prefix + ".scaled_rhs_norm"]                  = block.scaledRightHandSideNorm;
+        solveMetrics[prefix + ".scaled_true_norm"]                 = block.scaledTrueResidualNorm;
+        solveMetrics[prefix + ".fraction_global_squared_residual"] = block.fractionOfGlobalSquaredResidualNorm;
+        solveMetrics[prefix + ".global_relative_contribution"]     = block.contributionToGlobalRelativeResidual;
+    }
+    experiment::record_experiment_result(
+        "stellar_preconditioning_p0", "identity_linear_solve", common_parameters("linear_solve", problem.StateSize()),
+        std::move(solveMetrics)
+    );
+
+    const double reportedInitialDenominator = std::max(solveMeasurement.solverReportedInitialNorm, 1.0e-300);
+    for (std::size_t sample = 0; sample < solveMeasurement.reportedResidualHistory.size(); ++sample) {
+        const solver::IterationResidualMeasurement &residual = solveMeasurement.reportedResidualHistory[sample];
+        experiment::record_experiment_result(
+            "stellar_preconditioning_p0", "identity_fgmres_history_" + std::to_string(sample),
+            common_parameters("fgmres_residual_history", problem.StateSize()),
+            {{"history_sample", static_cast<double>(sample)},
+             {"iteration", static_cast<double>(residual.iteration)},
+             {"reported_residual_norm", residual.reportedNorm},
+             {"reported_relative_residual", residual.reportedNorm / reportedInitialDenominator},
+             {"final_measurement", residual.final ? 1.0 : 0.0}}
+        );
+    }
+
+    instrumentedJacobian.ResetStatistics();
+    instrumentedPreconditioner.ResetStatistics();
+    solver::FixedRightPreconditionedOperator rightPreconditionedProduct(
+        instrumentedJacobian, instrumentedPreconditioner
+    );
+    ArnoldiProgressOperator progressOperator(rightPreconditionedProduct, communicator, arnoldiDimension, 4);
+    announce(
+        communicator,
+        "P0 extended baseline: starting the " + std::to_string(arnoldiDimension) + "-vector Arnoldi measurement"
+    );
+    const solver::ArnoldiSpectralMeasurement spectrum = solver::measureArnoldiSpectrum(
+        progressOperator, knownDirection, communicator,
+        {.krylovDimension                  = arnoldiDimension,
+         .breakdownRelativeTolerance       = 1.0e-13,
+         .ritzConvergenceRelativeTolerance = 1.0e-7,
+         .reorthogonalize                  = true}
+    );
+    require_finite(spectrum.projectedLargestSingularValue);
+    require_finite(spectrum.centroidRealPart);
+    require_finite(spectrum.rmsClusterRadius);
+
+    experiment::record_experiment_result(
+        "stellar_preconditioning_p0", "identity_arnoldi_summary",
+        common_parameters("arnoldi_summary", problem.StateSize()),
+        {{"requested_krylov_dimension", static_cast<double>(spectrum.requestedDimension)},
+         {"achieved_krylov_dimension", static_cast<double>(spectrum.achievedDimension)},
+         {"invariant_subspace_found", spectrum.invariantSubspaceFound ? 1.0 : 0.0},
+         {"operator_applications", static_cast<double>(spectrum.operatorApplications)},
+         {"arnoldi_operator_application_seconds", spectrum.operatorApplicationSecondsMaximumRank},
+         {"arnoldi_operator_maximum_application_seconds", spectrum.operatorMaximumApplicationSecondsMaximumRank},
+         {"arnoldi_measurement_seconds", spectrum.measurementSecondsMaximumRank},
+         {"arnoldi_nonapplication_seconds", spectrum.nonApplicationSecondsMaximumRank},
+         {"experiment_elapsed_through_arnoldi_seconds", maximum_rank_seconds(experimentStart, communicator)},
+         {"converged_ritz_values", static_cast<double>(spectrum.convergedRitzValueCount)},
+         {"negative_real_part_ritz_values", static_cast<double>(spectrum.negativeRealPartCount)},
+         {"projected_largest_singular_value", spectrum.projectedLargestSingularValue},
+         {"projected_smallest_singular_value", spectrum.projectedSmallestSingularValue},
+         {"projected_condition_proxy", spectrum.projectedConditionProxy},
+         {"ritz_centroid_real", spectrum.centroidRealPart},
+         {"ritz_centroid_imaginary", spectrum.centroidImaginaryPart},
+         {"ritz_rms_distance_from_one", spectrum.rmsDistanceFromOne},
+         {"ritz_rms_cluster_radius", spectrum.rmsClusterRadius},
+         {"ritz_minimum_magnitude", spectrum.minimumMagnitude},
+         {"ritz_maximum_magnitude", spectrum.maximumMagnitude},
+         {"ritz_minimum_real_part", spectrum.minimumRealPart},
+         {"ritz_maximum_real_part", spectrum.maximumRealPart},
+         {"ritz_maximum_absolute_imaginary_part", spectrum.maximumAbsoluteImaginaryPart},
+         {"ritz_conjugate_pair_defect", spectrum.conjugatePairDefect},
+         {"projected_departure_from_normality", spectrum.projectedDepartureFromNormality},
+         {"projected_field_of_values_minimum_real_part", spectrum.projectedFieldOfValuesMinimumRealPart},
+         {"projected_field_of_values_maximum_real_part", spectrum.projectedFieldOfValuesMaximumRealPart},
+         {"measured_jacobian_applications", static_cast<double>(instrumentedJacobian.GetStatistics().applications)},
+         {"measured_jacobian_application_seconds", instrumentedJacobian.GetStatistics().totalSeconds},
+         {"measured_jacobian_maximum_application_seconds", instrumentedJacobian.GetStatistics().maximumSeconds},
+         {"measured_inverse_preconditioner_applications",
+          static_cast<double>(instrumentedPreconditioner.GetStatistics().applications)},
+         {"measured_inverse_preconditioner_application_seconds",
+          instrumentedPreconditioner.GetStatistics().totalSeconds}}
+    );
+
+    std::vector<solver::RitzValueMeasurement> orderedRitzValues = spectrum.ritzValues;
+    std::ranges::sort(orderedRitzValues, [](const auto &left, const auto &right) {
+        if (left.realPart != right.realPart) {
+            return left.realPart < right.realPart;
+        }
+        return left.imaginaryPart < right.imaginaryPart;
+    });
+    for (std::size_t index = 0; index < orderedRitzValues.size(); ++index) {
+        const solver::RitzValueMeasurement &ritz = orderedRitzValues[index];
+        experiment::record_experiment_result(
+            "stellar_preconditioning_p0", "identity_ritz_" + std::to_string(index),
+            common_parameters("ritz_value", problem.StateSize()),
+            {{"ritz_index", static_cast<double>(index)},
+             {"ritz_real", ritz.realPart},
+             {"ritz_imaginary", ritz.imaginaryPart},
+             {"ritz_magnitude", ritz.magnitude},
+             {"ritz_distance_from_one", ritz.distanceFromOne},
+             {"ritz_residual_estimate", ritz.residualEstimate},
+             {"ritz_relative_residual_estimate", ritz.relativeResidualEstimate},
+             {"ritz_converged", ritz.converged ? 1.0 : 0.0}}
+        );
+    }
+
+    int rank{0};
+    MPI_Comm_rank(communicator, &rank);
+    if (rank == 0) {
+        std::cout << "P0 identity baseline: " << solveMeasurement.outerIterations << " FGMRES iterations, "
+                  << spectrum.achievedDimension << " Arnoldi vectors, true relative residual "
+                  << solveMeasurement.directResidual.relativeResidual << '\n';
+    }
+}

libmeanfield/impl/mapping/domain_mapper.cpp

@@ -489,7 +489,8 @@ namespace mean_field::mapping {
         mfem::ElementTransformation &transformation,
         const mfem::IntegrationPoint &integration_point,
         Workspace &workspace,
-        CompactificationPointData &point_data
+        CompactificationPointData &point_data,
+        const mfem::DenseMatrix *inverse_mesh_jacobian
     ) const {
         const mfem::FiniteElement &element = compactification.GetElement();
         const mfem::Vector &dofs           = compactification.GetDofs();
@@ -509,13 +510,17 @@ namespace mean_field::mapping {
                 return MappingStatus::non_finite_input;
         }
 
-        transformation.SetIntPoint(&integration_point);
-
         workspace.m_compactification_shape.SetSize(dof_count);
         workspace.m_compactification_dshape.SetSize(dof_count, m_options.dimension);
 
         element.CalcShape(integration_point, workspace.m_compactification_shape);
-        element.CalcPhysDShape(transformation, workspace.m_compactification_dshape);
+        if (inverse_mesh_jacobian != nullptr) {
+            workspace.m_reference_dshape.SetSize(dof_count, m_options.dimension);
+            element.CalcDShape(integration_point, workspace.m_reference_dshape);
+            mfem::Mult(workspace.m_reference_dshape, *inverse_mesh_jacobian, workspace.m_compactification_dshape);
+        } else {
+            element.CalcPhysDShape(transformation, workspace.m_compactification_dshape);
+        }
 
         point_data.coordinate = dofs * workspace.m_compactification_shape;
         point_data.coordinate_gradient.SetSize(m_options.dimension);
@@ -539,10 +544,9 @@ namespace mean_field::mapping {
         const mfem::IntegrationPoint &integration_point,
         Workspace &workspace,
         mfem::Vector &value,
-        mfem::DenseMatrix &jacobian
+        mfem::DenseMatrix &jacobian,
+        const mfem::DenseMatrix *inverse_mesh_jacobian
     ) const {
-        transformation.SetIntPoint(&integration_point);
-
         const mfem::FiniteElement &element  = field.GetElement();
         const mfem::DenseMatrix &dof_matrix = field.GetDofMatrix();
 
@@ -550,7 +554,13 @@ namespace mean_field::mapping {
         workspace.m_mesh_dshape.SetSize(element.GetDof(), m_options.dimension);
 
         element.CalcShape(integration_point, workspace.m_shape);
-        element.CalcPhysDShape(transformation, workspace.m_mesh_dshape);
+        if (inverse_mesh_jacobian != nullptr) {
+            workspace.m_reference_dshape.SetSize(element.GetDof(), m_options.dimension);
+            element.CalcDShape(integration_point, workspace.m_reference_dshape);
+            mfem::Mult(workspace.m_reference_dshape, *inverse_mesh_jacobian, workspace.m_mesh_dshape);
+        } else {
+            element.CalcPhysDShape(transformation, workspace.m_mesh_dshape);
+        }
 
         value.SetSize(m_options.dimension);
         dof_matrix.MultTranspose(workspace.m_shape, value);
@@ -586,7 +596,7 @@ namespace mean_field::mapping {
 
         EvaluateField(
             element_data.displacement, transformation, integration_point, workspace, workspace.m_field_value,
-            workspace.m_field_jacobian
+            workspace.m_field_jacobian, nullptr
         );
 
         if (!vector_is_finite(context.reference_position) || !vector_is_finite(workspace.m_field_value) ||
@@ -608,7 +618,7 @@ namespace mean_field::mapping {
         if (context.compactified) {
             const MappingStatus coordinate_status = EvaluateCompactificationCoordinate(
                 element_data.compactification, transformation, integration_point, workspace,
-                workspace.m_compactification_point
+                workspace.m_compactification_point, nullptr
             );
 
             if (coordinate_status != MappingStatus::valid)
@@ -663,7 +673,6 @@ namespace mean_field::mapping {
         if (point_status != MappingStatus::valid)
             return point_status;
 
-        transformation.SetIntPoint(&integration_point);
         mfem::Mult(context.mapping.mapping_jacobian, transformation.Jacobian(), workspace.m_full_element_jacobian);
 
         context.quadrature.J_inv.SetSize(m_options.dimension, m_options.dimension);
@@ -766,6 +775,21 @@ namespace mean_field::mapping {
         const MappingPointContext &base_context,
         Workspace &workspace,
         MappingPointVariation &variation
+    ) const {
+        return EvaluatePointVariationImpl(
+            element_data, direction, transformation, integration_point, base_context, workspace, variation, nullptr
+        );
+    }
+
+    MappingStatus DomainMapper::EvaluatePointVariationImpl(
+        const ElementMappingData &element_data,
+        const ElementDisplacementData &direction,
+        mfem::ElementTransformation &transformation,
+        const mfem::IntegrationPoint &integration_point,
+        const MappingPointContext &base_context,
+        Workspace &workspace,
+        MappingPointVariation &variation,
+        const mfem::DenseMatrix *inverse_mesh_jacobian
     ) const {
         ValidateElementData(element_data);
         const ElementMappingData direction_data{
@@ -786,8 +810,13 @@ namespace mean_field::mapping {
                 "domain."
             );
 
+        if (inverse_mesh_jacobian == nullptr) {
+            transformation.SetIntPoint(&integration_point);
+        }
+
         EvaluateField(
-            direction, transformation, integration_point, workspace, workspace.m_field_value, workspace.m_field_jacobian
+            direction, transformation, integration_point, workspace, workspace.m_field_value,
+            workspace.m_field_jacobian, inverse_mesh_jacobian
         );
 
         if (!vector_is_finite(workspace.m_field_value) || !matrix_is_finite(workspace.m_field_jacobian))
@@ -799,7 +828,7 @@ namespace mean_field::mapping {
         if (base_context.compactified) {
             const MappingStatus coordinate_status = EvaluateCompactificationCoordinate(
                 element_data.compactification, transformation, integration_point, workspace,
-                workspace.m_compactification_point
+                workspace.m_compactification_point, inverse_mesh_jacobian
             );
 
             if (coordinate_status != MappingStatus::valid)
@@ -872,27 +901,28 @@ namespace mean_field::mapping {
         Workspace &workspace,
         VolumeMappingVariation &variation
     ) const {
-        const MappingStatus point_status = EvaluatePointVariation(
+        mfem::Mult(base_context.quadrature.J_inv, base_context.mapping.mapping_jacobian, workspace.m_matrix_temp_2);
+
+        const MappingStatus point_status = EvaluatePointVariationImpl(
             element_data, direction, transformation, integration_point, base_context.mapping, workspace,
-            variation.mapping
+            variation.mapping, &workspace.m_matrix_temp_2
         );
         if (point_status != MappingStatus::valid)
             return point_status;
 
-        transformation.SetIntPoint(&integration_point);
         mfem::Mult(
-            variation.mapping.mapping_jacobian_variation, transformation.Jacobian(), workspace.m_full_element_jacobian
+            base_context.quadrature.J_inv, variation.mapping.mapping_jacobian_variation, workspace.m_matrix_temp_1
         );
-        mfem::Mult(base_context.quadrature.J_inv, workspace.m_full_element_jacobian, workspace.m_matrix_temp_1);
 
         variation.inverse_element_jacobian_variation.SetSize(m_options.dimension, m_options.dimension);
         mfem::Mult(
-            workspace.m_matrix_temp_1, base_context.quadrature.J_inv, variation.inverse_element_jacobian_variation
+            workspace.m_matrix_temp_1, base_context.mapping.inverse_mapping_jacobian,
+            variation.inverse_element_jacobian_variation
         );
         variation.inverse_element_jacobian_variation *= -1.0;
 
-        variation.weight_variation =
-            integration_point.weight * transformation.Weight() * variation.mapping.mapping_determinant_variation;
+        variation.weight_variation = base_context.quadrature.weight / base_context.mapping.mapping_determinant *
+                                     variation.mapping.mapping_determinant_variation;
 
         if (!matrix_is_finite(variation.inverse_element_jacobian_variation) ||
             !std::isfinite(variation.weight_variation))

libmeanfield/impl/mapping/transformations.cpp

@@ -217,16 +217,22 @@ namespace mean_field::mapping {
         );
         MFEM_VERIFY(std::isfinite(determinant_variation), "The mapping determinant variation must be finite.");
 
-        mfem::DenseMatrix determinant_correction(dimension, dimension);
-        ComputeHDivMassTensor(context, determinant_correction);
-        determinant_correction *= determinant_variation / determinant;
+        const mfem::DenseMatrix &jacobian          = context.mapping_jacobian;
+        const mfem::DenseMatrix &jacobianVariation = variation.mapping_jacobian_variation;
+        const double inverseDeterminant            = 1.0 / determinant;
+        const double determinantScale              = determinant_variation * inverseDeterminant;
 
-        mfem::DenseMatrix right_jacobian_variation(dimension, dimension);
-        mfem::MultAtB(context.mapping_jacobian, variation.mapping_jacobian_variation, right_jacobian_variation);
-        mfem::MultAtB(variation.mapping_jacobian_variation, context.mapping_jacobian, mass_tensor_variation);
-
-        mass_tensor_variation += right_jacobian_variation;
-        mass_tensor_variation *= 1 / determinant;
-        mass_tensor_variation -= determinant_correction;
+        for (int row = 0; row < dimension; ++row) {
+            for (int column = 0; column < dimension; ++column) {
+                double gram{0.0};
+                double gramVariation{0.0};
+                for (int inner = 0; inner < dimension; ++inner) {
+                    gram += jacobian(inner, row) * jacobian(inner, column);
+                    gramVariation += jacobian(inner, row) * jacobianVariation(inner, column) +
+                                     jacobianVariation(inner, row) * jacobian(inner, column);
+                }
+                mass_tensor_variation(row, column) = inverseDeterminant * (gramVariation - determinantScale * gram);
+            }
+        }
     }
 } // namespace mean_field::mapping

libmeanfield/impl/models/polytropic.cpp

@@ -1,8 +1,12 @@
 module;
+
 #include <cmath>
 #include <format>
-#include <numbers>
+#include <stdexcept>
+#include <utility>
+
 module mean_field;
+
 import :model.structure.polytropic;
 
 namespace mean_field::models::structure {
@@ -24,64 +28,18 @@ namespace mean_field::models::structure {
     }
 
     StructureSeed PolytropicStructure::makeInitialSeed(const StructureSeedRequest &request) const {
-        validateSeedRequest(request);
+        const seed::RadialProfile profile = seed::generateLaneEmdenProfile(
+            m_equationOfState, dimensions::DensityValue{request.centralDensity}, request.radialSampleCount
+        );
 
-        const double polytropicIndex                        = m_equationOfState.polytropic_index();
-        const std::vector<LaneEmdenPoint> laneEmdenSolution = solveLaneEmden(polytropicIndex);
-        const double surfaceCoordinate                      = laneEmdenSolution.back().coordinate;
-        const double centralEnthalpy =
-            eos::evaluate<eos::quantity::SpecificEnthalpy>(m_equationOfState, eos::DensityValue{request.centralDensity})
-                .value();
-        const double radialScaleSquared =
-            centralEnthalpy / (4.0 * std::numbers::pi_v<double> * mean_field::utils::G * request.centralDensity);
-
-        if (!std::isfinite(radialScaleSquared) || radialScaleSquared <= 0.0) {
-            throw std::runtime_error(
-                "The polytropic Lane-Emden radial scale is not "
-                "finite and positive."
-            );
-        }
-
-        const double radialScale = std::sqrt(radialScaleSquared);
-
-        StructureSeed seed;
-
-        seed.radius.SetSize(request.radialSampleCount);
-        seed.density.SetSize(request.radialSampleCount);
-        seed.enthalpy.SetSize(request.radialSampleCount);
-
-        seed.stellarRadius             = radialScale * surfaceCoordinate;
-        seed.centralDensity            = request.centralDensity;
-        seed.centralEnthalpy           = centralEnthalpy;
-
-        std::size_t interpolationIndex = 0;
-
-        for (int sampleIndex = 0; sampleIndex < request.radialSampleCount; ++sampleIndex) {
-            const double sampleFraction =
-                static_cast<double>(sampleIndex) / static_cast<double>(request.radialSampleCount - 1);
-
-            const double dimensionlessRadius = sampleFraction * surfaceCoordinate;
-            const double laneEmdenValue =
-                interpolateLaneEmdenValue(laneEmdenSolution, dimensionlessRadius, interpolationIndex);
-            const double density      = request.centralDensity * std::pow(laneEmdenValue, polytropicIndex);
-
-            seed.radius(sampleIndex)  = radialScale * dimensionlessRadius;
-            seed.density(sampleIndex) = density;
-            seed.enthalpy(sampleIndex) =
-                eos::evaluate<eos::quantity::SpecificEnthalpy>(m_equationOfState, eos::DensityValue{density}).value();
-        }
-
-        seed.radius(0)              = 0.0;
-        seed.density(0)             = request.centralDensity;
-        seed.enthalpy(0)            = centralEnthalpy;
-
-        const int surfaceIndex      = request.radialSampleCount - 1;
-
-        seed.radius(surfaceIndex)   = seed.stellarRadius;
-        seed.density(surfaceIndex)  = 0.0;
-        seed.enthalpy(surfaceIndex) = 0.0;
-
-        return seed;
+        return {
+            .radius          = profile.radius,
+            .density         = profile.density,
+            .enthalpy        = profile.specificEnthalpy,
+            .stellarRadius   = profile.stellarRadius.value(),
+            .centralDensity  = profile.centralDensity.value(),
+            .centralEnthalpy = profile.centralSpecificEnthalpy.value()
+        };
     }
 
     void PolytropicStructure::validate() const {
@@ -90,8 +48,7 @@ namespace mean_field::models::structure {
         if (!std::isfinite(polytropicIndex) || polytropicIndex < 1.0 || polytropicIndex >= 5.0) {
             throw std::invalid_argument(
                 std::format(
-                    "PolytropicStructure requires a finite-radius "
-                    "polytrope with 1 <= n < 5. Instead n = {} was "
+                    "PolytropicStructure requires a finite-radius polytrope with 1 <= n < 5. Instead n = {} was "
                     "provided.",
                     polytropicIndex
                 )
@@ -101,163 +58,10 @@ namespace mean_field::models::structure {
         if (!std::isfinite(m_targetMass) || m_targetMass <= 0.0) {
             throw std::invalid_argument(
                 std::format(
-                    "The target stellar mass must be finite and "
-                    "positive. Instead a value of {} was provided.",
+                    "The target stellar mass must be finite and positive. Instead a value of {} was provided.",
                     m_targetMass
                 )
             );
         }
     }
-
-    void PolytropicStructure::validateSeedRequest(const StructureSeedRequest &request) {
-        if (!std::isfinite(request.centralDensity) || request.centralDensity <= 0.0) {
-            throw std::invalid_argument(
-                std::format(
-                    "The seed central density must be finite and "
-                    "positive. Instead a value of {} was provided.",
-                    request.centralDensity
-                )
-            );
-        }
-
-        if (request.radialSampleCount < 2) {
-            throw std::invalid_argument(
-                std::format(
-                    "A polytropic seed requires at least two radial "
-                    "samples. Instead {} samples were requested.",
-                    request.radialSampleCount
-                )
-            );
-        }
-    }
-
-    PolytropicStructure::LaneEmdenDerivative PolytropicStructure::evaluateLaneEmdenRhs(
-        const double coordinate,
-        const double value,
-        const double derivative,
-        const double polytropicIndex
-    ) {
-        const double nonnegativeValue = std::max(value, 0.0);
-
-        return {
-            .value      = derivative,
-            .derivative = -2.0 * derivative / coordinate - std::pow(nonnegativeValue, polytropicIndex)
-        };
-    }
-
-    PolytropicStructure::LaneEmdenPoint PolytropicStructure::takeLaneEmdenStep(
-        const LaneEmdenPoint &point,
-        const double step,
-        const double polytropicIndex
-    ) {
-        const LaneEmdenDerivative first =
-            evaluateLaneEmdenRhs(point.coordinate, point.value, point.derivative, polytropicIndex);
-
-        const LaneEmdenDerivative second = evaluateLaneEmdenRhs(
-            point.coordinate + 0.5 * step, point.value + 0.5 * step * first.value,
-            point.derivative + 0.5 * step * first.derivative, polytropicIndex
-        );
-
-        const LaneEmdenDerivative third = evaluateLaneEmdenRhs(
-            point.coordinate + 0.5 * step, point.value + 0.5 * step * second.value,
-            point.derivative + 0.5 * step * second.derivative, polytropicIndex
-        );
-
-        const LaneEmdenDerivative fourth = evaluateLaneEmdenRhs(
-            point.coordinate + step, point.value + step * third.value, point.derivative + step * third.derivative,
-            polytropicIndex
-        );
-
-        return {
-            .coordinate = point.coordinate + step,
-            .value = point.value + step / 6.0 * (first.value + 2.0 * second.value + 2.0 * third.value + fourth.value),
-            .derivative =
-                point.derivative +
-                step / 6.0 * (first.derivative + 2.0 * second.derivative + 2.0 * third.derivative + fourth.derivative)
-        };
-    }
-
-    std::vector<PolytropicStructure::LaneEmdenPoint> PolytropicStructure::solveLaneEmden(const double polytropicIndex) {
-        constexpr double initialCoordinate = 1.0e-6;
-        constexpr double integrationStep   = 1.0e-3;
-        constexpr int maximumStepCount     = 2'000'000;
-
-        const double coordinateSquared     = initialCoordinate * initialCoordinate;
-
-        const double coordinateCubed       = coordinateSquared * initialCoordinate;
-
-        const double coordinateFourth      = coordinateSquared * coordinateSquared;
-
-        LaneEmdenPoint point{
-            .coordinate = initialCoordinate,
-            .value      = 1.0 - coordinateSquared / 6.0 + polytropicIndex * coordinateFourth / 120.0,
-            .derivative = -initialCoordinate / 3.0 + polytropicIndex * coordinateCubed / 30.0
-        };
-
-        std::vector<LaneEmdenPoint> solution;
-        solution.reserve(8192);
-        solution.push_back({.coordinate = 0.0, .value = 1.0, .derivative = 0.0});
-        solution.push_back(point);
-
-        for (int stepIndex = 0; stepIndex < maximumStepCount; ++stepIndex) {
-            LaneEmdenPoint nextPoint = takeLaneEmdenStep(point, integrationStep, polytropicIndex);
-
-            if (!std::isfinite(nextPoint.value)) {
-                throw std::runtime_error(
-                    "The Lane-Emden integration produced a non-finite "
-                    "solution before reaching the stellar surface."
-                );
-            }
-
-            if (nextPoint.value <= 0.0) {
-                const double rootFraction = point.value / (point.value - nextPoint.value);
-
-                solution.push_back(
-                    {.coordinate = point.coordinate + rootFraction * (nextPoint.coordinate - point.coordinate),
-                     .value      = 0.0,
-                     .derivative = point.derivative + rootFraction * (nextPoint.derivative - point.derivative)}
-                );
-
-                return solution;
-            }
-
-            solution.push_back(nextPoint);
-            point = nextPoint;
-        }
-
-        throw std::runtime_error(
-            "The Lane-Emden integration did not reach its first zero "
-            "within the configured step limit."
-        );
-    }
-
-    double PolytropicStructure::interpolateLaneEmdenValue(
-        const std::vector<LaneEmdenPoint> &solution,
-        const double coordinate,
-        std::size_t &lowerIndex
-    ) {
-        while (lowerIndex + 1 < solution.size() && solution[lowerIndex + 1].coordinate < coordinate) {
-            ++lowerIndex;
-        }
-
-        if (lowerIndex + 1 >= solution.size()) {
-            return 0.0;
-        }
-
-        const LaneEmdenPoint &lower = solution[lowerIndex];
-        const LaneEmdenPoint &upper = solution[lowerIndex + 1];
-
-        const double interval       = upper.coordinate - lower.coordinate;
-
-        if (interval <= 0.0) {
-            throw std::runtime_error(
-                "The Lane-Emden interpolation grid is not strictly "
-                "increasing."
-            );
-        }
-
-        const double fraction = (coordinate - lower.coordinate) / interval;
-
-        return std::clamp(lower.value + fraction * (upper.value - lower.value), 0.0, 1.0);
-    }
-}; // namespace mean_field::models::structure
+} // namespace mean_field::models::structure

libmeanfield/impl/operators/gravity_field_jacobian.cpp

@@ -3,7 +3,6 @@ module;
 
 module mean_field;
 import :operators.gravity_field_jacobian;
-import :operators.kernels.gravity_field;
 import :utils.blocks;
 
 namespace {
@@ -202,7 +201,6 @@ namespace mean_field::operators {
         const context::gravity_field::GravityFieldGeometryContext &geometry_context =
             m_linearization_context.GetGeometryContext();
         const mfem::Vector &density          = m_linearization_context.GetDensityTrue();
-        const mfem::Vector &displacement     = geometry_context.GetDisplacementTrue();
         const mfem::Vector &gravity_gradient = m_linearization_context.GetGravityGradientTrue();
 
         const mfem::Vector density_direction = make_read_only_value_view(direction, m_state_offsets, density_block);
@@ -244,14 +242,13 @@ namespace mean_field::operators {
         geometry_context.GetMassOperator().Mult(gravity_gradient_direction, gravity_gradient_action);
         geometry_context.GetSourceOperator().Mult(density_direction, source_action);
 
-        kernels::apply_mapped_hdiv_mass_variation(
-            m_fem, m_domain_mapper, gravity_gradient, displacement, displacement_direction_true,
-            mass_variation_action_true
+        geometry_context.GetMassOperator().MultDisplacementVariationTrue(
+            gravity_gradient, displacement_direction_true, mass_variation_action_true
         );
         flux_map.gather(mass_variation_action_true, mass_variation_action);
 
-        kernels::apply_mapped_source_variation(
-            m_fem, m_domain_mapper, density, displacement, displacement_direction_true, source_variation_action_true
+        geometry_context.GetSourceOperator().MultDisplacementVariationTrue(
+            density, displacement_direction_true, source_variation_action_true
         );
         potential_map.gather(source_variation_action_true, source_variation_action);
 

libmeanfield/impl/operators/kernels/barotropic_closure_kernels.cpp

@@ -12,10 +12,11 @@ import :field.registry;
 import :utils.domain;
 
 namespace {
-    namespace eos       = mean_field::eos;
+    namespace dimensions = mean_field::dimensions;
+    namespace eos        = mean_field::eos;
 
-    using DomainSchema  = mean_field::utils::domain::CoreEnvelopeVacuumDomainSchema;
-    using ClosureDomain = mean_field::field::FieldDomainT<mean_field::field::Density>;
+    using DomainSchema   = mean_field::utils::domain::CoreEnvelopeVacuumDomainSchema;
+    using ClosureDomain  = mean_field::field::FieldDomainT<mean_field::field::Density>;
 
     enum class ClosureAction { residual, density, enthalpy };
 
@@ -339,7 +340,9 @@ namespace {
                         const double density = elementDensityInput * densityShape;
 
                         const double equationOfStateDensity =
-                            eos::evaluate<eos::quantity::Density>(barotrope, eos::SpecificEnthalpyValue{baseEnthalpy})
+                            eos::evaluate<dimensions::quantity::Density>(
+                                barotrope, dimensions::SpecificEnthalpyValue{baseEnthalpy}
+                            )
                                 .value();
 
                         integrand = density - equationOfStateDensity;
@@ -348,7 +351,7 @@ namespace {
 
                         const double densityDerivative =
                             eos::partialDerivative<eos::quantity::Density, eos::quantity::SpecificEnthalpy>(
-                                barotrope, eos::SpecificEnthalpyValue{baseEnthalpy}
+                                barotrope, dimensions::SpecificEnthalpyValue{baseEnthalpy}
                             )
                                 .value();
 
@@ -640,16 +643,18 @@ namespace mean_field::operators::kernels {
 
                 enthalpyElement.CalcShape(integrationPoint, enthalpyShape);
 
-                const double densityValue  = elementBaseDensity * densityShape;
+                const double densityValue           = elementBaseDensity * densityShape;
 
-                const double enthalpyValue = elementBaseEnthalpy * enthalpyShape;
+                const double enthalpyValue          = elementBaseEnthalpy * enthalpyShape;
 
-                const double equationOfStateDensity =
-                    eos::evaluate<eos::quantity::Density>(barotrope, eos::SpecificEnthalpyValue{enthalpyValue}).value();
+                const double equationOfStateDensity = eos::evaluate<dimensions::quantity::Density>(
+                                                          barotrope, dimensions::SpecificEnthalpyValue{enthalpyValue}
+                )
+                                                          .value();
 
-                const double closureValue        = densityValue - equationOfStateDensity;
+                const double closureValue           = densityValue - equationOfStateDensity;
 
-                const double geometryActionValue = closureValue * mappingVariation.weight_variation;
+                const double geometryActionValue    = closureValue * mappingVariation.weight_variation;
 
                 MFEM_VERIFY(
                     std::isfinite(closureValue) && std::isfinite(geometryActionValue),

libmeanfield/impl/operators/kernels/gravity_kernels.cpp

@@ -490,6 +490,8 @@ namespace mean_field::operators::kernels {
 
         mfem::DenseMatrix gravity_gradient_shape;
         mfem::DenseMatrix mass_tensor_variation;
+        mapping::VolumeMappingContext mapping_context;
+        mapping::VolumeMappingVariation mapping_variation;
 
         for (int element_id = 0; element_id < f.mesh->GetNE(); ++element_id) {
             const mfem::FiniteElement &gravity_gradient_element = *f.gravityFluxFes->GetFE(element_id);
@@ -555,7 +557,6 @@ namespace mean_field::operators::kernels {
                 const mfem::IntegrationPoint &integration_point = integration_rule.IntPoint(q);
                 transformation->SetIntPoint(&integration_point);
 
-                mapping::VolumeMappingContext mapping_context;
                 const mapping::MappingStatus status = domain_mapper.EvaluateVolume(
                     mapping_data, *transformation, integration_point, workspace, mapping_context
                 );
@@ -568,7 +569,6 @@ namespace mean_field::operators::kernels {
                         << ", status: " << static_cast<int>(status)
                 );
 
-                mapping::VolumeMappingVariation mapping_variation;
                 const mapping::MappingStatus variation_status = domain_mapper.EvaluateVolumeVariation(
                     mapping_data, displacement_variation_data, *transformation, integration_point, mapping_context,
                     workspace, mapping_variation

libmeanfield/impl/operators/kernels/pressure_force_kernels.cpp

@@ -12,9 +12,10 @@ module mean_field;
 import :operators.kernels.pressure_force;
 
 namespace {
-    namespace eos      = mean_field::eos;
+    namespace dimensions = mean_field::dimensions;
+    namespace eos        = mean_field::eos;
 
-    using DomainSchema = mean_field::utils::domain::CoreEnvelopeVacuumDomainSchema;
+    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);
@@ -416,14 +417,15 @@ namespace {
 
                 if (pressureForceAction == PressureForceAction::residual ||
                     pressureForceAction == PressureForceAction::displacement) {
-                    pressureFactor =
-                        eos::evaluate<eos::quantity::Pressure>(barotrope, eos::SpecificEnthalpyValue{enthalpyValue})
-                            .value();
+                    pressureFactor = eos::evaluate<dimensions::quantity::Pressure>(
+                                         barotrope, dimensions::SpecificEnthalpyValue{enthalpyValue}
+                    )
+                                         .value();
                 } else {
                     const double enthalpyVariationValue = elementEnthalpyVariation * enthalpyShape;
 
                     pressureFactor = eos::partialDerivative<eos::quantity::Pressure, eos::quantity::SpecificEnthalpy>(
-                                         barotrope, eos::SpecificEnthalpyValue{enthalpyValue}
+                                         barotrope, dimensions::SpecificEnthalpyValue{enthalpyValue}
                                      )
                                          .value() *
                                      enthalpyVariationValue;

libmeanfield/impl/operators/prepared_barotropic_closure.cpp

@@ -287,7 +287,6 @@ namespace mean_field::operators {
 
         mapping::DomainMapper::Workspace workspace(m_fem.mesh->Dimension());
 
-        mfem::Array<int> displacementDofs;
         mfem::Array<int> compactificationDofs;
 
         mfem::Vector elementBaseDensity;
@@ -311,18 +310,19 @@ namespace mean_field::operators {
 
             m_elements.emplace_back();
             ElementPAData &data            = m_elements.back();
+            data.elementId                 = elementId;
 
             data.densityDofTransformation  = m_fem.densityFes->GetElementDofs(elementId, data.densityDofs);
             data.enthalpyDofTransformation = m_fem.enthalpyFes->GetElementDofs(elementId, data.enthalpyDofs);
 
-            mfem::DofTransformation *displacementDofTransformation =
-                m_fem.displacementFes->GetElementVDofs(elementId, displacementDofs);
+            data.displacementDofTransformation =
+                m_fem.displacementFes->GetElementVDofs(elementId, data.displacementDofs);
             mfem::DofTransformation *compactificationDofTransformation =
                 m_fem.compactificationFes->GetElementDofs(elementId, compactificationDofs);
 
             baseDensityLocal.GetSubVector(data.densityDofs, elementBaseDensity);
             baseEnthalpyLocal.GetSubVector(data.enthalpyDofs, elementBaseEnthalpy);
-            displacementLocal.GetSubVector(displacementDofs, elementDisplacement);
+            displacementLocal.GetSubVector(data.displacementDofs, elementDisplacement);
             m_fem.compactificationCoordinate->GetSubVector(compactificationDofs, elementCompactification);
 
             if (data.densityDofTransformation != nullptr) {
@@ -331,8 +331,8 @@ namespace mean_field::operators {
             if (data.enthalpyDofTransformation != nullptr) {
                 data.enthalpyDofTransformation->InvTransformPrimal(elementBaseEnthalpy);
             }
-            if (displacementDofTransformation != nullptr) {
-                displacementDofTransformation->InvTransformPrimal(elementDisplacement);
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->InvTransformPrimal(elementDisplacement);
             }
             if (compactificationDofTransformation != nullptr) {
                 compactificationDofTransformation->InvTransformPrimal(elementCompactification);
@@ -363,6 +363,9 @@ namespace mean_field::operators {
 
             data.densityBasis.SetSize(quadraturePointCount, densityDofCount);
             data.enthalpyBasis.SetSize(quadraturePointCount, enthalpyDofCount);
+            data.inverseElementJacobians.SetSize(
+                quadraturePointCount, m_fem.mesh->Dimension() * m_fem.mesh->Dimension()
+            );
             data.weightedResidual.SetSize(quadraturePointCount);
             data.quadratureWeights.SetSize(quadraturePointCount);
             data.weightedEnthalpyDerivative.SetSize(quadraturePointCount);
@@ -388,6 +391,18 @@ namespace mean_field::operators {
                         << ", quadrature point: " << quadraturePoint << ", status: " << static_cast<int>(mappingStatus)
                 );
 
+                MFEM_VERIFY(
+                    !mappingContext.mapping.compactified,
+                    "Prepared barotropic closure support unexpectedly includes a compactified element."
+                );
+
+                for (int row = 0; row < m_fem.mesh->Dimension(); ++row) {
+                    for (int column = 0; column < m_fem.mesh->Dimension(); ++column) {
+                        data.inverseElementJacobians(quadraturePoint, row * m_fem.mesh->Dimension() + column) =
+                            mappingContext.quadrature.J_inv(row, column);
+                    }
+                }
+
                 densityElement.CalcShape(integrationPoint, densityShape);
                 enthalpyElement.CalcShape(integrationPoint, enthalpyShape);
 
@@ -401,7 +416,7 @@ namespace mean_field::operators {
                 const double density          = elementBaseDensity * densityShape;
                 const double enthalpy         = elementBaseEnthalpy * enthalpyShape;
                 const double quadratureWeight = mappingContext.quadrature.weight;
-                const eos::SpecificEnthalpyValue specificEnthalpy{enthalpy};
+                const dimensions::SpecificEnthalpyValue specificEnthalpy{enthalpy};
                 const double eosDensity =
                     eos::evaluate<eos::quantity::Density>(m_equationOfState, specificEnthalpy).value();
                 const double enthalpyDerivative =
@@ -485,10 +500,7 @@ namespace mean_field::operators {
 
         ApplyThermodynamicActionFull(m_densityVariationTrue, m_enthalpyVariationTrue, m_fullThermodynamicAction);
 
-        kernels::apply_barotropic_closure_displacement_action(
-            m_fem, m_domainMapper, m_equationOfState, m_baseDensityTrue, m_baseEnthalpyTrue, m_baseDisplacementTrue,
-            m_displacementVariationTrue, m_fullDisplacementAction
-        );
+        ApplyDisplacementActionFull(m_displacementVariationTrue, m_fullDisplacementAction);
 
         MFEM_VERIFY(
             m_fullThermodynamicAction.Size() == m_densityMap.full_size() &&
@@ -589,6 +601,86 @@ namespace mean_field::operators {
         local_to_true(*m_fem.densityFes, localAction, actionTrue);
     }
 
+    void PreparedBarotropicClosureOperator::ApplyDisplacementActionFull(
+        const mfem::Vector &displacementVariationTrue,
+        mfem::Vector &actionTrue
+    ) const {
+        MFEM_VERIFY(
+            displacementVariationTrue.Size() == m_displacementMap.full_size(),
+            "The full displacement variation has the wrong size."
+        );
+
+        true_to_local(*m_fem.displacementFes, displacementVariationTrue, m_displacementVariationLocal);
+
+        m_localDisplacementAction.SetSize(m_fem.densityFes->GetVSize());
+        m_localDisplacementAction = 0.0;
+
+        const int dimension       = m_fem.mesh->Dimension();
+
+        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 mapping::ElementDisplacementData directionData =
+                mapping::ElementDisplacementDataFromElementVDofs(displacementElement, m_elementDisplacementVariation);
+            const mfem::DenseMatrix &directionDofs      = directionData.GetDofMatrix();
+            mfem::ElementTransformation *transformation = m_fem.mesh->GetElementTransformation(data.elementId);
+            MFEM_VERIFY(
+                transformation != nullptr,
+                "Prepared barotropic closure displacement action received a null element transformation."
+            );
+            const mfem::FiniteElement &densityElement  = *m_fem.densityFes->GetFE(data.elementId);
+            const mfem::FiniteElement &enthalpyElement = *m_fem.enthalpyFes->GetFE(data.elementId);
+            const mfem::IntegrationRule &integrationRule =
+                get_eos_rule(m_fem, m_equationOfState, densityElement, enthalpyElement, *transformation);
+
+            MFEM_VERIFY(
+                data.inverseElementJacobians.Height() == integrationRule.GetNPoints() &&
+                    data.inverseElementJacobians.Width() == dimension * dimension,
+                "Prepared barotropic closure inverse-Jacobian data has an incompatible size."
+            );
+
+            m_referenceDShape.SetSize(displacementElement.GetDof(), dimension);
+            m_referenceDisplacementJacobian.SetSize(dimension, dimension);
+            m_quadratureDisplacementAction.SetSize(integrationRule.GetNPoints());
+
+            for (int quadraturePoint = 0; quadraturePoint < integrationRule.GetNPoints(); ++quadraturePoint) {
+                const mfem::IntegrationPoint &integrationPoint = integrationRule.IntPoint(quadraturePoint);
+                displacementElement.CalcDShape(integrationPoint, m_referenceDShape);
+                mfem::MultAtB(directionDofs, m_referenceDShape, m_referenceDisplacementJacobian);
+
+                double logarithmicJacobianVariation{0.0};
+                for (int row = 0; row < dimension; ++row) {
+                    for (int column = 0; column < dimension; ++column) {
+                        logarithmicJacobianVariation +=
+                            data.inverseElementJacobians(quadraturePoint, row * dimension + column) *
+                            m_referenceDisplacementJacobian(column, row);
+                    }
+                }
+
+                m_quadratureDisplacementAction(quadraturePoint) =
+                    data.weightedResidual(quadraturePoint) * logarithmicJacobianVariation;
+                MFEM_VERIFY(
+                    std::isfinite(m_quadratureDisplacementAction(quadraturePoint)),
+                    "Prepared barotropic closure displacement action encountered a non-finite quadrature value."
+                );
+            }
+
+            m_elementDisplacementAction.SetSize(data.densityDofs.Size());
+            data.densityBasis.MultTranspose(m_quadratureDisplacementAction, m_elementDisplacementAction);
+
+            if (data.densityDofTransformation != nullptr) {
+                data.densityDofTransformation->TransformDual(m_elementDisplacementAction);
+            }
+            m_localDisplacementAction.AddElementVector(data.densityDofs, m_elementDisplacementAction);
+        }
+
+        local_to_true(*m_fem.densityFes, m_localDisplacementAction, actionTrue);
+    }
+
     bool PreparedBarotropicClosureOperator::IsPrepared() const noexcept {
         return m_isPrepared && m_context.IsPrepared();
     }

libmeanfield/impl/operators/prepared_central_density_stellar_equilibrium.cpp

@@ -0,0 +1,223 @@
+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_gravity_displacement_force.cpp

@@ -8,6 +8,8 @@ import :operators.kernels.gravity_displacement_force;
 import :operators.prepared_gravity_displacement_force;
 
 namespace {
+    using DomainSchema = mean_field::utils::domain::CoreEnvelopeVacuumDomainSchema;
+
     [[nodiscard]] bool relevant_revisions_match(
         const mean_field::operators::context::gravity_field::GravityFieldRevisions &left,
         const mean_field::operators::context::gravity_field::GravityFieldRevisions &right
@@ -15,6 +17,71 @@ namespace {
         return left.discretization == right.discretization && left.displacement == right.displacement &&
                left.density == right.density && left.gravity_gradient == right.gravity_gradient;
     }
+
+    [[nodiscard]] bool is_vacuum_attribute(const int attribute) {
+        return DomainSchema::template attribute_belongs_to<mean_field::utils::domain::Vacuum>(attribute);
+    }
+
+    void true_to_local(
+        const mfem::ParFiniteElementSpace &finiteElementSpace,
+        const mfem::Vector &trueVector,
+        mfem::Vector &localVector
+    ) {
+        localVector.SetSize(finiteElementSpace.GetVSize());
+        const mfem::Operator *prolongation = finiteElementSpace.GetProlongationMatrix();
+        if (prolongation != nullptr) {
+            prolongation->Mult(trueVector, localVector);
+        } else {
+            localVector = trueVector;
+        }
+    }
+
+    void local_to_true(
+        const mfem::ParFiniteElementSpace &finiteElementSpace,
+        const mfem::Vector &localVector,
+        mfem::Vector &trueVector
+    ) {
+        trueVector.SetSize(finiteElementSpace.GetTrueVSize());
+        trueVector                         = 0.0;
+        const mfem::Operator *prolongation = finiteElementSpace.GetProlongationMatrix();
+        if (prolongation != nullptr) {
+            prolongation->MultTranspose(localVector, trueVector);
+        } else {
+            trueVector = localVector;
+        }
+    }
+
+    [[nodiscard]] int vector_dof_index(
+        const mfem::Ordering::Type ordering,
+        const int scalarDof,
+        const int component,
+        const int scalarDofCount,
+        const int dimension
+    ) {
+        if (ordering == mfem::Ordering::byNODES) {
+            return scalarDof + component * scalarDofCount;
+        }
+        MFEM_VERIFY(ordering == mfem::Ordering::byVDIM, "Unsupported displacement ordering.");
+        return scalarDof * dimension + component;
+    }
+
+    [[nodiscard]] const mfem::IntegrationRule &get_gravity_force_rule(
+        const mean_field::fem::FEM &f,
+        const mfem::ElementTransformation &transformation
+    ) {
+        using DisplacementField = mean_field::field::Field<mean_field::field::Displacement>;
+        const mean_field::quadrature::Query query =
+            DisplacementField::make_query<mean_field::field::Displacement::Form::GravityForce>(
+                mean_field::quadrature::QuadratureRole::discretization, transformation.OrderW(), {},
+                mean_field::utils::DOMAINS::STELLAR, mean_field::quadrature::MappingKind::general
+            );
+        const mean_field::quadrature::MfemRule rule = f.quadratureFactory->get(query, transformation.GetGeometryType());
+        MFEM_VERIFY(
+            rule.integration_rule != nullptr,
+            "The quadrature policy did not return a gravity-displacement-force integration rule."
+        );
+        return *rule.integration_rule;
+    }
 } // namespace
 
 namespace mean_field::operators {
@@ -41,6 +108,122 @@ namespace mean_field::operators {
         );
     }
 
+    void PreparedGravityDisplacementForceOperator::PrepareElementData() {
+        m_elements.clear();
+        m_elements.reserve(m_fem.mesh->GetNE());
+
+        mfem::Vector baseDensityLocal;
+        mfem::Vector baseGravityGradientLocal;
+        mfem::Vector baseDisplacementLocal;
+        true_to_local(*m_fem.densityFes, m_gravityContext.GetDensityTrue(), baseDensityLocal);
+        true_to_local(*m_fem.gravityFluxFes, m_gravityContext.GetGravityGradientTrue(), baseGravityGradientLocal);
+        true_to_local(
+            *m_fem.displacementFes, m_gravityContext.GetGeometryContext().GetDisplacementTrue(), baseDisplacementLocal
+        );
+
+        mapping::DomainMapper::Workspace workspace(m_domainMapper.GetDimension());
+        mapping::VolumeMappingContext mappingContext;
+        mfem::Array<int> compactificationDofs;
+        mfem::Vector elementBaseDensity;
+        mfem::Vector elementBaseGravityGradient;
+        mfem::Vector elementBaseDisplacement;
+        mfem::Vector elementCompactification;
+        mfem::Vector densityShape;
+        mfem::Vector baseGravityReferenceValue;
+        mfem::DenseMatrix gravityGradientShape;
+
+        const int dimension = m_domainMapper.GetDimension();
+
+        for (int elementId = 0; elementId < m_fem.mesh->GetNE(); ++elementId) {
+            mfem::ElementTransformation *transformation = m_fem.mesh->GetElementTransformation(elementId);
+            MFEM_VERIFY(transformation != nullptr, "Prepared gravity force received a null transformation.");
+            if (is_vacuum_attribute(transformation->Attribute)) {
+                continue;
+            }
+
+            m_elements.emplace_back();
+            ElementPAData &data           = m_elements.back();
+            data.elementId                = elementId;
+            data.densityDofTransformation = m_fem.densityFes->GetElementDofs(elementId, data.densityDofs);
+            data.gravityGradientDofTransformation =
+                m_fem.gravityFluxFes->GetElementVDofs(elementId, data.gravityGradientDofs);
+            data.displacementDofTransformation =
+                m_fem.displacementFes->GetElementVDofs(elementId, data.displacementDofs);
+            mfem::DofTransformation *compactificationDofTransformation =
+                m_fem.compactificationFes->GetElementDofs(elementId, compactificationDofs);
+
+            baseDensityLocal.GetSubVector(data.densityDofs, elementBaseDensity);
+            baseGravityGradientLocal.GetSubVector(data.gravityGradientDofs, elementBaseGravityGradient);
+            baseDisplacementLocal.GetSubVector(data.displacementDofs, elementBaseDisplacement);
+            m_fem.compactificationCoordinate->GetSubVector(compactificationDofs, elementCompactification);
+            if (data.densityDofTransformation != nullptr) {
+                data.densityDofTransformation->InvTransformPrimal(elementBaseDensity);
+            }
+            if (data.gravityGradientDofTransformation != nullptr) {
+                data.gravityGradientDofTransformation->InvTransformPrimal(elementBaseGravityGradient);
+            }
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->InvTransformPrimal(elementBaseDisplacement);
+            }
+            if (compactificationDofTransformation != nullptr) {
+                compactificationDofTransformation->InvTransformPrimal(elementCompactification);
+            }
+
+            const mfem::FiniteElement &densityElement          = *m_fem.densityFes->GetFE(elementId);
+            const mfem::FiniteElement &gravityGradientElement  = *m_fem.gravityFluxFes->GetFE(elementId);
+            const mfem::FiniteElement &displacementElement     = *m_fem.displacementFes->GetFE(elementId);
+            const mfem::FiniteElement &compactificationElement = *m_fem.compactificationFes->GetFE(elementId);
+            data.integrationRule                               = &get_gravity_force_rule(m_fem, *transformation);
+
+            const mapping::ElementDisplacementData displacementData =
+                mapping::ElementDisplacementDataFromElementVDofs(displacementElement, elementBaseDisplacement);
+            const mapping::ElementCompactificationData compactificationData(
+                compactificationElement, elementCompactification
+            );
+            const mapping::ElementMappingData mappingData{
+                .displacement = displacementData, .compactification = compactificationData
+            };
+
+            const int quadraturePointCount = data.integrationRule->GetNPoints();
+            data.mappingJacobians.SetSize(quadraturePointCount, dimension * dimension);
+            data.inverseMeshJacobians.SetSize(quadraturePointCount, dimension * dimension);
+            data.baseGravityReferenceValues.SetSize(quadraturePointCount, dimension);
+            data.baseDensityValues.SetSize(quadraturePointCount);
+            data.referenceWeights.SetSize(quadraturePointCount);
+            densityShape.SetSize(densityElement.GetDof());
+            gravityGradientShape.SetSize(gravityGradientElement.GetDof(), dimension);
+            baseGravityReferenceValue.SetSize(dimension);
+
+            for (int quadraturePoint = 0; quadraturePoint < quadraturePointCount; ++quadraturePoint) {
+                const mfem::IntegrationPoint &integrationPoint = data.integrationRule->IntPoint(quadraturePoint);
+                const mapping::MappingStatus status            = m_domainMapper.EvaluateVolume(
+                    mappingData, *transformation, integrationPoint, workspace, mappingContext
+                );
+                MFEM_VERIFY(
+                    status == mapping::MappingStatus::valid && !mappingContext.mapping.compactified,
+                    "Prepared gravity force encountered an invalid stellar mapping."
+                );
+
+                densityElement.CalcShape(integrationPoint, densityShape);
+                gravityGradientElement.CalcVShape(*transformation, gravityGradientShape);
+                gravityGradientShape.MultTranspose(elementBaseGravityGradient, baseGravityReferenceValue);
+                data.baseDensityValues(quadraturePoint)      = elementBaseDensity * densityShape;
+                data.referenceWeights(quadraturePoint)       = integrationPoint.weight * transformation->Weight();
+
+                const mfem::DenseMatrix &inverseMeshJacobian = transformation->InverseJacobian();
+                for (int row = 0; row < dimension; ++row) {
+                    data.baseGravityReferenceValues(quadraturePoint, row) = baseGravityReferenceValue(row);
+                    for (int column = 0; column < dimension; ++column) {
+                        const int entry = row * dimension + column;
+                        data.mappingJacobians(quadraturePoint, entry) =
+                            mappingContext.mapping.mapping_jacobian(row, column);
+                        data.inverseMeshJacobians(quadraturePoint, entry) = inverseMeshJacobian(row, column);
+                    }
+                }
+            }
+        }
+    }
+
     PreparedGravityDisplacementForceReport PreparedGravityDisplacementForceOperator::Prepare() {
         MFEM_VERIFY(
             m_gravityContext.IsPrepared(), "PreparedGravityDisplacementForceOperator requires the shared "
@@ -59,6 +242,7 @@ namespace mean_field::operators {
         );
         m_cachedResidual.SetSize(m_gravityContext.GetDisplacementMap().reduced_size());
         m_gravityContext.GetDisplacementMap().gather(m_actionTrue, m_cachedResidual);
+        PrepareElementData();
 
         m_preparedRevisions = requestedRevisions;
         ++m_residualPreparationCount;
@@ -130,6 +314,117 @@ namespace mean_field::operators {
         ++m_displacementJacobianStatistics.applications;
     }
 
+    void PreparedGravityDisplacementForceOperator::ApplyPreparedCompleteJacobianActionTrue(
+        const mfem::Vector &densityVariationTrue,
+        const mfem::Vector &displacementVariationTrue,
+        const mfem::Vector &gravityGradientVariationTrue,
+        mfem::Vector &actionTrue
+    ) const {
+        true_to_local(*m_fem.densityFes, densityVariationTrue, m_densityVariationLocal);
+        true_to_local(*m_fem.gravityFluxFes, gravityGradientVariationTrue, m_gravityGradientVariationLocal);
+        true_to_local(*m_fem.displacementFes, displacementVariationTrue, m_displacementVariationLocal);
+        m_localAction.SetSize(m_fem.displacementFes->GetVSize());
+        m_localAction                       = 0.0;
+
+        const int dimension                 = m_domainMapper.GetDimension();
+        const mfem::Ordering::Type ordering = m_fem.displacementFes->GetOrdering();
+
+        for (const ElementPAData &data : m_elements) {
+            MFEM_VERIFY(data.integrationRule != nullptr, "Prepared gravity force has no integration rule.");
+
+            m_densityVariationLocal.GetSubVector(data.densityDofs, m_elementDensityVariation);
+            m_gravityGradientVariationLocal.GetSubVector(data.gravityGradientDofs, m_elementGravityGradientVariation);
+            m_displacementVariationLocal.GetSubVector(data.displacementDofs, m_elementDisplacementVariation);
+            if (data.densityDofTransformation != nullptr) {
+                data.densityDofTransformation->InvTransformPrimal(m_elementDensityVariation);
+            }
+            if (data.gravityGradientDofTransformation != nullptr) {
+                data.gravityGradientDofTransformation->InvTransformPrimal(m_elementGravityGradientVariation);
+            }
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->InvTransformPrimal(m_elementDisplacementVariation);
+            }
+
+            const mfem::FiniteElement &densityElement         = *m_fem.densityFes->GetFE(data.elementId);
+            const mfem::FiniteElement &gravityGradientElement = *m_fem.gravityFluxFes->GetFE(data.elementId);
+            const mfem::FiniteElement &displacementElement    = *m_fem.displacementFes->GetFE(data.elementId);
+            mfem::ElementTransformation *transformation       = m_fem.mesh->GetElementTransformation(data.elementId);
+            MFEM_VERIFY(transformation != nullptr, "Prepared gravity force received a null transformation.");
+
+            const mapping::ElementDisplacementData directionData =
+                mapping::ElementDisplacementDataFromElementVDofs(displacementElement, m_elementDisplacementVariation);
+            const mfem::DenseMatrix &directionDofs = directionData.GetDofMatrix();
+            const int scalarDisplacementDofCount   = displacementElement.GetDof();
+
+            m_densityShape.SetSize(densityElement.GetDof());
+            m_displacementShape.SetSize(scalarDisplacementDofCount);
+            m_gravityGradientShape.SetSize(gravityGradientElement.GetDof(), dimension);
+            m_referenceDisplacementDShape.SetSize(scalarDisplacementDofCount, dimension);
+            m_referenceDisplacementJacobian.SetSize(dimension, dimension);
+            m_displacementJacobianVariation.SetSize(dimension, dimension);
+            m_mappingJacobian.SetSize(dimension, dimension);
+            m_inverseMeshJacobian.SetSize(dimension, dimension);
+            m_baseGravityReferenceValue.SetSize(dimension);
+            m_gravityVariationReferenceValue.SetSize(dimension);
+            m_mappedBaseGravity.SetSize(dimension);
+            m_mappedGravityVariation.SetSize(dimension);
+            m_mappedGeometryVariation.SetSize(dimension);
+            m_forceValue.SetSize(dimension);
+            m_elementAction.SetSize(data.displacementDofs.Size());
+            m_elementAction = 0.0;
+
+            for (int quadraturePoint = 0; quadraturePoint < data.integrationRule->GetNPoints(); ++quadraturePoint) {
+                const mfem::IntegrationPoint &integrationPoint = data.integrationRule->IntPoint(quadraturePoint);
+                densityElement.CalcShape(integrationPoint, m_densityShape);
+                displacementElement.CalcShape(integrationPoint, m_displacementShape);
+                displacementElement.CalcDShape(integrationPoint, m_referenceDisplacementDShape);
+                mfem::MultAtB(directionDofs, m_referenceDisplacementDShape, m_referenceDisplacementJacobian);
+
+                transformation->SetIntPoint(&integrationPoint);
+                gravityGradientElement.CalcVShape(*transformation, m_gravityGradientShape);
+                m_gravityGradientShape.MultTranspose(
+                    m_elementGravityGradientVariation, m_gravityVariationReferenceValue
+                );
+
+                for (int row = 0; row < dimension; ++row) {
+                    m_baseGravityReferenceValue(row) = data.baseGravityReferenceValues(quadraturePoint, row);
+                    for (int column = 0; column < dimension; ++column) {
+                        const int entry                    = row * dimension + column;
+                        m_mappingJacobian(row, column)     = data.mappingJacobians(quadraturePoint, entry);
+                        m_inverseMeshJacobian(row, column) = data.inverseMeshJacobians(quadraturePoint, entry);
+                    }
+                }
+                mfem::Mult(m_referenceDisplacementJacobian, m_inverseMeshJacobian, m_displacementJacobianVariation);
+                m_mappingJacobian.Mult(m_baseGravityReferenceValue, m_mappedBaseGravity);
+                m_mappingJacobian.Mult(m_gravityVariationReferenceValue, m_mappedGravityVariation);
+                m_displacementJacobianVariation.Mult(m_baseGravityReferenceValue, m_mappedGeometryVariation);
+
+                const double densityVariationValue = m_elementDensityVariation * m_densityShape;
+                const double baseDensityValue      = data.baseDensityValues(quadraturePoint);
+                m_forceValue                       = 0.0;
+                m_forceValue.Add(densityVariationValue, m_mappedBaseGravity);
+                m_forceValue.Add(baseDensityValue, m_mappedGravityVariation);
+                m_forceValue.Add(baseDensityValue, m_mappedGeometryVariation);
+                m_forceValue *= data.referenceWeights(quadraturePoint);
+
+                for (int scalarDof = 0; scalarDof < scalarDisplacementDofCount; ++scalarDof) {
+                    for (int component = 0; component < dimension; ++component) {
+                        const int vectorDof =
+                            vector_dof_index(ordering, scalarDof, component, scalarDisplacementDofCount, dimension);
+                        m_elementAction(vectorDof) += m_displacementShape(scalarDof) * m_forceValue(component);
+                    }
+                }
+            }
+
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->TransformDual(m_elementAction);
+            }
+            m_localAction.AddElementVector(data.displacementDofs, m_elementAction);
+        }
+
+        local_to_true(*m_fem.displacementFes, m_localAction, actionTrue);
+    }
+
     void PreparedGravityDisplacementForceOperator::ApplyCompleteJacobianAction(
         const mfem::Vector &densityVariation,
         const mfem::Vector &displacementVariation,
@@ -145,10 +440,8 @@ namespace mean_field::operators {
         m_gravityContext.GetGravityGradientMap().scatter(gravityGradientVariation, m_gravityGradientVariationTrue);
         m_gravityContext.GetDisplacementMap().scatter(displacementVariation, m_displacementVariationTrue);
 
-        kernels::apply_gravity_displacement_force_complete_action(
-            m_fem, m_domainMapper, m_gravityContext.GetDensityTrue(), m_densityVariationTrue,
-            m_gravityContext.GetGravityGradientTrue(), m_gravityGradientVariationTrue, m_displacementVariationTrue,
-            m_gravityContext.GetGeometryContext().GetDisplacementTrue(), m_actionTrue
+        ApplyPreparedCompleteJacobianActionTrue(
+            m_densityVariationTrue, m_displacementVariationTrue, m_gravityGradientVariationTrue, m_actionTrue
         );
         action.SetSize(m_gravityContext.GetDisplacementMap().reduced_size());
         m_gravityContext.GetDisplacementMap().gather(m_actionTrue, action);

libmeanfield/impl/operators/prepared_gravity_source.cpp

@@ -174,9 +174,15 @@ namespace {
                 "non-finite mapping determinant."
             );
 
+            m_inverse_element_jacobian = mapping_context.quadrature.J_inv;
+
             return 4.0 * std::numbers::pi * mean_field::utils::G * mapping_determinant;
         }
 
+        [[nodiscard]] const mfem::DenseMatrix &GetInverseElementJacobian() const noexcept {
+            return m_inverse_element_jacobian;
+        }
+
     private:
         void LoadElement(const int element_id) {
             if (element_id == m_cached_element_id) {
@@ -230,6 +236,7 @@ namespace {
         std::unique_ptr<mean_field::mapping::ElementCompactificationData> m_compactification_data;
 
         mean_field::mapping::DomainMapper::Workspace m_workspace;
+        mfem::DenseMatrix m_inverse_element_jacobian;
         int m_cached_element_id{-1};
     };
 } // namespace
@@ -336,6 +343,9 @@ namespace mean_field::operators {
             data.potential_dof_transformation =
                 m_fem.gravityPotentialFes->GetElementDofs(element_id, data.potential_dofs);
 
+            data.displacement_dof_transformation =
+                m_fem.displacementFes->GetElementVDofs(element_id, data.displacement_dofs);
+
             const mfem::FiniteElement &density_element   = *m_fem.densityFes->GetFE(element_id);
 
             const mfem::FiniteElement &potential_element = *m_fem.gravityPotentialFes->GetFE(element_id);
@@ -344,6 +354,7 @@ namespace mean_field::operators {
 
             const mfem::IntegrationRule &integration_rule =
                 get_source_rule(m_fem, density_element, potential_element, transformation);
+            data.integration_rule            = &integration_rule;
 
             const int quadrature_point_count = integration_rule.GetNPoints();
 
@@ -355,6 +366,9 @@ namespace mean_field::operators {
 
             data.potential_basis.SetSize(quadrature_point_count, potential_dof_count);
 
+            const int dimension = m_fem.mesh->Dimension();
+            data.inverse_element_jacobians.SetSize(quadrature_point_count, dimension * dimension);
+
             data.quadrature_data.SetSize(quadrature_point_count);
 
             mfem::Vector density_shape(density_dof_count);
@@ -381,6 +395,14 @@ namespace mean_field::operators {
 
                 const double coefficient_value = source_coefficient.Eval(transformation, integration_point);
 
+                const mfem::DenseMatrix &inverse_element_jacobian = source_coefficient.GetInverseElementJacobian();
+                for (int row = 0; row < dimension; ++row) {
+                    for (int column = 0; column < dimension; ++column) {
+                        data.inverse_element_jacobians(quadrature_point, row * dimension + column) =
+                            inverse_element_jacobian(row, column);
+                    }
+                }
+
                 transformation.SetIntPoint(&integration_point);
 
                 const double quadrature_value = integration_point.weight * transformation.Weight() * coefficient_value;
@@ -463,6 +485,98 @@ namespace mean_field::operators {
         m_potential_map.gather(m_action_true, action);
     }
 
+    void PreparedMappedGravitySourceOperator::MultDisplacementVariationTrue(
+        const mfem::Vector &densityTrue,
+        const mfem::Vector &displacementVariationTrue,
+        mfem::Vector &actionVariationTrue
+    ) const {
+        MFEM_VERIFY(
+            m_is_prepared,
+            "PreparedMappedGravitySourceOperator must be prepared before applying a displacement variation."
+        );
+        MFEM_VERIFY(
+            densityTrue.Size() == m_fem.densityFes->GetTrueVSize(), "The full density vector has the wrong size."
+        );
+        MFEM_VERIFY(
+            displacementVariationTrue.Size() == m_fem.displacementFes->GetTrueVSize(),
+            "The full displacement variation has the wrong size."
+        );
+
+        true_to_local(*m_fem.densityFes, densityTrue, m_density_local);
+        true_to_local(*m_fem.displacementFes, displacementVariationTrue, m_displacement_variation_local);
+
+        m_local_variation_action.SetSize(m_fem.gravityPotentialFes->GetVSize());
+        m_local_variation_action = 0.0;
+
+        const int dimension      = m_fem.mesh->Dimension();
+
+        for (const ElementPAData &data : m_elements) {
+            MFEM_VERIFY(
+                data.integration_rule != nullptr,
+                "Prepared gravity source displacement variation has no integration rule."
+            );
+
+            m_density_local.GetSubVector(data.density_dofs, m_element_density);
+            m_displacement_variation_local.GetSubVector(data.displacement_dofs, m_element_displacement_variation);
+
+            if (data.density_dof_transformation != nullptr) {
+                data.density_dof_transformation->InvTransformPrimal(m_element_density);
+            }
+            if (data.displacement_dof_transformation != nullptr) {
+                data.displacement_dof_transformation->InvTransformPrimal(m_element_displacement_variation);
+            }
+
+            const mfem::FiniteElement &displacement_element       = *m_fem.displacementFes->GetFE(data.element_id);
+            const mapping::ElementDisplacementData direction_data = mapping::ElementDisplacementDataFromElementVDofs(
+                displacement_element, m_element_displacement_variation
+            );
+            const mfem::DenseMatrix &direction_dofs = direction_data.GetDofMatrix();
+
+            MFEM_VERIFY(
+                data.inverse_element_jacobians.Height() == data.integration_rule->GetNPoints() &&
+                    data.inverse_element_jacobians.Width() == dimension * dimension,
+                "Prepared gravity source inverse-Jacobian data has an incompatible size."
+            );
+
+            m_reference_displacement_dshape.SetSize(displacement_element.GetDof(), dimension);
+            m_reference_displacement_jacobian.SetSize(dimension, dimension);
+            m_quadrature_variation_action.SetSize(data.integration_rule->GetNPoints());
+            data.density_basis.Mult(m_element_density, m_quadrature_variation_action);
+
+            for (int quadrature_point = 0; quadrature_point < data.integration_rule->GetNPoints(); ++quadrature_point) {
+                const mfem::IntegrationPoint &integration_point = data.integration_rule->IntPoint(quadrature_point);
+                displacement_element.CalcDShape(integration_point, m_reference_displacement_dshape);
+                mfem::MultAtB(direction_dofs, m_reference_displacement_dshape, m_reference_displacement_jacobian);
+
+                double logarithmic_jacobian_variation{0.0};
+                for (int row = 0; row < dimension; ++row) {
+                    for (int column = 0; column < dimension; ++column) {
+                        logarithmic_jacobian_variation +=
+                            data.inverse_element_jacobians(quadrature_point, row * dimension + column) *
+                            m_reference_displacement_jacobian(column, row);
+                    }
+                }
+
+                m_quadrature_variation_action(quadrature_point) *=
+                    data.quadrature_data(quadrature_point) * logarithmic_jacobian_variation;
+                MFEM_VERIFY(
+                    std::isfinite(m_quadrature_variation_action(quadrature_point)),
+                    "Prepared gravity source displacement variation encountered a non-finite quadrature value."
+                );
+            }
+
+            m_element_variation_action.SetSize(data.potential_dofs.Size());
+            data.potential_basis.MultTranspose(m_quadrature_variation_action, m_element_variation_action);
+
+            if (data.potential_dof_transformation != nullptr) {
+                data.potential_dof_transformation->TransformDual(m_element_variation_action);
+            }
+            m_local_variation_action.AddElementVector(data.potential_dofs, m_element_variation_action);
+        }
+
+        local_to_true(*m_fem.gravityPotentialFes, m_local_variation_action, actionVariationTrue);
+    }
+
     void PreparedMappedGravitySourceOperator::MultTranspose(
         const mfem::Vector &potential,
         mfem::Vector &action

libmeanfield/impl/operators/prepared_hdiv_mass.cpp

@@ -35,6 +35,128 @@ namespace {
         }
     }
 
+    void local_to_true(
+        const mfem::ParFiniteElementSpace &finite_element_space,
+        const mfem::Vector &local_vector,
+        mfem::Vector &true_vector
+    ) {
+        true_vector.SetSize(finite_element_space.GetTrueVSize());
+        true_vector                        = 0.0;
+        const mfem::Operator *prolongation = finite_element_space.GetProlongationMatrix();
+        if (prolongation != nullptr) {
+            prolongation->MultTranspose(local_vector, true_vector);
+        } else {
+            true_vector = local_vector;
+        }
+    }
+
+    mean_field::quadrature::MappingKind get_mapping_kind(
+        const mean_field::mapping::DomainMapper &domain_mapper,
+        const mfem::ElementTransformation &transformation
+    ) {
+        return domain_mapper.IsCompactifiedElement(transformation) ? mean_field::quadrature::MappingKind::kelvin
+                                                                   : mean_field::quadrature::MappingKind::general;
+    }
+
+    const mfem::IntegrationRule &get_hdiv_mass_rule(
+        const mean_field::fem::FEM &f,
+        const mean_field::mapping::DomainMapper &domain_mapper,
+        const mfem::FiniteElement &element,
+        const mfem::ElementTransformation &transformation
+    ) {
+        using GravityField = mean_field::field::Field<mean_field::field::Gravity>;
+        const mean_field::quadrature::Query query =
+            GravityField::make_query<mean_field::field::Gravity::Form::HDivMass>(
+                mean_field::quadrature::QuadratureRole::discretization, transformation.OrderW(), {},
+                mean_field::utils::DOMAINS::ALL, get_mapping_kind(domain_mapper, transformation)
+            );
+        const auto resolution = f.quadratureFactory->get(query, transformation.GetGeometryType());
+        MFEM_VERIFY(
+            resolution.integration_rule != nullptr,
+            "The quadrature policy did not return an H(div) mass integration rule."
+        );
+        return *resolution.integration_rule;
+    }
+
+    int frozen_mapping_width(const int dimension) {
+        return 3 * dimension + 4 * dimension * dimension + 3;
+    }
+
+    void freeze_mapping_context(
+        const mean_field::mapping::VolumeMappingContext &context,
+        const int quadrature_point,
+        mfem::DenseMatrix &data
+    ) {
+        const int dimension                   = context.mapping.reference_position.Size();
+        const int displacement_jacobian_start = 3 * dimension;
+        const int mapping_jacobian_start      = displacement_jacobian_start + dimension * dimension;
+        const int inverse_mapping_start       = mapping_jacobian_start + dimension * dimension;
+        const int inverse_element_start       = inverse_mapping_start + dimension * dimension;
+        const int scalar_start                = inverse_element_start + dimension * dimension;
+
+        for (int component = 0; component < dimension; ++component) {
+            data(quadrature_point, component)                 = context.mapping.reference_position(component);
+            data(quadrature_point, dimension + component)     = context.mapping.displaced_position(component);
+            data(quadrature_point, 2 * dimension + component) = context.mapping.physical_position(component);
+        }
+        for (int row = 0; row < dimension; ++row) {
+            for (int column = 0; column < dimension; ++column) {
+                const int entry = row * dimension + column;
+                data(quadrature_point, displacement_jacobian_start + entry) =
+                    context.mapping.displacement_jacobian(row, column);
+                data(quadrature_point, mapping_jacobian_start + entry) = context.mapping.mapping_jacobian(row, column);
+                data(quadrature_point, inverse_mapping_start + entry) =
+                    context.mapping.inverse_mapping_jacobian(row, column);
+                data(quadrature_point, inverse_element_start + entry) = context.quadrature.J_inv(row, column);
+            }
+        }
+        data(quadrature_point, scalar_start)     = context.mapping.mapping_determinant;
+        data(quadrature_point, scalar_start + 1) = context.quadrature.weight;
+        data(quadrature_point, scalar_start + 2) = context.mapping.compactified ? 1.0 : 0.0;
+    }
+
+    void thaw_mapping_context(
+        const mfem::DenseMatrix &data,
+        const int quadrature_point,
+        const int dimension,
+        mean_field::mapping::VolumeMappingContext &context
+    ) {
+        const int displacement_jacobian_start = 3 * dimension;
+        const int mapping_jacobian_start      = displacement_jacobian_start + dimension * dimension;
+        const int inverse_mapping_start       = mapping_jacobian_start + dimension * dimension;
+        const int inverse_element_start       = inverse_mapping_start + dimension * dimension;
+        const int scalar_start                = inverse_element_start + dimension * dimension;
+
+        context.mapping.reference_position.SetSize(dimension);
+        context.mapping.displaced_position.SetSize(dimension);
+        context.mapping.physical_position.SetSize(dimension);
+        context.mapping.displacement_jacobian.SetSize(dimension, dimension);
+        context.mapping.mapping_jacobian.SetSize(dimension, dimension);
+        context.mapping.inverse_mapping_jacobian.SetSize(dimension, dimension);
+        context.quadrature.J_inv.SetSize(dimension, dimension);
+
+        for (int component = 0; component < dimension; ++component) {
+            context.mapping.reference_position(component) = data(quadrature_point, component);
+            context.mapping.displaced_position(component) = data(quadrature_point, dimension + component);
+            context.mapping.physical_position(component)  = data(quadrature_point, 2 * dimension + component);
+        }
+        for (int row = 0; row < dimension; ++row) {
+            for (int column = 0; column < dimension; ++column) {
+                const int entry = row * dimension + column;
+                context.mapping.displacement_jacobian(row, column) =
+                    data(quadrature_point, displacement_jacobian_start + entry);
+                context.mapping.mapping_jacobian(row, column) = data(quadrature_point, mapping_jacobian_start + entry);
+                context.mapping.inverse_mapping_jacobian(row, column) =
+                    data(quadrature_point, inverse_mapping_start + entry);
+                context.quadrature.J_inv(row, column) = data(quadrature_point, inverse_element_start + entry);
+            }
+        }
+        context.mapping.mapping_determinant = data(quadrature_point, scalar_start);
+        context.mapping.compactified        = data(quadrature_point, scalar_start + 2) != 0.0;
+        context.quadrature.detJ             = context.mapping.mapping_determinant;
+        context.quadrature.weight           = data(quadrature_point, scalar_start + 1);
+    }
+
     int find_representative_element(
         const mean_field::fem::FEM &f,
         const mfem::Array<int> &marker
@@ -238,7 +360,8 @@ namespace mean_field::operators {
               field::make_field_dof_map<
                   field::Displacement,
                   DomainSchema>(*f.displacementFes)
-          ) {
+          ),
+          m_variationWorkspace(domain_mapper.GetDimension()) {
         MFEM_VERIFY(f.mesh != nullptr, "PreparedMappedHDivMassOperator requires a mesh.");
         MFEM_VERIFY(
             f.gravityFluxFes != nullptr, "PreparedMappedHDivMassOperator requires the "
@@ -285,6 +408,78 @@ namespace mean_field::operators {
         validate_uniform_domain_discretization(f, m_vacuum_marker, vacuum_element_id);
     }
 
+    void PreparedMappedHDivMassOperator::PrepareVariationData() {
+        m_variationElements.clear();
+        m_variationElements.reserve(m_fem.mesh->GetNE());
+
+        mfem::Vector displacementLocal;
+        true_to_local(*m_fem.displacementFes, m_displacement_true, displacementLocal);
+
+        mfem::Vector elementDisplacement;
+        mfem::Vector elementCompactification;
+        mapping::VolumeMappingContext mappingContext;
+
+        for (int elementId = 0; elementId < m_fem.mesh->GetNE(); ++elementId) {
+            m_variationElements.emplace_back();
+            ElementVariationData &data = m_variationElements.back();
+            data.elementId             = elementId;
+
+            data.gravityGradientDofTransformation =
+                m_fem.gravityFluxFes->GetElementVDofs(elementId, data.gravityGradientDofs);
+            data.displacementDofTransformation =
+                m_fem.displacementFes->GetElementVDofs(elementId, data.displacementDofs);
+            mfem::DofTransformation *compactificationDofTransformation =
+                m_fem.compactificationFes->GetElementDofs(elementId, data.compactificationDofs);
+
+            displacementLocal.GetSubVector(data.displacementDofs, elementDisplacement);
+            m_fem.compactificationCoordinate->GetSubVector(data.compactificationDofs, elementCompactification);
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->InvTransformPrimal(elementDisplacement);
+            }
+            if (compactificationDofTransformation != nullptr) {
+                compactificationDofTransformation->InvTransformPrimal(elementCompactification);
+            }
+            data.baseDisplacement                              = elementDisplacement;
+            data.compactification                              = elementCompactification;
+
+            const mfem::FiniteElement &gravityGradientElement  = *m_fem.gravityFluxFes->GetFE(elementId);
+            const mfem::FiniteElement &displacementElement     = *m_fem.displacementFes->GetFE(elementId);
+            const mfem::FiniteElement &compactificationElement = *m_fem.compactificationFes->GetFE(elementId);
+            mfem::ElementTransformation *transformation        = m_fem.mesh->GetElementTransformation(elementId);
+            MFEM_VERIFY(
+                transformation != nullptr, "Prepared H(div) variation data received a null element transformation."
+            );
+
+            data.integrationRule = &get_hdiv_mass_rule(m_fem, m_domain_mapper, gravityGradientElement, *transformation);
+            data.frozenMappingData.SetSize(
+                data.integrationRule->GetNPoints(), frozen_mapping_width(m_domain_mapper.GetDimension())
+            );
+
+            const mapping::ElementDisplacementData displacementData =
+                mapping::ElementDisplacementDataFromElementVDofs(displacementElement, data.baseDisplacement);
+            const mapping::ElementCompactificationData compactificationData(
+                compactificationElement, data.compactification
+            );
+            const mapping::ElementMappingData mappingData{
+                .displacement = displacementData, .compactification = compactificationData
+            };
+
+            for (int quadraturePoint = 0; quadraturePoint < data.integrationRule->GetNPoints(); ++quadraturePoint) {
+                const mfem::IntegrationPoint &integrationPoint = data.integrationRule->IntPoint(quadraturePoint);
+                const mapping::MappingStatus status            = m_domain_mapper.EvaluateVolume(
+                    mappingData, *transformation, integrationPoint, m_variationWorkspace, mappingContext
+                );
+                MFEM_VERIFY(
+                    status == mapping::MappingStatus::valid,
+                    "Prepared H(div) variation data encountered an invalid mapping. Element: "
+                        << elementId << ", quadrature point: " << quadraturePoint
+                        << ", status: " << static_cast<int>(status)
+                );
+                freeze_mapping_context(mappingContext, quadraturePoint, data.frozenMappingData);
+            }
+        }
+    }
+
     void PreparedMappedHDivMassOperator::Prepare(const mfem::Vector &displacement) {
         MFEM_VERIFY(
             displacement.Size() == m_displacement_map.reduced_size(),
@@ -346,6 +541,8 @@ namespace mean_field::operators {
         m_stellar_mass_form->Assemble();
         m_vacuum_mass_form->Assemble();
 
+        PrepareVariationData();
+
         m_is_prepared = true;
         ++m_preparation_count;
     }
@@ -378,6 +575,110 @@ namespace mean_field::operators {
         m_flux_map.gather(m_action_true, action);
     }
 
+    void PreparedMappedHDivMassOperator::MultDisplacementVariationTrue(
+        const mfem::Vector &gravityGradientTrue,
+        const mfem::Vector &displacementVariationTrue,
+        mfem::Vector &actionVariationTrue
+    ) const {
+        MFEM_VERIFY(
+            m_is_prepared, "PreparedMappedHDivMassOperator must be prepared before applying a displacement variation."
+        );
+        MFEM_VERIFY(
+            gravityGradientTrue.Size() == m_fem.gravityFluxFes->GetTrueVSize(),
+            "The full gravity-gradient vector has the wrong size."
+        );
+        MFEM_VERIFY(
+            displacementVariationTrue.Size() == m_fem.displacementFes->GetTrueVSize(),
+            "The full displacement variation has the wrong size."
+        );
+
+        true_to_local(*m_fem.gravityFluxFes, gravityGradientTrue, m_gravityGradientLocal);
+        true_to_local(*m_fem.displacementFes, displacementVariationTrue, m_displacementVariationLocal);
+        m_localVariationAction.SetSize(m_fem.gravityFluxFes->GetVSize());
+        m_localVariationAction = 0.0;
+
+        const int dimension    = m_domain_mapper.GetDimension();
+
+        for (const ElementVariationData &data : m_variationElements) {
+            MFEM_VERIFY(
+                data.integrationRule != nullptr &&
+                    data.frozenMappingData.Height() == data.integrationRule->GetNPoints() &&
+                    data.frozenMappingData.Width() == frozen_mapping_width(dimension),
+                "Prepared H(div) variation data is incomplete."
+            );
+
+            m_gravityGradientLocal.GetSubVector(data.gravityGradientDofs, m_elementGravityGradient);
+            m_displacementVariationLocal.GetSubVector(data.displacementDofs, m_elementDisplacementVariation);
+            if (data.gravityGradientDofTransformation != nullptr) {
+                data.gravityGradientDofTransformation->InvTransformPrimal(m_elementGravityGradient);
+            }
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->InvTransformPrimal(m_elementDisplacementVariation);
+            }
+
+            const mfem::FiniteElement &gravityGradientElement  = *m_fem.gravityFluxFes->GetFE(data.elementId);
+            const mfem::FiniteElement &displacementElement     = *m_fem.displacementFes->GetFE(data.elementId);
+            const mfem::FiniteElement &compactificationElement = *m_fem.compactificationFes->GetFE(data.elementId);
+            mfem::ElementTransformation *transformation        = m_fem.mesh->GetElementTransformation(data.elementId);
+            MFEM_VERIFY(
+                transformation != nullptr,
+                "Prepared H(div) displacement variation received a null element transformation."
+            );
+
+            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
+            };
+
+            m_elementVariationAction.SetSize(gravityGradientElement.GetDof());
+            m_elementVariationAction = 0.0;
+            m_gravityGradientValue.SetSize(dimension);
+            m_massTensorVariationAction.SetSize(dimension);
+            m_gravityGradientShape.SetSize(gravityGradientElement.GetDof(), dimension);
+            m_massTensorVariation.SetSize(dimension, dimension);
+
+            for (int quadraturePoint = 0; quadraturePoint < data.integrationRule->GetNPoints(); ++quadraturePoint) {
+                const mfem::IntegrationPoint &integrationPoint = data.integrationRule->IntPoint(quadraturePoint);
+                thaw_mapping_context(data.frozenMappingData, quadraturePoint, dimension, m_baseMappingContext);
+
+                const mapping::MappingStatus status = m_domain_mapper.EvaluateVolumeVariation(
+                    mappingData, directionData, *transformation, integrationPoint, m_baseMappingContext,
+                    m_variationWorkspace, m_mappingVariation
+                );
+                MFEM_VERIFY(
+                    status == mapping::MappingStatus::valid,
+                    "Prepared H(div) displacement variation encountered an invalid mapping variation. Element: "
+                        << data.elementId << ", quadrature point: " << quadraturePoint
+                        << ", status: " << static_cast<int>(status)
+                );
+
+                mapping::ComputeHDivMassTensorVariation(
+                    m_baseMappingContext.mapping, m_mappingVariation.mapping, m_massTensorVariation
+                );
+
+                transformation->SetIntPoint(&integrationPoint);
+                gravityGradientElement.CalcVShape(*transformation, m_gravityGradientShape);
+                m_gravityGradientShape.MultTranspose(m_elementGravityGradient, m_gravityGradientValue);
+                m_massTensorVariation.Mult(m_gravityGradientValue, m_massTensorVariationAction);
+                const double referenceWeight = integrationPoint.weight * transformation->Weight();
+                m_gravityGradientShape.AddMult(m_massTensorVariationAction, m_elementVariationAction, referenceWeight);
+            }
+
+            if (data.gravityGradientDofTransformation != nullptr) {
+                data.gravityGradientDofTransformation->TransformDual(m_elementVariationAction);
+            }
+            m_localVariationAction.AddElementVector(data.gravityGradientDofs, m_elementVariationAction);
+        }
+
+        local_to_true(*m_fem.gravityFluxFes, m_localVariationAction, actionVariationTrue);
+    }
+
     void PreparedMappedHDivMassOperator::AssembleDiagonal(mfem::Vector &diagonal) const {
         mfem::Vector true_diagonal;
         AssembleTrueDiagonal(true_diagonal);

libmeanfield/impl/operators/prepared_hydrostatic_equilibrium.cpp

@@ -992,6 +992,7 @@ namespace mean_field::operators {
         mfem::Vector elementDisplacementVariation;
         mfem::Vector weightedQuadratureVariation;
         mfem::Vector elementAction;
+        mapping::VolumeMappingVariation variation;
 
         for (const ElementPAData &data : m_elements) {
             MFEM_VERIFY(
@@ -1040,11 +1041,7 @@ namespace mean_field::operators {
             for (int quadraturePoint = 0; quadraturePoint < quadraturePointCount; ++quadraturePoint) {
                 const mfem::IntegrationPoint &integrationPoint = data.integrationRule->IntPoint(quadraturePoint);
 
-                transformation->SetIntPoint(&integrationPoint);
-
-                mapping::VolumeMappingVariation variation;
-
-                const mapping::MappingStatus mappingStatus = m_domainMapper.EvaluateVolumeVariation(
+                const mapping::MappingStatus mappingStatus     = m_domainMapper.EvaluateVolumeVariation(
                     mappingData, directionData, *transformation, integrationPoint,
                     data.baseMappingContexts[quadraturePoint], workspace, variation
                 );

libmeanfield/impl/operators/prepared_mass_normalization.cpp

@@ -42,6 +42,25 @@ namespace {
         }
     }
 
+    void local_to_true(
+        const mfem::ParFiniteElementSpace &finiteElementSpace,
+        const mfem::Vector &localVector,
+        mfem::Vector &trueVector
+    ) {
+        MFEM_VERIFY(localVector.Size() == finiteElementSpace.GetVSize(), "Local vector has the wrong size.");
+
+        trueVector.SetSize(finiteElementSpace.GetTrueVSize());
+        trueVector                         = 0.0;
+
+        const mfem::Operator *prolongation = finiteElementSpace.GetProlongationMatrix();
+
+        if (prolongation != nullptr) {
+            prolongation->MultTranspose(localVector, trueVector);
+        } else {
+            trueVector = localVector;
+        }
+    }
+
     const mfem::IntegrationRule &get_mass_normalization_rule(
         const mean_field::fem::FEM &f,
         const mfem::FiniteElement &densityElement,
@@ -212,6 +231,13 @@ namespace mean_field::operators {
         return report;
     }
 
+    PreparedMassNormalizationReport PreparedMassNormalizationOperator::Prepare(
+        const models::CompiledFixedMass &constraint,
+        const MassNormalizationDependencies &dependencies
+    ) {
+        return Prepare({.targetMass = constraint.targetMass().value()}, dependencies);
+    }
+
     void PreparedMassNormalizationOperator::BuildStaticPlan() {
         m_elements.clear();
         m_elements.reserve(m_fem.mesh->GetNE());
@@ -432,6 +458,7 @@ namespace mean_field::operators {
         true_to_local(*m_fem.displacementFes, displacementVariation, displacementVariationLocal);
 
         mapping::DomainMapper::Workspace workspace(m_fem.mesh->Dimension());
+        mapping::VolumeMappingVariation variation;
 
         mfem::Vector elementDisplacementVariation;
         double localAction = 0.0;
@@ -464,8 +491,6 @@ namespace mean_field::operators {
             mfem::ElementTransformation *transformation = m_fem.mesh->GetElementTransformation(data.elementId);
 
             for (const QuadraturePointData &point : data.quadraturePoints) {
-                mapping::VolumeMappingVariation variation;
-
                 const mapping::MappingStatus status = m_domainMapper.EvaluateVolumeVariation(
                     mappingData, directionData, *transformation, point.integrationPoint, point.mappingContext,
                     workspace, variation
@@ -558,6 +583,137 @@ namespace mean_field::operators {
         ++m_actionStatistics.completeApplications;
     }
 
+    void PreparedMassNormalizationOperator::ApplyJacobian(
+        const FixedMassJacobianInput &input,
+        mfem::Vector &action
+    ) const {
+        ApplyCompleteJacobianAction(input.densityVariation, input.displacementVariation, action);
+    }
+
+    void PreparedMassNormalizationOperator::AssembleDensityTransposeAction(
+        const double residualDual,
+        mfem::Vector &densityDual
+    ) const {
+        mfem::Vector localDual(m_fem.densityFes->GetVSize());
+        localDual = 0.0;
+
+        mfem::Vector elementDual;
+
+        for (const ElementPAData &data : m_elements) {
+            elementDual.SetSize(data.densityDofs.Size());
+            elementDual = 0.0;
+
+            for (const QuadraturePointData &point : data.quadraturePoints) {
+                elementDual.Add(residualDual * point.mappingContext.quadrature.weight, point.densityShape);
+            }
+
+            if (data.densityDofTransformation != nullptr) {
+                data.densityDofTransformation->TransformDual(elementDual);
+            }
+
+            localDual.AddElementVector(data.densityDofs, elementDual);
+        }
+
+        mfem::Vector trueDual;
+        local_to_true(*m_fem.densityFes, localDual, trueDual);
+
+        densityDual.SetSize(m_gravityContext.GetDensityMap().reduced_size());
+        m_gravityContext.GetDensityMap().gather(trueDual, densityDual);
+    }
+
+    void PreparedMassNormalizationOperator::AssembleDisplacementTransposeAction(
+        const double residualDual,
+        mfem::Vector &displacementDual
+    ) const {
+        mfem::Vector localDual(m_fem.displacementFes->GetVSize());
+        localDual = 0.0;
+
+        mapping::DomainMapper::Workspace workspace(m_fem.mesh->Dimension());
+        mapping::VolumeMappingVariation variation;
+        mfem::Vector elementDirection;
+        mfem::Vector elementDual;
+
+        for (const ElementPAData &data : m_elements) {
+            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::ElementCompactificationData compactificationData(
+                compactificationElement, data.compactification
+            );
+            const mapping::ElementMappingData mappingData{
+                .displacement = baseDisplacementData, .compactification = compactificationData
+            };
+
+            mfem::ElementTransformation *transformation = m_fem.mesh->GetElementTransformation(data.elementId);
+
+            elementDirection.SetSize(data.displacementDofs.Size());
+            elementDual.SetSize(data.displacementDofs.Size());
+            elementDual = 0.0;
+
+            for (int elementDof = 0; elementDof < elementDirection.Size(); ++elementDof) {
+                elementDirection             = 0.0;
+                elementDirection(elementDof) = 1.0;
+
+                const mapping::ElementDisplacementData directionData =
+                    mapping::ElementDisplacementDataFromElementVDofs(displacementElement, elementDirection);
+
+                double elementDofAction = 0.0;
+
+                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,
+                        "Stateless mapping variation failed in the mass-normalization transpose action. Element: "
+                            << data.elementId << ", status: " << static_cast<int>(status)
+                    );
+
+                    elementDofAction += point.density * variation.weight_variation;
+                }
+
+                elementDual(elementDof) = residualDual * elementDofAction;
+            }
+
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->TransformDual(elementDual);
+            }
+
+            localDual.AddElementVector(data.displacementDofs, elementDual);
+        }
+
+        mfem::Vector trueDual;
+        local_to_true(*m_fem.displacementFes, localDual, trueDual);
+
+        displacementDual.SetSize(m_gravityContext.GetDisplacementMap().reduced_size());
+        m_gravityContext.GetDisplacementMap().gather(trueDual, displacementDual);
+    }
+
+    void PreparedMassNormalizationOperator::ApplyCompleteJacobianTransposeAction(
+        const double residualDual,
+        mfem::Vector &densityDual,
+        mfem::Vector &displacementDual
+    ) const {
+        VerifyPrepared();
+        MFEM_VERIFY(std::isfinite(residualDual), "Mass-normalization transpose action received a non-finite dual.");
+
+        AssembleDensityTransposeAction(residualDual, densityDual);
+        AssembleDisplacementTransposeAction(residualDual, displacementDual);
+        ++m_actionStatistics.transposeApplications;
+    }
+
+    void PreparedMassNormalizationOperator::ApplyJacobianTranspose(
+        const mfem::Vector &residualDual,
+        FixedMassJacobianTransposeOutput output
+    ) const {
+        MFEM_VERIFY(residualDual.Size() == 1, "Fixed-mass transpose action requires one residual dual value.");
+        ApplyCompleteJacobianTransposeAction(residualDual(0), output.densityDual, output.displacementDual);
+    }
+
     double PreparedMassNormalizationOperator::GlobalSum(const double localValue) const {
         double globalValue = 0.0;
         MPI_Allreduce(&localValue, &globalValue, 1, MPI_DOUBLE, MPI_SUM, m_fem.mesh->GetComm());
@@ -720,6 +876,44 @@ namespace mean_field::operators {
         action(m_layout.offset(massResidual)) = massAction(0);
     }
 
+    void PreparedMassNormalizationJacobianOperator::MultTranspose(
+        const mfem::Vector &residualDual,
+        mfem::Vector &stateDual
+    ) const {
+        MFEM_VERIFY(
+            m_preparedOperator.IsPrepared(),
+            "Prepared mass-normalization MFEM adapter requires a prepared row operator."
+        );
+        MFEM_VERIFY(
+            residualDual.Size() == Height(),
+            "Prepared mass-normalization MFEM adapter received a residual dual with the wrong size."
+        );
+
+        using Form                  = utils::blocks::barotropic_equilibrium_form;
+        constexpr auto densityValue = utils::blocks::get_value_block<Form>(utils::blocks::density_field.mass_term);
+        constexpr auto displacementValue =
+            utils::blocks::get_value_block<Form>(utils::blocks::displacement_field.geometry_term);
+        constexpr auto massResidual =
+            utils::blocks::get_residual_block<Form>(utils::blocks::fixed_total_mass_constraint.mass_normalization_term);
+
+        mfem::Vector densityDual;
+        mfem::Vector displacementDual;
+        m_preparedOperator.ApplyCompleteJacobianTransposeAction(
+            residualDual(m_layout.offset(massResidual)), densityDual, displacementDual
+        );
+
+        stateDual.SetSize(Width());
+        stateDual = 0.0;
+
+        mfem::Vector densityBlock(stateDual.GetData() + m_layout.offset(densityValue), m_layout.size(densityValue));
+        densityBlock = densityDual;
+
+        mfem::Vector displacementBlock(
+            stateDual.GetData() + m_layout.offset(displacementValue), m_layout.size(displacementValue)
+        );
+        displacementBlock = displacementDual;
+    }
+
     const MassNormalizationLayout &PreparedMassNormalizationJacobianOperator::GetLayout() const noexcept {
         return m_layout;
     }

libmeanfield/impl/operators/prepared_pressure_force.cpp

@@ -619,7 +619,7 @@ namespace mean_field::operators {
             for (int quadraturePoint = 0; quadraturePoint < quadraturePointCount; ++quadraturePoint) {
                 const double enthalpy = quadratureEnthalpy(quadraturePoint);
 
-                const eos::SpecificEnthalpyValue specificEnthalpy{enthalpy};
+                const dimensions::SpecificEnthalpyValue specificEnthalpy{enthalpy};
                 const double pressure =
                     eos::evaluate<eos::quantity::Pressure>(m_equationOfState, specificEnthalpy).value();
 
@@ -813,11 +813,13 @@ namespace mean_field::operators {
 
         localAction = 0.0;
 
-        mapping::DomainMapper::Workspace workspace(m_fem.mesh->Dimension());
-
         mfem::Vector elementDisplacementVariation;
         mfem::Vector elementAction;
 
+        mfem::DenseMatrix referenceDisplacementDShape;
+        mfem::DenseMatrix referenceDisplacementJacobian;
+        mfem::DenseMatrix inverseElementJacobianVariation;
+        mfem::DenseMatrix matrixTemporary;
         mfem::DenseMatrix physicalTestGradientVariation;
 
         const int dimension                             = m_fem.mesh->Dimension();
@@ -850,13 +852,11 @@ namespace mean_field::operators {
             const mapping::ElementDisplacementData directionData =
                 mapping::ElementDisplacementDataFromElementVDofs(displacementElement, elementDisplacementVariation);
 
-            const mapping::ElementMappingData mappingData{
-                .displacement = *data.baseDisplacementData, .compactification = *data.compactificationData
-            };
+            const int quadraturePointCount         = data.integrationRule->GetNPoints();
 
-            const int quadraturePointCount       = data.integrationRule->GetNPoints();
+            const int scalarDisplacementDofCount   = displacementElement.GetDof();
 
-            const int scalarDisplacementDofCount = displacementElement.GetDof();
+            const mfem::DenseMatrix &directionDofs = directionData.GetDofMatrix();
 
             MFEM_VERIFY(
                 static_cast<int>(data.baseMappingContexts.size()) == quadraturePointCount &&
@@ -869,31 +869,32 @@ namespace mean_field::operators {
 
             elementAction = 0.0;
 
+            referenceDisplacementDShape.SetSize(scalarDisplacementDofCount, dimension);
+            referenceDisplacementJacobian.SetSize(dimension, dimension);
+            inverseElementJacobianVariation.SetSize(dimension, dimension);
+            matrixTemporary.SetSize(dimension, dimension);
             physicalTestGradientVariation.SetSize(scalarDisplacementDofCount, dimension);
 
             for (int quadraturePoint = 0; quadraturePoint < quadraturePointCount; ++quadraturePoint) {
                 const mfem::IntegrationPoint &integrationPoint = data.integrationRule->IntPoint(quadraturePoint);
 
-                transformation->SetIntPoint(&integrationPoint);
+                displacementElement.CalcDShape(integrationPoint, referenceDisplacementDShape);
+                mfem::MultAtB(directionDofs, referenceDisplacementDShape, referenceDisplacementJacobian);
 
-                mapping::VolumeMappingVariation variation;
+                const mfem::DenseMatrix &inverseElementJacobian =
+                    data.baseMappingContexts[quadraturePoint].quadrature.J_inv;
+                mfem::Mult(inverseElementJacobian, referenceDisplacementJacobian, matrixTemporary);
 
-                const mapping::MappingStatus mappingStatus = m_domainMapper.EvaluateVolumeVariation(
-                    mappingData, directionData, *transformation, integrationPoint,
-                    data.baseMappingContexts[quadraturePoint], workspace, variation
-                );
+                double logarithmicJacobianVariation{0.0};
+                for (int component = 0; component < dimension; ++component) {
+                    logarithmicJacobianVariation += matrixTemporary(component, component);
+                }
 
-                MFEM_VERIFY(
-                    mappingStatus == mapping::MappingStatus::valid,
-                    "Stateless mapping variation failed while applying "
-                    "the prepared pressure-force displacement Jacobian. "
-                    "Element: "
-                        << data.elementId << ", attribute: " << transformation->Attribute
-                        << ", quadrature point: " << quadraturePoint << ", status: " << static_cast<int>(mappingStatus)
-                );
+                mfem::Mult(matrixTemporary, inverseElementJacobian, inverseElementJacobianVariation);
+                inverseElementJacobianVariation *= -1.0;
 
                 mfem::Mult(
-                    data.referenceTestGradients[quadraturePoint], variation.inverse_element_jacobian_variation,
+                    data.referenceTestGradients[quadraturePoint], inverseElementJacobianVariation,
                     physicalTestGradientVariation
                 );
 
@@ -905,12 +906,13 @@ namespace mean_field::operators {
                             displacementOrdering, scalarDof, component, scalarDisplacementDofCount, dimension
                         );
 
-                        const double gradientWeightVariation =
-                            data.quadratureWeights(quadraturePoint) *
-                                physicalTestGradientVariation(scalarDof, component) +
-                            variation.weight_variation * physicalTestGradient(scalarDof, component);
+                        const double gradientWeightVariation = data.quadratureWeights(quadraturePoint) *
+                                                                   physicalTestGradientVariation(scalarDof, component) +
+                                                               data.quadratureWeights(quadraturePoint) *
+                                                                   logarithmicJacobianVariation *
+                                                                   physicalTestGradient(scalarDof, component);
 
-                        const double contribution = data.pressure(quadraturePoint) * gradientWeightVariation;
+                        const double contribution            = data.pressure(quadraturePoint) * gradientWeightVariation;
 
                         MFEM_VERIFY(
                             std::isfinite(gradientWeightVariation) && std::isfinite(contribution),

libmeanfield/impl/operators/prepared_rotation_displacement_force.cpp

@@ -1,5 +1,6 @@
 module;
 
+#include <array>
 #include <mfem.hpp>
 
 module mean_field;
@@ -7,6 +8,75 @@ module mean_field;
 import :operators.kernels.rotational_displacement_force;
 import :operators.prepared_rotational_displacement_force;
 
+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 true_to_local(
+        const mfem::ParFiniteElementSpace &finiteElementSpace,
+        const mfem::Vector &trueVector,
+        mfem::Vector &localVector
+    ) {
+        localVector.SetSize(finiteElementSpace.GetVSize());
+        const mfem::Operator *prolongation = finiteElementSpace.GetProlongationMatrix();
+        if (prolongation != nullptr) {
+            prolongation->Mult(trueVector, localVector);
+        } else {
+            localVector = trueVector;
+        }
+    }
+
+    void local_to_true(
+        const mfem::ParFiniteElementSpace &finiteElementSpace,
+        const mfem::Vector &localVector,
+        mfem::Vector &trueVector
+    ) {
+        trueVector.SetSize(finiteElementSpace.GetTrueVSize());
+        trueVector                         = 0.0;
+        const mfem::Operator *prolongation = finiteElementSpace.GetProlongationMatrix();
+        if (prolongation != nullptr) {
+            prolongation->MultTranspose(localVector, trueVector);
+        } else {
+            trueVector = localVector;
+        }
+    }
+
+    [[nodiscard]] int vector_dof_index(
+        const mfem::Ordering::Type ordering,
+        const int scalarDof,
+        const int component,
+        const int scalarDofCount,
+        const int dimension
+    ) {
+        if (ordering == mfem::Ordering::byNODES) {
+            return scalarDof + component * scalarDofCount;
+        }
+        MFEM_VERIFY(ordering == mfem::Ordering::byVDIM, "Unsupported displacement ordering.");
+        return scalarDof * dimension + component;
+    }
+
+    [[nodiscard]] const mfem::IntegrationRule &get_rotation_force_rule(
+        const mean_field::fem::FEM &f,
+        const mfem::ElementTransformation &transformation
+    ) {
+        using DisplacementField = mean_field::field::Field<mean_field::field::Displacement>;
+        const mean_field::quadrature::Query query =
+            DisplacementField::make_query<mean_field::field::Displacement::Form::CentrifugalForce>(
+                mean_field::quadrature::QuadratureRole::discretization, transformation.OrderW(), std::array<int, 1>{1},
+                mean_field::utils::DOMAINS::STELLAR, mean_field::quadrature::MappingKind::general
+            );
+        const mean_field::quadrature::MfemRule rule = f.quadratureFactory->get(query, transformation.GetGeometryType());
+        MFEM_VERIFY(
+            rule.integration_rule != nullptr,
+            "The quadrature policy did not return a rotational-displacement-force integration rule."
+        );
+        return *rule.integration_rule;
+    }
+} // namespace
+
 namespace mean_field::operators {
     PreparedRotationalDisplacementForceOperator::PreparedRotationalDisplacementForceOperator(
         const fem::FEM &f,
@@ -49,6 +119,105 @@ namespace mean_field::operators {
         );
     }
 
+    void PreparedRotationalDisplacementForceOperator::PrepareElementData() {
+        MFEM_VERIFY(m_rotation.has_value(), "Prepared rotational force has no frozen rotation state.");
+
+        m_elements.clear();
+        m_elements.reserve(m_fem.mesh->GetNE());
+
+        mfem::Vector baseDensityLocal;
+        mfem::Vector baseDisplacementLocal;
+        true_to_local(*m_fem.densityFes, m_context.GetBaseDensityTrue(), baseDensityLocal);
+        true_to_local(*m_fem.displacementFes, m_context.GetDisplacementTrue(), baseDisplacementLocal);
+
+        mapping::DomainMapper::Workspace workspace(m_domainMapper.GetDimension());
+        mapping::VolumeMappingContext mappingContext;
+        mfem::Array<int> compactificationDofs;
+        mfem::Vector elementBaseDensity;
+        mfem::Vector elementBaseDisplacement;
+        mfem::Vector elementCompactification;
+        mfem::Vector densityShape;
+        mfem::Vector potentialGradient;
+
+        const int dimension = m_domainMapper.GetDimension();
+
+        for (int elementId = 0; elementId < m_fem.mesh->GetNE(); ++elementId) {
+            mfem::ElementTransformation *transformation = m_fem.mesh->GetElementTransformation(elementId);
+            MFEM_VERIFY(transformation != nullptr, "Prepared rotational force received a null transformation.");
+            if (is_vacuum_attribute(transformation->Attribute)) {
+                continue;
+            }
+
+            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);
+            mfem::DofTransformation *compactificationDofTransformation =
+                m_fem.compactificationFes->GetElementDofs(elementId, compactificationDofs);
+
+            baseDensityLocal.GetSubVector(data.densityDofs, elementBaseDensity);
+            baseDisplacementLocal.GetSubVector(data.displacementDofs, elementBaseDisplacement);
+            m_fem.compactificationCoordinate->GetSubVector(compactificationDofs, elementCompactification);
+            if (data.densityDofTransformation != nullptr) {
+                data.densityDofTransformation->InvTransformPrimal(elementBaseDensity);
+            }
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->InvTransformPrimal(elementBaseDisplacement);
+            }
+            if (compactificationDofTransformation != nullptr) {
+                compactificationDofTransformation->InvTransformPrimal(elementCompactification);
+            }
+
+            const mfem::FiniteElement &densityElement          = *m_fem.densityFes->GetFE(elementId);
+            const mfem::FiniteElement &displacementElement     = *m_fem.displacementFes->GetFE(elementId);
+            const mfem::FiniteElement &compactificationElement = *m_fem.compactificationFes->GetFE(elementId);
+            data.integrationRule                               = &get_rotation_force_rule(m_fem, *transformation);
+
+            const mapping::ElementDisplacementData displacementData =
+                mapping::ElementDisplacementDataFromElementVDofs(displacementElement, elementBaseDisplacement);
+            const mapping::ElementCompactificationData compactificationData(
+                compactificationElement, elementCompactification
+            );
+            const mapping::ElementMappingData mappingData{
+                .displacement = displacementData, .compactification = compactificationData
+            };
+
+            const int quadraturePointCount = data.integrationRule->GetNPoints();
+            data.inverseElementJacobians.SetSize(quadraturePointCount, dimension * dimension);
+            data.centrifugalAccelerations.SetSize(quadraturePointCount, dimension);
+            data.baseDensityValues.SetSize(quadraturePointCount);
+            data.quadratureWeights.SetSize(quadraturePointCount);
+            densityShape.SetSize(densityElement.GetDof());
+            potentialGradient.SetSize(dimension);
+
+            for (int quadraturePoint = 0; quadraturePoint < quadraturePointCount; ++quadraturePoint) {
+                const mfem::IntegrationPoint &integrationPoint = data.integrationRule->IntPoint(quadraturePoint);
+                const mapping::MappingStatus status            = m_domainMapper.EvaluateVolume(
+                    mappingData, *transformation, integrationPoint, workspace, mappingContext
+                );
+                MFEM_VERIFY(
+                    status == mapping::MappingStatus::valid && !mappingContext.mapping.compactified,
+                    "Prepared rotational force encountered an invalid stellar mapping."
+                );
+
+                densityElement.CalcShape(integrationPoint, densityShape);
+                m_rotation->potential_gradient(mappingContext.mapping.physical_position, potentialGradient);
+                data.baseDensityValues(quadraturePoint) = elementBaseDensity * densityShape;
+                data.quadratureWeights(quadraturePoint) = mappingContext.quadrature.weight;
+
+                for (int row = 0; row < dimension; ++row) {
+                    data.centrifugalAccelerations(quadraturePoint, row) = -potentialGradient(row);
+                    for (int column = 0; column < dimension; ++column) {
+                        data.inverseElementJacobians(quadraturePoint, row * dimension + column) =
+                            mappingContext.quadrature.J_inv(row, column);
+                    }
+                }
+            }
+        }
+    }
+
     PreparedRotationalDisplacementForceReport PreparedRotationalDisplacementForceOperator::Prepare(
         const context::rotational_displacement_force::RotationalDisplacementForceStateView &state,
         const context::rotational_displacement_force::RotationalDisplacementForceDependencies &dependencies,
@@ -83,6 +252,7 @@ namespace mean_field::operators {
             );
             m_cachedResidual.SetSize(m_context.GetDisplacementMap().reduced_size());
             m_context.GetDisplacementMap().gather(m_actionTrue, m_cachedResidual);
+            PrepareElementData();
 
             ++m_residualPreparationCount;
             report.preparedResidual = true;
@@ -143,6 +313,97 @@ namespace mean_field::operators {
         ++m_displacementJacobianStatistics.applications;
     }
 
+    void PreparedRotationalDisplacementForceOperator::ApplyPreparedCompleteJacobianActionTrue(
+        const mfem::Vector &densityVariationTrue,
+        const mfem::Vector &displacementVariationTrue,
+        mfem::Vector &actionTrue
+    ) const {
+        true_to_local(*m_fem.densityFes, densityVariationTrue, m_densityVariationLocal);
+        true_to_local(*m_fem.displacementFes, displacementVariationTrue, m_displacementVariationLocal);
+        m_localAction.SetSize(m_fem.displacementFes->GetVSize());
+        m_localAction                       = 0.0;
+
+        const int dimension                 = m_domainMapper.GetDimension();
+        const mfem::Ordering::Type ordering = m_fem.displacementFes->GetOrdering();
+
+        for (const ElementPAData &data : m_elements) {
+            MFEM_VERIFY(data.integrationRule != nullptr, "Prepared rotational force has no integration rule.");
+
+            m_densityVariationLocal.GetSubVector(data.densityDofs, m_elementDensityVariation);
+            m_displacementVariationLocal.GetSubVector(data.displacementDofs, m_elementDisplacementVariation);
+            if (data.densityDofTransformation != nullptr) {
+                data.densityDofTransformation->InvTransformPrimal(m_elementDensityVariation);
+            }
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->InvTransformPrimal(m_elementDisplacementVariation);
+            }
+
+            const mfem::FiniteElement &densityElement      = *m_fem.densityFes->GetFE(data.elementId);
+            const mfem::FiniteElement &displacementElement = *m_fem.displacementFes->GetFE(data.elementId);
+            const mapping::ElementDisplacementData directionData =
+                mapping::ElementDisplacementDataFromElementVDofs(displacementElement, m_elementDisplacementVariation);
+            const mfem::DenseMatrix &directionDofs = directionData.GetDofMatrix();
+            const int scalarDisplacementDofCount   = displacementElement.GetDof();
+
+            m_densityShape.SetSize(densityElement.GetDof());
+            m_displacementShape.SetSize(scalarDisplacementDofCount);
+            m_referenceDisplacementDShape.SetSize(scalarDisplacementDofCount, dimension);
+            m_referenceDisplacementJacobian.SetSize(dimension, dimension);
+            m_physicalPositionVariation.SetSize(dimension);
+            m_centrifugalAcceleration.SetSize(dimension);
+            m_centrifugalAccelerationVariation.SetSize(dimension);
+            m_weightedForce.SetSize(dimension);
+            m_elementAction.SetSize(data.displacementDofs.Size());
+            m_elementAction = 0.0;
+
+            for (int quadraturePoint = 0; quadraturePoint < data.integrationRule->GetNPoints(); ++quadraturePoint) {
+                const mfem::IntegrationPoint &integrationPoint = data.integrationRule->IntPoint(quadraturePoint);
+                densityElement.CalcShape(integrationPoint, m_densityShape);
+                displacementElement.CalcShape(integrationPoint, m_displacementShape);
+                displacementElement.CalcDShape(integrationPoint, m_referenceDisplacementDShape);
+                mfem::MultAtB(directionDofs, m_referenceDisplacementDShape, m_referenceDisplacementJacobian);
+                directionDofs.MultTranspose(m_displacementShape, m_physicalPositionVariation);
+                m_rotation->potential_gradient_directional_derivative(
+                    m_physicalPositionVariation, m_centrifugalAccelerationVariation
+                );
+                m_centrifugalAccelerationVariation *= -1.0;
+
+                double logarithmicJacobianVariation{0.0};
+                for (int row = 0; row < dimension; ++row) {
+                    m_centrifugalAcceleration(row) = data.centrifugalAccelerations(quadraturePoint, row);
+                    for (int column = 0; column < dimension; ++column) {
+                        logarithmicJacobianVariation +=
+                            data.inverseElementJacobians(quadraturePoint, row * dimension + column) *
+                            m_referenceDisplacementJacobian(column, row);
+                    }
+                }
+
+                const double densityVariationValue = m_elementDensityVariation * m_densityShape;
+                const double baseDensityValue      = data.baseDensityValues(quadraturePoint);
+                m_weightedForce                    = 0.0;
+                m_weightedForce.Add(densityVariationValue, m_centrifugalAcceleration);
+                m_weightedForce.Add(baseDensityValue, m_centrifugalAccelerationVariation);
+                m_weightedForce.Add(baseDensityValue * logarithmicJacobianVariation, m_centrifugalAcceleration);
+                m_weightedForce *= data.quadratureWeights(quadraturePoint);
+
+                for (int scalarDof = 0; scalarDof < scalarDisplacementDofCount; ++scalarDof) {
+                    for (int component = 0; component < dimension; ++component) {
+                        const int vectorDof =
+                            vector_dof_index(ordering, scalarDof, component, scalarDisplacementDofCount, dimension);
+                        m_elementAction(vectorDof) += m_displacementShape(scalarDof) * m_weightedForce(component);
+                    }
+                }
+            }
+
+            if (data.displacementDofTransformation != nullptr) {
+                data.displacementDofTransformation->TransformDual(m_elementAction);
+            }
+            m_localAction.AddElementVector(data.displacementDofs, m_elementAction);
+        }
+
+        local_to_true(*m_fem.displacementFes, m_localAction, actionTrue);
+    }
+
     void PreparedRotationalDisplacementForceOperator::ApplyCompleteJacobianAction(
         const mfem::Vector &densityVariation,
         const mfem::Vector &displacementVariation,
@@ -155,10 +416,7 @@ namespace mean_field::operators {
         m_context.GetDensityMap().scatter(densityVariation, m_densityVariationTrue);
         m_context.GetDisplacementMap().scatter(displacementVariation, m_displacementVariationTrue);
 
-        kernels::apply_rotational_displacement_force_complete_action(
-            m_fem, m_domainMapper, *m_rotation, m_context.GetBaseDensityTrue(), m_densityVariationTrue,
-            m_displacementVariationTrue, m_context.GetDisplacementTrue(), m_actionTrue
-        );
+        ApplyPreparedCompleteJacobianActionTrue(m_densityVariationTrue, m_displacementVariationTrue, m_actionTrue);
         action.SetSize(m_context.GetDisplacementMap().reduced_size());
         m_context.GetDisplacementMap().gather(m_actionTrue, action);
 

libmeanfield/impl/operators/prepared_stellar_equilibrium.cpp

@@ -22,26 +22,34 @@ namespace {
         );
     }
 
-    [[nodiscard]] mean_field::operators::StellarEquilibriumLayout make_layout(
+    using StellarRootForm = mean_field::utils::blocks::surface_deformed_stellar_equilibrium_form;
+
+    [[nodiscard]] std::array<
+        int,
+        StellarRootForm::value_block_count>
+    make_value_sizes(
         const mean_field::field::FieldDofMap &densityMap,
         const int surfaceDeformationParameterCount,
         const mean_field::field::FieldDofMap &gravityFluxMap,
         const mean_field::field::FieldDofMap &gravityPotentialMap,
         const mean_field::field::FieldDofMap &enthalpyMap
     ) {
-        using Form = mean_field::utils::blocks::surface_deformed_stellar_equilibrium_form;
+        return {densityMap.reduced_size(),          surfaceDeformationParameterCount, gravityFluxMap.reduced_size(),
+                gravityPotentialMap.reduced_size(), enthalpyMap.reduced_size(),       1};
+    }
 
-        const std::array<int, Form::value_block_count> valueSizes{
-            densityMap.reduced_size(),          surfaceDeformationParameterCount, gravityFluxMap.reduced_size(),
-            gravityPotentialMap.reduced_size(), enthalpyMap.reduced_size(),       1
-        };
-
-        const std::array<int, Form::residual_block_count> residualSizes{
-            gravityFluxMap.reduced_size(),    gravityPotentialMap.reduced_size(), densityMap.reduced_size(),
-            surfaceDeformationParameterCount, enthalpyMap.reduced_size(),         1
-        };
-
-        return {valueSizes, residualSizes};
+    [[nodiscard]] std::array<
+        int,
+        StellarRootForm::residual_block_count>
+    make_residual_sizes(
+        const mean_field::field::FieldDofMap &densityMap,
+        const int surfaceDeformationParameterCount,
+        const mean_field::field::FieldDofMap &gravityFluxMap,
+        const mean_field::field::FieldDofMap &gravityPotentialMap,
+        const mean_field::field::FieldDofMap &enthalpyMap
+    ) {
+        return {gravityFluxMap.reduced_size(),    gravityPotentialMap.reduced_size(), densityMap.reduced_size(),
+                surfaceDeformationParameterCount, enthalpyMap.reduced_size(),         1};
     }
 
     [[nodiscard]] mfem::Array<int> make_gravity_state_offsets(
@@ -70,47 +78,6 @@ namespace {
         return offsets;
     }
 
-    template <int index>
-    [[nodiscard]] mfem::Vector make_value_view(
-        const mfem::Vector &vector,
-        const mean_field::operators::StellarEquilibriumLayout &layout,
-        const mean_field::utils::blocks::value_block<index> block
-    ) {
-        MFEM_VERIFY(
-            vector.Size() == layout.value_offsets().Last(),
-            "The coupled vector does not match the stellar-equilibrium value layout."
-        );
-
-        return mfem::Vector(const_cast<mfem::real_t *>(vector.GetData()) + layout.offset(block), layout.size(block));
-    }
-
-    template <int index>
-    [[nodiscard]] mfem::Vector make_residual_view(
-        mfem::Vector &vector,
-        const mean_field::operators::StellarEquilibriumLayout &layout,
-        const mean_field::utils::blocks::residual_block<index> block
-    ) {
-        MFEM_VERIFY(
-            vector.Size() == layout.residual_offsets().Last(),
-            "The coupled vector does not match the stellar-equilibrium residual layout."
-        );
-
-        return mfem::Vector(vector.GetData() + layout.offset(block), layout.size(block));
-    }
-
-    template <int index>
-    void assign_residual_block(
-        mfem::Vector &coupledResidual,
-        const mean_field::operators::StellarEquilibriumLayout &layout,
-        const mean_field::utils::blocks::residual_block<index> block,
-        const mfem::Vector &blockResidual,
-        const char *message
-    ) {
-        MFEM_VERIFY(layout.size(block) == blockResidual.Size(), message);
-        mfem::Vector destination = make_residual_view(coupledResidual, layout, block);
-        destination              = blockResidual;
-    }
-
     void assign_gravity_block(
         mfem::Vector &gravityState,
         const mfem::Array<int> &offsets,
@@ -264,7 +231,8 @@ namespace mean_field::operators {
         field::FieldDofMap enthalpyMap;
         field::FieldBoundaryDofMap pressureSurfaceRows;
 
-        StellarEquilibriumLayout layout;
+        std::array<int, StellarRootForm::value_block_count> valueSizes;
+        std::array<int, StellarRootForm::residual_block_count> residualSizes;
         mfem::Array<int> gravityStateOffsets;
         mfem::Array<int> gravityResidualOffsets;
 
@@ -307,7 +275,14 @@ namespace mean_field::operators {
                       enthalpyMap
                   )
               ),
-              layout(make_layout(
+              valueSizes(make_value_sizes(
+                  densityMap,
+                  domainDeformation.parameterCount(),
+                  gravityFluxMap,
+                  gravityPotentialMap,
+                  enthalpyMap
+              )),
+              residualSizes(make_residual_sizes(
                   densityMap,
                   domainDeformation.parameterCount(),
                   gravityFluxMap,
@@ -339,7 +314,7 @@ namespace mean_field::operators {
         fem::FEM &f,
         const mapping::DomainMapper &domainMapper,
         const eos::Polytrope &equationOfState,
-        const double targetMass,
+        models::CompiledFixedMass fixedMassConstraint,
         const PressureSurfaceConstraintView surfaceConstraint,
         deformation::PreparedDomainDeformationRuntime domainDeformation
     )
@@ -347,7 +322,7 @@ namespace mean_field::operators {
               f,
               domainMapper,
               equationOfState,
-              targetMass,
+              std::move(fixedMassConstraint),
               surfaceConstraint,
               MakeConstructionData(
                   f,
@@ -360,15 +335,31 @@ namespace mean_field::operators {
         fem::FEM &f,
         const mapping::DomainMapper &domainMapper,
         const eos::Polytrope &equationOfState,
-        const double targetMass,
+        models::CompiledFixedMass fixedMassConstraint,
         const PressureSurfaceConstraintView surfaceConstraint,
         ConstructionData constructionData
     )
         : mfem::Operator(
-              constructionData.layout.residual_offsets().Last(),
-              constructionData.layout.value_offsets().Last()
+              StellarEquilibriumLayout(
+                  constructionData.valueSizes,
+                  constructionData.residualSizes
+              )
+                  .residual_offsets()
+                  .Last(),
+              StellarEquilibriumLayout(
+                  constructionData.valueSizes,
+                  constructionData.residualSizes
+              )
+                  .value_offsets()
+                  .Last()
+          ),
+          m_rootManifest(
+              constructionData.valueSizes,
+              constructionData.residualSizes,
+              fixedMassConstraint.targetMass().value(),
+              surfaceConstraint.descriptor().targetPressure,
+              constructionData.pressureSurfaceRows.size()
           ),
-          m_layout(constructionData.layout),
           m_gravityStateOffsets(constructionData.gravityStateOffsets),
           m_gravityContext(
               f,
@@ -413,14 +404,11 @@ namespace mean_field::operators {
               surfaceConstraint
           ),
           m_domainDeformation(std::move(constructionData.domainDeformation)),
-          m_targetMass(targetMass) {
-        MFEM_VERIFY(
-            std::isfinite(m_targetMass) && m_targetMass > 0.0,
-            "PreparedStellarEquilibriumOperator requires a finite, positive target mass."
-        );
+          m_fixedMassConstraint(std::move(fixedMassConstraint)) {
 
         MFEM_VERIFY(
-            Width() == m_layout.value_offsets().Last() && Height() == m_layout.residual_offsets().Last(),
+            Width() == m_rootManifest.layout().value_offsets().Last() &&
+                Height() == m_rootManifest.layout().residual_offsets().Last(),
             "PreparedStellarEquilibriumOperator has inconsistent block dimensions."
         );
 
@@ -502,27 +490,18 @@ namespace mean_field::operators {
             );
         }
 
-        m_isPrepared                = false;
+        m_isPrepared                      = false;
 
-        using Form                  = utils::blocks::surface_deformed_stellar_equilibrium_form;
-        constexpr auto densityValue = utils::blocks::get_value_block<Form>(utils::blocks::density_field.mass_term);
-        constexpr auto surfaceDeformationValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::surface_deformation_field.parameters_term);
-        constexpr auto gravityGradientValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::gravity_field.gradient_term);
-        constexpr auto gravityPotentialValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::gravity_field.poisson_term);
-        constexpr auto enthalpyValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::enthalpy_field.specific_term);
-        constexpr auto bernoulliValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::barotropic_constant_field.mass_normalization_term);
+        const auto rootState              = m_rootManifest.stateView(state);
 
-        const mfem::Vector reducedDensity               = make_value_view(state, m_layout, densityValue);
-        const mfem::Vector surfaceDeformationParameters = make_value_view(state, m_layout, surfaceDeformationValue);
-        const mfem::Vector gravityGradient              = make_value_view(state, m_layout, gravityGradientValue);
-        const mfem::Vector gravityPotential             = make_value_view(state, m_layout, gravityPotentialValue);
-        const mfem::Vector reducedEnthalpy              = make_value_view(state, m_layout, enthalpyValue);
-        const mfem::Vector bernoulli                    = make_value_view(state, m_layout, bernoulliValue);
+        const mfem::Vector reducedDensity = rootState.block(utils::blocks::density_field.mass_term);
+        const mfem::Vector surfaceDeformationParameters =
+            rootState.block(utils::blocks::surface_deformation_field.parameters_term);
+        const mfem::Vector gravityGradient  = rootState.block(utils::blocks::gravity_field.gradient_term);
+        const mfem::Vector gravityPotential = rootState.block(utils::blocks::gravity_field.poisson_term);
+        const mfem::Vector reducedEnthalpy  = rootState.block(utils::blocks::enthalpy_field.specific_term);
+        const mfem::Vector bernoulli =
+            rootState.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term);
 
         const bool generatedGeometryChanged =
             !wasPrepared || dependencies.discretization != m_preparedDependencies.discretization ||
@@ -569,7 +548,7 @@ namespace mean_field::operators {
         );
 
         report.massNormalization = m_massNormalizationOperator.Prepare(
-            {.targetMass = m_targetMass}, make_mass_dependencies(dependencies, m_generatedDisplacementDependency)
+            m_fixedMassConstraint, make_mass_dependencies(dependencies, m_generatedDisplacementDependency)
         );
 
         report.surfaceConstraint = m_surfaceConstraintOperator.Prepare(
@@ -588,21 +567,6 @@ namespace mean_field::operators {
     }
 
     void PreparedStellarEquilibriumOperator::AssembleResidual() {
-        using Form = utils::blocks::surface_deformed_stellar_equilibrium_form;
-
-        constexpr auto gravityGradientResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::gravity_field.gradient_term);
-        constexpr auto gravityPotentialResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::gravity_field.poisson_term);
-        constexpr auto densityResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::density_field.mass_term);
-        constexpr auto surfaceShapeResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::surface_deformation_field.shape_equilibrium_term);
-        constexpr auto enthalpyResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::enthalpy_field.specific_term);
-        constexpr auto massResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::barotropic_constant_field.mass_normalization_term);
-
         mfem::Vector gravity;
         mfem::Vector closure;
         mfem::Vector surfaceShape;
@@ -621,41 +585,29 @@ namespace mean_field::operators {
         m_massNormalizationOperator.BuildResidual(mass);
 
         m_cachedResidual.SetSize(Height());
-        m_cachedResidual = 0.0;
+        m_cachedResidual        = 0.0;
+        const auto residualView = m_rootManifest.residualView(m_cachedResidual);
 
         MFEM_VERIFY(
-            gravity.Size() == m_layout.size(gravityGradientResidual) + m_layout.size(gravityPotentialResidual),
+            gravity.Size() == residualView.block(utils::blocks::gravity_field.gradient_term).Size() +
+                                  residualView.block(utils::blocks::gravity_field.poisson_term).Size(),
             "The gravity residual has the wrong size."
         );
 
-        mfem::Vector gravityGradient(gravity.GetData(), m_layout.size(gravityGradientResidual));
+        mfem::Vector gravityGradient(
+            gravity.GetData(), residualView.block(utils::blocks::gravity_field.gradient_term).Size()
+        );
         mfem::Vector gravityPotential(
-            gravity.GetData() + m_layout.size(gravityGradientResidual), m_layout.size(gravityPotentialResidual)
+            gravity.GetData() + gravityGradient.Size(),
+            residualView.block(utils::blocks::gravity_field.poisson_term).Size()
         );
 
-        assign_residual_block(
-            m_cachedResidual, m_layout, gravityGradientResidual, gravityGradient,
-            "The gravity-gradient residual has the wrong size."
-        );
-        assign_residual_block(
-            m_cachedResidual, m_layout, gravityPotentialResidual, gravityPotential,
-            "The gravity-potential residual has the wrong size."
-        );
-        assign_residual_block(
-            m_cachedResidual, m_layout, densityResidual, closure, "The closure residual has the wrong size."
-        );
-        assign_residual_block(
-            m_cachedResidual, m_layout, surfaceShapeResidual, surfaceShape,
-            "The surface-shape residual has the wrong size."
-        );
-
-        assign_residual_block(
-            m_cachedResidual, m_layout, enthalpyResidual, hydrostatic, "The hydrostatic residual has the wrong size."
-        );
-
-        assign_residual_block(
-            m_cachedResidual, m_layout, massResidual, mass, "The mass-normalization residual has the wrong size."
-        );
+        residualView.assign(utils::blocks::gravity_field.gradient_term, gravityGradient);
+        residualView.assign(utils::blocks::gravity_field.poisson_term, gravityPotential);
+        residualView.assign(utils::blocks::density_field.mass_term, closure);
+        residualView.assign(utils::blocks::surface_deformation_field.shape_equilibrium_term, surfaceShape);
+        residualView.assign(utils::blocks::enthalpy_field.specific_term, hydrostatic);
+        residualView.assign(utils::blocks::fixed_total_mass_constraint.mass_normalization_term, mass);
 
         ++m_statistics.residualAssemblies;
     }
@@ -679,39 +631,16 @@ namespace mean_field::operators {
             direction, "PreparedStellarEquilibriumOperator received a non-finite Jacobian direction."
         );
 
-        using Form                  = utils::blocks::surface_deformed_stellar_equilibrium_form;
+        const auto rootDirection                   = m_rootManifest.directionView(direction);
 
-        constexpr auto densityValue = utils::blocks::get_value_block<Form>(utils::blocks::density_field.mass_term);
-        constexpr auto surfaceDeformationValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::surface_deformation_field.parameters_term);
-        constexpr auto gravityGradientValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::gravity_field.gradient_term);
-        constexpr auto gravityPotentialValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::gravity_field.poisson_term);
-        constexpr auto enthalpyValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::enthalpy_field.specific_term);
-        constexpr auto bernoulliValue =
-            utils::blocks::get_value_block<Form>(utils::blocks::barotropic_constant_field.mass_normalization_term);
-
-        constexpr auto gravityGradientResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::gravity_field.gradient_term);
-        constexpr auto gravityPotentialResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::gravity_field.poisson_term);
-        constexpr auto densityResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::density_field.mass_term);
-        constexpr auto surfaceShapeResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::surface_deformation_field.shape_equilibrium_term);
-        constexpr auto enthalpyResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::enthalpy_field.specific_term);
-        constexpr auto massResidual =
-            utils::blocks::get_residual_block<Form>(utils::blocks::barotropic_constant_field.mass_normalization_term);
-
-        const mfem::Vector reducedDensityDirection     = make_value_view(direction, m_layout, densityValue);
-        const mfem::Vector surfaceDeformationDirection = make_value_view(direction, m_layout, surfaceDeformationValue);
-        const mfem::Vector gravityGradientDirection    = make_value_view(direction, m_layout, gravityGradientValue);
-        const mfem::Vector gravityPotentialDirection   = make_value_view(direction, m_layout, gravityPotentialValue);
-        const mfem::Vector reducedEnthalpyDirection    = make_value_view(direction, m_layout, enthalpyValue);
-        const mfem::Vector bernoulliDirection          = make_value_view(direction, m_layout, bernoulliValue);
+        const mfem::Vector reducedDensityDirection = rootDirection.block(utils::blocks::density_field.mass_term);
+        const mfem::Vector surfaceDeformationDirection =
+            rootDirection.block(utils::blocks::surface_deformation_field.parameters_term);
+        const mfem::Vector gravityGradientDirection  = rootDirection.block(utils::blocks::gravity_field.gradient_term);
+        const mfem::Vector gravityPotentialDirection = rootDirection.block(utils::blocks::gravity_field.poisson_term);
+        const mfem::Vector reducedEnthalpyDirection  = rootDirection.block(utils::blocks::enthalpy_field.specific_term);
+        const mfem::Vector bernoulliDirection =
+            rootDirection.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term);
 
         m_domainDeformation.applyJacobian(
             m_surfaceDeformationParameters, surfaceDeformationDirection, m_volumeDisplacementDirection
@@ -757,41 +686,29 @@ namespace mean_field::operators {
         );
 
         action.SetSize(Height());
-        action = 0.0;
+        action                = 0.0;
+        const auto actionView = m_rootManifest.residualView(action);
 
         MFEM_VERIFY(
-            gravityAction.Size() == m_layout.size(gravityGradientResidual) + m_layout.size(gravityPotentialResidual),
+            gravityAction.Size() == actionView.block(utils::blocks::gravity_field.gradient_term).Size() +
+                                        actionView.block(utils::blocks::gravity_field.poisson_term).Size(),
             "The gravity Jacobian action has the wrong size."
         );
 
-        mfem::Vector gravityGradientAction(gravityAction.GetData(), m_layout.size(gravityGradientResidual));
+        mfem::Vector gravityGradientAction(
+            gravityAction.GetData(), actionView.block(utils::blocks::gravity_field.gradient_term).Size()
+        );
         mfem::Vector gravityPotentialAction(
-            gravityAction.GetData() + m_layout.size(gravityGradientResidual), m_layout.size(gravityPotentialResidual)
+            gravityAction.GetData() + gravityGradientAction.Size(),
+            actionView.block(utils::blocks::gravity_field.poisson_term).Size()
         );
 
-        assign_residual_block(
-            action, m_layout, gravityGradientResidual, gravityGradientAction,
-            "The gravity-gradient Jacobian action has the wrong size."
-        );
-        assign_residual_block(
-            action, m_layout, gravityPotentialResidual, gravityPotentialAction,
-            "The gravity-potential Jacobian action has the wrong size."
-        );
-        assign_residual_block(
-            action, m_layout, densityResidual, closureAction, "The closure Jacobian action has the wrong size."
-        );
-        assign_residual_block(
-            action, m_layout, surfaceShapeResidual, m_surfaceShapeAction,
-            "The surface-shape Jacobian action has the wrong size."
-        );
-
-        assign_residual_block(
-            action, m_layout, enthalpyResidual, hydrostaticAction, "The hydrostatic Jacobian action has the wrong size."
-        );
-
-        assign_residual_block(
-            action, m_layout, massResidual, massAction, "The mass-normalization Jacobian action has the wrong size."
-        );
+        actionView.assign(utils::blocks::gravity_field.gradient_term, gravityGradientAction);
+        actionView.assign(utils::blocks::gravity_field.poisson_term, gravityPotentialAction);
+        actionView.assign(utils::blocks::density_field.mass_term, closureAction);
+        actionView.assign(utils::blocks::surface_deformation_field.shape_equilibrium_term, m_surfaceShapeAction);
+        actionView.assign(utils::blocks::enthalpy_field.specific_term, hydrostaticAction);
+        actionView.assign(utils::blocks::fixed_total_mass_constraint.mass_normalization_term, massAction);
 
         ++m_statistics.jacobianApplications;
     }
@@ -803,11 +720,30 @@ namespace mean_field::operators {
     }
 
     double PreparedStellarEquilibriumOperator::GetTargetMass() const noexcept {
-        return m_targetMass;
+        return m_fixedMassConstraint.targetMass().value();
     }
 
     const StellarEquilibriumLayout &PreparedStellarEquilibriumOperator::GetLayout() const noexcept {
-        return m_layout;
+        return m_rootManifest.layout();
+    }
+
+    const StellarEquilibriumRootManifest &PreparedStellarEquilibriumOperator::GetRootManifest() const noexcept {
+        return m_rootManifest;
+    }
+
+    RootStateView<utils::blocks::surface_deformed_stellar_equilibrium_form>
+    PreparedStellarEquilibriumOperator::GetRootStateView(const mfem::Vector &state) const {
+        return m_rootManifest.stateView(state);
+    }
+
+    ResidualView<utils::blocks::surface_deformed_stellar_equilibrium_form>
+    PreparedStellarEquilibriumOperator::GetResidualView(mfem::Vector &residual) const {
+        return m_rootManifest.residualView(residual);
+    }
+
+    RootConstraintReport PreparedStellarEquilibriumOperator::GetFixedMassReport() const {
+        VerifyPrepared();
+        return m_rootManifest.fixedMassReport(m_massNormalizationOperator.GetCurrentMass());
     }
 
     const StellarEquilibriumDependencies &PreparedStellarEquilibriumOperator::GetDependencies() const {

libmeanfield/impl/physics/gravity.cpp

@@ -102,7 +102,7 @@ namespace mean_field::physics {
 
     GravitySolution solve_gravity_field(
         fem::FEM &f,
-        const utils::Args &args,
+        const GravitySolveOptions &options,
         const mfem::GridFunction &rho,
         const mfem::GridFunction &displacement
     ) {
@@ -132,7 +132,15 @@ namespace mean_field::physics {
             "Vec_H1 "
             "space."
         );
-        MFEM_VERIFY(args.p.max_iters > 0, "Gravity solve requires a positive MINRES iteration limit.");
+        MFEM_VERIFY(
+            std::isfinite(options.relativeTolerance) && options.relativeTolerance >= 0.0,
+            "Gravity solve requires a finite, nonnegative relative tolerance."
+        );
+        MFEM_VERIFY(
+            std::isfinite(options.absoluteTolerance) && options.absoluteTolerance >= 0.0,
+            "Gravity solve requires a finite, nonnegative absolute tolerance."
+        );
+        MFEM_VERIFY(options.maximumIterations > 0, "Gravity solve requires a positive MINRES iteration limit.");
 
         using form = utils::blocks::gravity_field_form;
 
@@ -206,9 +214,9 @@ namespace mean_field::physics {
         mfem::MINRESSolver minres(f.mesh->GetComm());
         minres.SetOperator(reduced_operator);
         minres.SetPreconditioner(reduced_preconditioner);
-        minres.SetRelTol(args.p.rtol);
-        minres.SetAbsTol(args.p.atol);
-        minres.SetMaxIter(args.p.max_iters);
+        minres.SetRelTol(options.relativeTolerance);
+        minres.SetAbsTol(options.absoluteTolerance);
+        minres.SetMaxIter(options.maximumIterations);
         // minres.SetPrintLevel(args.verbose ? 1 : 0);
         minres.SetPrintLevel(0);
         minres.Mult(right_hand_side, gravity_state);
@@ -222,4 +230,21 @@ namespace mean_field::physics {
 
         return solution;
     }
+
+    GravitySolution solve_gravity_field(
+        fem::FEM &f,
+        const utils::Args &args,
+        const mfem::GridFunction &rho,
+        const mfem::GridFunction &displacement
+    ) {
+        return solve_gravity_field(
+            f,
+            GravitySolveOptions{
+                .relativeTolerance = args.p.rtol,
+                .absoluteTolerance = args.p.atol,
+                .maximumIterations = args.p.max_iters
+            },
+            rho, displacement
+        );
+    }
 } // namespace mean_field::physics

libmeanfield/impl/seed/lane_emden.cpp

@@ -0,0 +1,248 @@
+module;
+
+#include <algorithm>
+#include <cmath>
+#include <numbers>
+#include <optional>
+#include <stdexcept>
+#include <vector>
+
+#include <mfem.hpp>
+
+module mean_field;
+
+import :seed.lane_emden;
+import :utils.misc;
+
+namespace {
+    struct LaneEmdenPoint final {
+        double coordinate{0.0};
+        double value{0.0};
+        double derivative{0.0};
+    };
+
+    struct LaneEmdenDerivative final {
+        double value{0.0};
+        double derivative{0.0};
+    };
+
+    [[nodiscard]] LaneEmdenDerivative evaluate_lane_emden_rhs(
+        const double coordinate,
+        const double value,
+        const double derivative,
+        const double polytropicIndex
+    ) {
+        const double nonnegativeValue = std::max(value, 0.0);
+        return {
+            .value      = derivative,
+            .derivative = -2.0 * derivative / coordinate - std::pow(nonnegativeValue, polytropicIndex)
+        };
+    }
+
+    [[nodiscard]] LaneEmdenPoint take_lane_emden_step(
+        const LaneEmdenPoint &point,
+        const double step,
+        const double polytropicIndex
+    ) {
+        const LaneEmdenDerivative first =
+            evaluate_lane_emden_rhs(point.coordinate, point.value, point.derivative, polytropicIndex);
+        const LaneEmdenDerivative second = evaluate_lane_emden_rhs(
+            point.coordinate + 0.5 * step, point.value + 0.5 * step * first.value,
+            point.derivative + 0.5 * step * first.derivative, polytropicIndex
+        );
+        const LaneEmdenDerivative third = evaluate_lane_emden_rhs(
+            point.coordinate + 0.5 * step, point.value + 0.5 * step * second.value,
+            point.derivative + 0.5 * step * second.derivative, polytropicIndex
+        );
+        const LaneEmdenDerivative fourth = evaluate_lane_emden_rhs(
+            point.coordinate + step, point.value + step * third.value, point.derivative + step * third.derivative,
+            polytropicIndex
+        );
+
+        return {
+            .coordinate = point.coordinate + step,
+            .value = point.value + step / 6.0 * (first.value + 2.0 * second.value + 2.0 * third.value + fourth.value),
+            .derivative =
+                point.derivative +
+                step / 6.0 * (first.derivative + 2.0 * second.derivative + 2.0 * third.derivative + fourth.derivative)
+        };
+    }
+
+    [[nodiscard]] std::vector<LaneEmdenPoint> solve_lane_emden(
+        const double polytropicIndex,
+        const double coordinateLimit,
+        const double integrationStep
+    ) {
+        if (!std::isfinite(polytropicIndex) || polytropicIndex < 0.0) {
+            throw std::invalid_argument("Lane-Emden integration requires a finite, nonnegative polytropic index.");
+        }
+        if (!std::isfinite(coordinateLimit) || coordinateLimit <= 0.0) {
+            throw std::invalid_argument("The Lane-Emden coordinate limit must be finite and positive.");
+        }
+        if (!std::isfinite(integrationStep) || integrationStep <= 0.0) {
+            throw std::invalid_argument("The Lane-Emden integration step must be finite and positive.");
+        }
+
+        constexpr int maximumStepCount = 2'000'000;
+        if (std::ceil(coordinateLimit / integrationStep) > static_cast<double>(maximumStepCount)) {
+            throw std::invalid_argument("The requested Lane-Emden interval exceeds the integration step limit.");
+        }
+
+        const double initialCoordinate = std::min(1.0e-6, coordinateLimit);
+
+        const double coordinateSquared = initialCoordinate * initialCoordinate;
+        const double coordinateCubed   = coordinateSquared * initialCoordinate;
+        const double coordinateFourth  = coordinateSquared * coordinateSquared;
+
+        LaneEmdenPoint point{
+            .coordinate = initialCoordinate,
+            .value      = 1.0 - coordinateSquared / 6.0 + polytropicIndex * coordinateFourth / 120.0,
+            .derivative = -initialCoordinate / 3.0 + polytropicIndex * coordinateCubed / 30.0
+        };
+
+        std::vector<LaneEmdenPoint> solution;
+        solution.reserve(8192);
+        solution.push_back({.coordinate = 0.0, .value = 1.0, .derivative = 0.0});
+        solution.push_back(point);
+
+        for (int stepIndex = 0; stepIndex < maximumStepCount && point.coordinate < coordinateLimit; ++stepIndex) {
+            const double step        = std::min(integrationStep, coordinateLimit - point.coordinate);
+            LaneEmdenPoint nextPoint = take_lane_emden_step(point, step, polytropicIndex);
+            if (!std::isfinite(nextPoint.value)) {
+                throw std::runtime_error(
+                    "The Lane-Emden integration produced a non-finite solution before reaching its termination."
+                );
+            }
+            if (nextPoint.value <= 0.0) {
+                const double rootFraction = point.value / (point.value - nextPoint.value);
+                solution.push_back(
+                    {.coordinate = point.coordinate + rootFraction * (nextPoint.coordinate - point.coordinate),
+                     .value      = 0.0,
+                     .derivative = point.derivative + rootFraction * (nextPoint.derivative - point.derivative)}
+                );
+                return solution;
+            }
+            solution.push_back(nextPoint);
+            point = nextPoint;
+        }
+
+        if (point.coordinate < coordinateLimit) {
+            throw std::runtime_error("The Lane-Emden integration exceeded its step limit.");
+        }
+        return solution;
+    }
+
+    [[nodiscard]] double interpolate_lane_emden_value(
+        const std::vector<LaneEmdenPoint> &solution,
+        const double coordinate,
+        std::size_t &lowerIndex
+    ) {
+        while (lowerIndex + 1 < solution.size() && solution[lowerIndex + 1].coordinate < coordinate) {
+            ++lowerIndex;
+        }
+        if (lowerIndex + 1 >= solution.size()) {
+            return 0.0;
+        }
+
+        const LaneEmdenPoint &lower = solution[lowerIndex];
+        const LaneEmdenPoint &upper = solution[lowerIndex + 1];
+        const double interval       = upper.coordinate - lower.coordinate;
+        if (interval <= 0.0) {
+            throw std::runtime_error("The Lane-Emden interpolation grid is not strictly increasing.");
+        }
+        const double fraction = (coordinate - lower.coordinate) / interval;
+        return std::clamp(lower.value + fraction * (upper.value - lower.value), 0.0, 1.0);
+    }
+} // namespace
+
+namespace mean_field::seed {
+    DimensionlessLaneEmdenSolution integrateLaneEmden(
+        const double polytropicIndex,
+        const double coordinateLimit,
+        const double integrationStep
+    ) {
+        const std::vector<LaneEmdenPoint> points = solve_lane_emden(polytropicIndex, coordinateLimit, integrationStep);
+
+        DimensionlessLaneEmdenSolution solution{
+            .coordinate          = mfem::Vector(static_cast<int>(points.size())),
+            .theta               = mfem::Vector(static_cast<int>(points.size())),
+            .thetaDerivative     = mfem::Vector(static_cast<int>(points.size())),
+            .firstZeroCoordinate = std::nullopt
+        };
+        for (int index = 0; index < static_cast<int>(points.size()); ++index) {
+            solution.coordinate(index)      = points[static_cast<std::size_t>(index)].coordinate;
+            solution.theta(index)           = points[static_cast<std::size_t>(index)].value;
+            solution.thetaDerivative(index) = points[static_cast<std::size_t>(index)].derivative;
+        }
+        if (points.back().value == 0.0) {
+            solution.firstZeroCoordinate = points.back().coordinate;
+        }
+        return solution;
+    }
+
+    RadialProfile generateLaneEmdenProfile(
+        const eos::Polytrope &equationOfState,
+        const dimensions::DensityValue centralDensity,
+        const int radialSampleCount
+    ) {
+        if (!std::isfinite(centralDensity.value()) || centralDensity.value() <= 0.0) {
+            throw std::invalid_argument("A Lane-Emden seed central density must be finite and positive.");
+        }
+        if (radialSampleCount < 2) {
+            throw std::invalid_argument("A Lane-Emden seed requires at least two radial samples.");
+        }
+
+        const double polytropicIndex = equationOfState.polytropic_index();
+        if (!std::isfinite(polytropicIndex) || polytropicIndex < 1.0 || polytropicIndex >= 5.0) {
+            throw std::invalid_argument("Lane-Emden seeds require a finite-radius polytrope with 1 <= n < 5.");
+        }
+
+        constexpr double seedCoordinateLimit = 2'000.0;
+        constexpr double integrationStep     = 1.0e-3;
+        const std::vector solution           = solve_lane_emden(polytropicIndex, seedCoordinateLimit, integrationStep);
+        if (solution.back().value != 0.0) {
+            throw std::runtime_error("The Lane-Emden integration did not reach its first zero within the step limit.");
+        }
+        const double surfaceCoordinate = solution.back().coordinate;
+        const dimensions::SpecificEnthalpyValue centralEnthalpy =
+            eos::evaluate<dimensions::quantity::SpecificEnthalpy>(equationOfState, centralDensity);
+        const double radialScaleSquared = centralEnthalpy.value() / (4.0 * std::numbers::pi_v<double> *
+                                                                     mean_field::utils::G * centralDensity.value());
+        if (!std::isfinite(radialScaleSquared) || radialScaleSquared <= 0.0) {
+            throw std::runtime_error("The polytropic Lane-Emden radial scale is not finite and positive.");
+        }
+
+        const double radialScale = std::sqrt(radialScaleSquared);
+        RadialProfile profile{
+            .radius                  = mfem::Vector(radialSampleCount),
+            .density                 = mfem::Vector(radialSampleCount),
+            .specificEnthalpy        = mfem::Vector(radialSampleCount),
+            .stellarRadius           = dimensions::LengthValue{radialScale * surfaceCoordinate},
+            .centralDensity          = centralDensity,
+            .centralSpecificEnthalpy = centralEnthalpy
+        };
+
+        std::size_t interpolationIndex = 0;
+        for (int sampleIndex = 0; sampleIndex < radialSampleCount; ++sampleIndex) {
+            const double fraction = static_cast<double>(sampleIndex) / static_cast<double>(radialSampleCount - 1);
+            const double dimensionlessRadius = fraction * surfaceCoordinate;
+            const double laneEmdenValue =
+                interpolate_lane_emden_value(solution, dimensionlessRadius, interpolationIndex);
+            const dimensions::DensityValue density{centralDensity.value() * std::pow(laneEmdenValue, polytropicIndex)};
+
+            profile.radius(sampleIndex)  = radialScale * dimensionlessRadius;
+            profile.density(sampleIndex) = density.value();
+            profile.specificEnthalpy(sampleIndex) =
+                eos::evaluate<dimensions::quantity::SpecificEnthalpy>(equationOfState, density).value();
+        }
+
+        profile.radius(0)                      = 0.0;
+        profile.density(0)                     = centralDensity.value();
+        profile.specificEnthalpy(0)            = centralEnthalpy.value();
+        const int surfaceIndex                 = radialSampleCount - 1;
+        profile.radius(surfaceIndex)           = profile.stellarRadius.value();
+        profile.density(surfaceIndex)          = 0.0;
+        profile.specificEnthalpy(surfaceIndex) = 0.0;
+        return profile;
+    }
+} // namespace mean_field::seed

libmeanfield/impl/seed/stellar_equilibrium_projection.cpp

@@ -0,0 +1,209 @@
+module;
+
+#include <algorithm>
+#include <cmath>
+#include <limits>
+#include <stdexcept>
+
+#include <mfem.hpp>
+#include <mpi.h>
+
+module mean_field;
+
+import :field.mfem;
+import :seed.stellar_equilibrium_projection;
+import :utils.domain;
+import :utils.misc;
+
+namespace {
+    using DomainSchema = mean_field::utils::domain::CoreEnvelopeVacuumDomainSchema;
+
+    void validate_profile(const mean_field::seed::RadialProfile &profile) {
+        const int sampleCount = profile.radius.Size();
+        if (sampleCount < 2 || profile.density.Size() != sampleCount ||
+            profile.specificEnthalpy.Size() != sampleCount) {
+            throw std::invalid_argument("A radial seed projection requires equally sized profiles with two samples.");
+        }
+        if (!std::isfinite(profile.stellarRadius.value()) || profile.stellarRadius.value() <= 0.0 ||
+            !std::isfinite(profile.centralDensity.value()) || profile.centralDensity.value() <= 0.0 ||
+            !std::isfinite(profile.centralSpecificEnthalpy.value()) || profile.centralSpecificEnthalpy.value() <= 0.0) {
+            throw std::invalid_argument("A radial seed projection requires finite, positive physical scales.");
+        }
+
+        for (int index = 0; index < sampleCount; ++index) {
+            if (!std::isfinite(profile.radius(index)) || !std::isfinite(profile.density(index)) ||
+                !std::isfinite(profile.specificEnthalpy(index)) || profile.density(index) < 0.0 ||
+                profile.specificEnthalpy(index) < 0.0) {
+                throw std::invalid_argument("A radial seed projection received a non-finite or negative profile.");
+            }
+            if (index > 0 && profile.radius(index) <= profile.radius(index - 1)) {
+                throw std::invalid_argument("A radial seed projection requires strictly increasing radii.");
+            }
+        }
+
+        const int surfaceIndex   = sampleCount - 1;
+        const double radialScale = std::max(profile.stellarRadius.value(), 1.0);
+        if (std::abs(profile.radius(0)) > 64.0 * std::numeric_limits<double>::epsilon() * radialScale ||
+            std::abs(profile.radius(surfaceIndex) - profile.stellarRadius.value()) >
+                64.0 * std::numeric_limits<double>::epsilon() * radialScale ||
+            profile.density(0) != profile.centralDensity.value() ||
+            profile.specificEnthalpy(0) != profile.centralSpecificEnthalpy.value() ||
+            profile.density(surfaceIndex) != 0.0 || profile.specificEnthalpy(surfaceIndex) != 0.0) {
+            throw std::invalid_argument("A radial seed projection received inconsistent center or surface metadata.");
+        }
+    }
+
+    [[nodiscard]] double interpolate_profile(
+        const mfem::Vector &radius,
+        const mfem::Vector &values,
+        const double requestedRadius
+    ) {
+        if (requestedRadius <= radius(0)) {
+            return values(0);
+        }
+        const int finalIndex = radius.Size() - 1;
+        if (requestedRadius >= radius(finalIndex)) {
+            return values(finalIndex);
+        }
+
+        int lowerIndex = 0;
+        int upperIndex = finalIndex;
+        while (upperIndex - lowerIndex > 1) {
+            const int middleIndex = lowerIndex + (upperIndex - lowerIndex) / 2;
+            if (radius(middleIndex) <= requestedRadius) {
+                lowerIndex = middleIndex;
+            } else {
+                upperIndex = middleIndex;
+            }
+        }
+
+        const double fraction = (requestedRadius - radius(lowerIndex)) / (radius(upperIndex) - radius(lowerIndex));
+        return (1.0 - fraction) * values(lowerIndex) + fraction * values(upperIndex);
+    }
+
+    struct SurfaceRadiusRange final {
+        double minimum;
+        double maximum;
+    };
+
+    [[nodiscard]] SurfaceRadiusRange measure_surface_radius(const mean_field::fem::FEM &finiteElementModel) {
+        if (finiteElementModel.surfaceDeformationFes == nullptr) {
+            throw std::invalid_argument("Radial seed projection requires the surface-deformation space.");
+        }
+
+        mfem::ParFiniteElementSpace &surfaceSpace = *finiteElementModel.surfaceDeformationFes;
+        const mean_field::field::ScalarBoundaryDofMap surfaceMap =
+            mean_field::field::make_stellar_surface_scalar_dof_map<DomainSchema>(surfaceSpace);
+        mfem::Vector radiusSquared(surfaceMap.local_size());
+        radiusSquared = 0.0;
+
+        mfem::ParGridFunction coordinateField(&surfaceSpace);
+        for (int component = 0; component < surfaceSpace.GetMesh()->SpaceDimension(); ++component) {
+            mfem::FunctionCoefficient coordinateCoefficient([component](const mfem::Vector &position) {
+                return position(component);
+            });
+            coordinateField.ProjectCoefficient(coordinateCoefficient);
+            mfem::Vector coordinateTrue;
+            coordinateField.GetTrueDofs(coordinateTrue);
+            const mfem::Vector surfaceCoordinate = surfaceMap.gather(coordinateTrue);
+            for (int index = 0; index < radiusSquared.Size(); ++index) {
+                radiusSquared(index) += surfaceCoordinate(index) * surfaceCoordinate(index);
+            }
+        }
+
+        double localMinimum = std::numeric_limits<double>::infinity();
+        double localMaximum = 0.0;
+        for (int index = 0; index < radiusSquared.Size(); ++index) {
+            const double radius = std::sqrt(radiusSquared(index));
+            localMinimum        = std::min(localMinimum, radius);
+            localMaximum        = std::max(localMaximum, radius);
+        }
+
+        double globalMinimum = 0.0;
+        double globalMaximum = 0.0;
+        MPI_Allreduce(&localMinimum, &globalMinimum, 1, MPI_DOUBLE, MPI_MIN, surfaceSpace.GetComm());
+        MPI_Allreduce(&localMaximum, &globalMaximum, 1, MPI_DOUBLE, MPI_MAX, surfaceSpace.GetComm());
+        if (!std::isfinite(globalMinimum) || !std::isfinite(globalMaximum) || globalMinimum <= 0.0 ||
+            globalMaximum < globalMinimum) {
+            throw std::runtime_error("The stellar surface has no finite, positive radial extent.");
+        }
+        return {.minimum = globalMinimum, .maximum = globalMaximum};
+    }
+} // namespace
+
+namespace mean_field::seed::detail {
+    ProjectedRadialFields projectRadialFields(
+        const equilibrium::StellarDiscretization &discretization,
+        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});
+        const double relativeMismatch =
+            std::max(std::abs(surfaceRadius.minimum - targetRadius), std::abs(surfaceRadius.maximum - targetRadius)) /
+            comparisonScale;
+        if (relativeMismatch > options.surfaceRadiusRelativeTolerance) {
+            throw std::invalid_argument(
+                "The radial seed surface does not coincide with the spherical reference discretization."
+            );
+        }
+
+        if (finiteElementModel.densityFes == nullptr || finiteElementModel.enthalpyFes == nullptr ||
+            finiteElementModel.displacementFes == nullptr || finiteElementModel.gravityFluxFes == nullptr ||
+            finiteElementModel.gravityPotentialFes == nullptr) {
+            throw std::invalid_argument("Radial seed projection requires the complete equilibrium discretization.");
+        }
+
+        mfem::FunctionCoefficient densityCoefficient([&profile](const mfem::Vector &position) {
+            return interpolate_profile(profile.radius, profile.density, position.Norml2());
+        });
+        mfem::FunctionCoefficient enthalpyCoefficient([&profile](const mfem::Vector &position) {
+            return interpolate_profile(profile.radius, profile.specificEnthalpy, position.Norml2());
+        });
+
+        mfem::ParGridFunction densityField(finiteElementModel.densityFes.get());
+        mfem::ParGridFunction enthalpyField(finiteElementModel.enthalpyFes.get());
+        mfem::ParGridFunction displacementField(finiteElementModel.displacementFes.get());
+        densityField      = 0.0;
+        enthalpyField     = 0.0;
+        displacementField = 0.0;
+        densityField.ProjectCoefficient(densityCoefficient);
+        enthalpyField.ProjectCoefficient(enthalpyCoefficient);
+
+        const physics::GravitySolution gravitySolution =
+            physics::solve_gravity_field(finiteElementModel, options.gravity, densityField, displacementField);
+
+        const field::FieldDofGridFunctionAdapter densityAdapter =
+            field::make_field_dof_grid_function_adapter<field::Density, DomainSchema>(*finiteElementModel.densityFes);
+        const field::FieldDofGridFunctionAdapter enthalpyAdapter =
+            field::make_field_dof_grid_function_adapter<field::Enthalpy, DomainSchema>(*finiteElementModel.enthalpyFes);
+        const field::FieldDofGridFunctionAdapter gravityFluxAdapter =
+            field::make_field_dof_grid_function_adapter<field::Gravity, DomainSchema>(
+                *finiteElementModel.gravityFluxFes
+            );
+        const field::FieldDofGridFunctionAdapter gravityPotentialAdapter =
+            field::make_field_dof_grid_function_adapter<field::Gravity, DomainSchema>(
+                *finiteElementModel.gravityPotentialFes
+            );
+
+        return {
+            .density           = densityAdapter.gather(densityField),
+            .gravityGradient   = gravityFluxAdapter.gather(gravitySolution.gradPhi),
+            .gravityPotential  = gravityPotentialAdapter.gather(gravitySolution.phi),
+            .specificEnthalpy  = enthalpyAdapter.gather(enthalpyField),
+            .bernoulliConstant = -utils::G * targetMass.value() / targetRadius
+        };
+    }
+} // namespace mean_field::seed::detail

libmeanfield/impl/solver/preconditioning_diagnostics.cpp

@@ -0,0 +1,638 @@
+module;
+
+#include <algorithm>
+#include <chrono>
+#include <cmath>
+#include <complex>
+#include <cstdint>
+#include <limits>
+#include <memory>
+#include <ranges>
+#include <stdexcept>
+#include <string>
+#include <utility>
+#include <vector>
+
+#include <Eigen/Dense>
+#include <Eigen/Eigenvalues>
+#include <Eigen/SVD>
+#include <mfem.hpp>
+#include <mpi.h>
+
+module mean_field;
+
+import :solver.preconditioning_diagnostics;
+
+namespace {
+    using Clock = std::chrono::steady_clock;
+
+    [[nodiscard]] double seconds_between(
+        const Clock::time_point start,
+        const Clock::time_point finish
+    ) {
+        return std::chrono::duration<double>(finish - start).count();
+    }
+
+    void verify_finite_vector(
+        const mfem::Vector &vector,
+        const char *message
+    ) {
+        for (int index = 0; index < vector.Size(); ++index) {
+            if (!std::isfinite(vector(index))) {
+                throw std::invalid_argument(message);
+            }
+        }
+    }
+
+    [[nodiscard]] double global_dot(
+        const mfem::Vector &left,
+        const mfem::Vector &right,
+        const MPI_Comm communicator
+    ) {
+        if (communicator == MPI_COMM_NULL) {
+            throw std::invalid_argument("Preconditioning diagnostics require a valid MPI communicator.");
+        }
+        if (left.Size() != right.Size()) {
+            throw std::invalid_argument("A distributed inner product received vectors with different sizes.");
+        }
+
+        const double localValue = left * right;
+        double globalValue      = 0.0;
+        MPI_Allreduce(&localValue, &globalValue, 1, MPI_DOUBLE, MPI_SUM, communicator);
+        return globalValue;
+    }
+
+    [[nodiscard]] double global_norm(
+        const mfem::Vector &vector,
+        const MPI_Comm communicator
+    ) {
+        return std::sqrt(std::max(global_dot(vector, vector, communicator), 0.0));
+    }
+
+    [[nodiscard]] mean_field::solver::OperatorApplicationStatistics maximum_rank_statistics(
+        const mean_field::solver::OperatorApplicationStatistics &local,
+        const MPI_Comm communicator
+    ) {
+        unsigned long long localApplications = static_cast<unsigned long long>(local.applications);
+        unsigned long long maximumApplications{0};
+        MPI_Allreduce(&localApplications, &maximumApplications, 1, MPI_UNSIGNED_LONG_LONG, MPI_MAX, communicator);
+
+        mean_field::solver::OperatorApplicationStatistics result;
+        result.applications = static_cast<std::uint64_t>(maximumApplications);
+        MPI_Allreduce(&local.totalSeconds, &result.totalSeconds, 1, MPI_DOUBLE, MPI_MAX, communicator);
+        MPI_Allreduce(&local.maximumSeconds, &result.maximumSeconds, 1, MPI_DOUBLE, MPI_MAX, communicator);
+        return result;
+    }
+
+    [[nodiscard]] double maximum_rank_value(
+        const double localValue,
+        const MPI_Comm communicator
+    ) {
+        double result = 0.0;
+        MPI_Allreduce(&localValue, &result, 1, MPI_DOUBLE, MPI_MAX, communicator);
+        return result;
+    }
+
+    [[nodiscard]] mean_field::solver::PreconditionerLifecycleStatistics maximum_rank_lifecycle_statistics(
+        const mean_field::solver::PreconditionerLifecycleStatistics &local,
+        const MPI_Comm communicator
+    ) {
+        unsigned long long localSetups    = static_cast<unsigned long long>(local.setups);
+        unsigned long long localRefreshes = static_cast<unsigned long long>(local.refreshes);
+        unsigned long long maximumSetups{0};
+        unsigned long long maximumRefreshes{0};
+        MPI_Allreduce(&localSetups, &maximumSetups, 1, MPI_UNSIGNED_LONG_LONG, MPI_MAX, communicator);
+        MPI_Allreduce(&localRefreshes, &maximumRefreshes, 1, MPI_UNSIGNED_LONG_LONG, MPI_MAX, communicator);
+
+        mean_field::solver::PreconditionerLifecycleStatistics result;
+        result.setups    = static_cast<std::uint64_t>(maximumSetups);
+        result.refreshes = static_cast<std::uint64_t>(maximumRefreshes);
+        MPI_Allreduce(&local.setupSeconds, &result.setupSeconds, 1, MPI_DOUBLE, MPI_MAX, communicator);
+        MPI_Allreduce(&local.refreshSeconds, &result.refreshSeconds, 1, MPI_DOUBLE, MPI_MAX, communicator);
+        return result;
+    }
+
+    [[nodiscard]] Eigen::MatrixXd copy_hessenberg(
+        const Eigen::MatrixXd &source,
+        const int rowCount,
+        const int columnCount
+    ) {
+        return source.topLeftCorner(rowCount, columnCount);
+    }
+} // namespace
+
+namespace mean_field::solver {
+    InstrumentedOperator::InstrumentedOperator(const mfem::Operator &operation)
+        : mfem::Operator(
+              operation.Height(),
+              operation.Width()
+          ),
+          m_operation(std::addressof(operation)) {
+    }
+
+    void InstrumentedOperator::Mult(
+        const mfem::Vector &input,
+        mfem::Vector &output
+    ) const {
+        const Clock::time_point start = Clock::now();
+        m_operation->Mult(input, output);
+        const double elapsed = seconds_between(start, Clock::now());
+
+        ++m_statistics.applications;
+        m_statistics.totalSeconds += elapsed;
+        m_statistics.maximumSeconds = std::max(m_statistics.maximumSeconds, elapsed);
+    }
+
+    void InstrumentedOperator::ResetStatistics() const noexcept {
+        m_statistics = {};
+    }
+
+    const OperatorApplicationStatistics &InstrumentedOperator::GetStatistics() const noexcept {
+        return m_statistics;
+    }
+
+    const mfem::Operator &InstrumentedOperator::GetOperation() const noexcept {
+        return *m_operation;
+    }
+
+    InstrumentedPreconditioner::InstrumentedPreconditioner(mfem::Solver &preconditioner)
+        : mfem::Solver(
+              preconditioner.Height(),
+              preconditioner.Width(),
+              preconditioner.iterative_mode
+          ),
+          m_preconditioner(std::addressof(preconditioner)) {
+    }
+
+    void InstrumentedPreconditioner::SetOperator(const mfem::Operator &operation) {
+        const Clock::time_point start = Clock::now();
+        m_preconditioner->SetOperator(operation);
+        m_lifecycleStatistics.setupSeconds += seconds_between(start, Clock::now());
+        ++m_lifecycleStatistics.setups;
+        if (m_preconditioner->Height() != Height() || m_preconditioner->Width() != Width()) {
+            throw std::invalid_argument("An instrumented preconditioner changed dimensions during SetOperator.");
+        }
+    }
+
+    void InstrumentedPreconditioner::Mult(
+        const mfem::Vector &input,
+        mfem::Vector &output
+    ) const {
+        const Clock::time_point start = Clock::now();
+        m_preconditioner->Mult(input, output);
+        const double elapsed = seconds_between(start, Clock::now());
+
+        ++m_statistics.applications;
+        m_statistics.totalSeconds += elapsed;
+        m_statistics.maximumSeconds = std::max(m_statistics.maximumSeconds, elapsed);
+    }
+
+    void InstrumentedPreconditioner::ResetStatistics() const noexcept {
+        m_statistics = {};
+    }
+
+    const OperatorApplicationStatistics &InstrumentedPreconditioner::GetStatistics() const noexcept {
+        return m_statistics;
+    }
+
+    const PreconditionerLifecycleStatistics &InstrumentedPreconditioner::GetLifecycleStatistics() const noexcept {
+        return m_lifecycleStatistics;
+    }
+
+    const mfem::Solver &InstrumentedPreconditioner::GetPreconditioner() const noexcept {
+        return *m_preconditioner;
+    }
+
+    IdentityPreconditioner::IdentityPreconditioner(const int size) : mfem::Solver(size) {
+        if (size <= 0) {
+            throw std::invalid_argument("An identity preconditioner requires a positive dimension.");
+        }
+    }
+
+    void IdentityPreconditioner::SetOperator(const mfem::Operator &operation) {
+        if (operation.Height() != Height() || operation.Width() != Width()) {
+            throw std::invalid_argument("The identity preconditioner received an incompatible operator.");
+        }
+    }
+
+    void IdentityPreconditioner::Mult(
+        const mfem::Vector &input,
+        mfem::Vector &output
+    ) const {
+        if (input.Size() != Width()) {
+            throw std::invalid_argument("The identity preconditioner received an input with the wrong size.");
+        }
+        output = input;
+    }
+
+    FixedRightPreconditionedOperator::FixedRightPreconditionedOperator(
+        const mfem::Operator &jacobian,
+        const mfem::Solver &inversePreconditioner
+    )
+        : mfem::Operator(
+              jacobian.Height(),
+              inversePreconditioner.Width()
+          ),
+          m_jacobian(std::addressof(jacobian)),
+          m_inversePreconditioner(std::addressof(inversePreconditioner)),
+          m_preconditionedDirection(inversePreconditioner.Height()) {
+        if (jacobian.Height() != jacobian.Width()) {
+            throw std::invalid_argument("A preconditioned stellar Jacobian must be square.");
+        }
+        if (inversePreconditioner.Height() != jacobian.Width() || inversePreconditioner.Width() != jacobian.Height()) {
+            throw std::invalid_argument("The inverse preconditioner does not map residuals into Jacobian states.");
+        }
+        if (Height() != Width()) {
+            throw std::invalid_argument("The fixed right-preconditioned product must be square.");
+        }
+    }
+
+    void FixedRightPreconditionedOperator::Mult(
+        const mfem::Vector &input,
+        mfem::Vector &output
+    ) const {
+        if (input.Size() != Width()) {
+            throw std::invalid_argument("The right-preconditioned operator received an input with the wrong size.");
+        }
+        m_inversePreconditioner->Mult(input, m_preconditionedDirection);
+        m_jacobian->Mult(m_preconditionedDirection, output);
+    }
+
+    const mfem::Operator &FixedRightPreconditionedOperator::GetJacobian() const noexcept {
+        return *m_jacobian;
+    }
+
+    const mfem::Solver &FixedRightPreconditionedOperator::GetInversePreconditioner() const noexcept {
+        return *m_inversePreconditioner;
+    }
+
+    void ResidualHistoryMonitor::Reset() {
+        mfem::IterativeSolverMonitor::Reset();
+        m_history.clear();
+    }
+
+    void ResidualHistoryMonitor::MonitorResidual(
+        const int iteration,
+        const double norm,
+        const mfem::Vector &,
+        const bool final
+    ) {
+        m_history.push_back({.iteration = iteration, .reportedNorm = norm, .final = final});
+    }
+
+    const std::vector<IterationResidualMeasurement> &ResidualHistoryMonitor::GetHistory() const noexcept {
+        return m_history;
+    }
+
+    DirectResidualMeasurement measureDirectResidual(
+        const mfem::Operator &jacobian,
+        const mfem::Vector &rightHandSide,
+        const mfem::Vector &solution,
+        const std::span<const operators::RootBlockDescriptor> residualBlocks,
+        const MPI_Comm communicator,
+        const double denominatorFloor
+    ) {
+        if (jacobian.Height() != jacobian.Width() || rightHandSide.Size() != jacobian.Height() ||
+            solution.Size() != jacobian.Width()) {
+            throw std::invalid_argument("Direct residual measurement received incompatible linear-system dimensions.");
+        }
+        if (!std::isfinite(denominatorFloor) || denominatorFloor <= 0.0) {
+            throw std::invalid_argument("The direct-residual denominator floor must be finite and positive.");
+        }
+        verify_finite_vector(rightHandSide, "Direct residual measurement received a non-finite right-hand side.");
+        verify_finite_vector(solution, "Direct residual measurement received a non-finite solution.");
+
+        int expectedOffset = 0;
+        for (const operators::RootBlockDescriptor &block : residualBlocks) {
+            if (block.kind != operators::RootBlockKind::residual || block.offset != expectedOffset || block.size < 0 ||
+                block.offset + block.size > jacobian.Height() || !std::isfinite(block.scale) || block.scale <= 0.0) {
+                throw std::invalid_argument("Residual block descriptors do not form the canonical equation layout.");
+            }
+            expectedOffset += block.size;
+        }
+        if (expectedOffset != jacobian.Height()) {
+            throw std::invalid_argument("Residual block descriptors do not cover the complete equation vector.");
+        }
+
+        mfem::Vector action(jacobian.Height());
+        jacobian.Mult(solution, action);
+        if (action.Size() != rightHandSide.Size()) {
+            throw std::runtime_error("The Jacobian returned an action with the wrong size.");
+        }
+        mfem::Vector trueResidual(rightHandSide);
+        trueResidual -= action;
+        verify_finite_vector(trueResidual, "Direct residual measurement produced a non-finite residual.");
+
+        DirectResidualMeasurement measurement;
+        measurement.rightHandSideNorm = global_norm(rightHandSide, communicator);
+        measurement.trueResidualNorm  = global_norm(trueResidual, communicator);
+        const double denominator      = std::max(measurement.rightHandSideNorm, denominatorFloor);
+        measurement.relativeResidual  = measurement.trueResidualNorm / denominator;
+        measurement.blocks.reserve(residualBlocks.size());
+
+        for (const operators::RootBlockDescriptor &block : residualBlocks) {
+            const mfem::Vector blockRightHandSide(
+                const_cast<mfem::real_t *>(rightHandSide.GetData()) + block.offset, block.size
+            );
+            const mfem::Vector blockResidual(trueResidual.GetData() + block.offset, block.size);
+            const double blockRightHandSideNorm = global_norm(blockRightHandSide, communicator);
+            const double blockResidualNorm      = global_norm(blockResidual, communicator);
+            const double blockDenominator       = std::max(blockRightHandSideNorm, denominatorFloor);
+            const double globalResidualFraction =
+                measurement.trueResidualNorm > denominatorFloor
+                    ? blockResidualNorm * blockResidualNorm /
+                          (measurement.trueResidualNorm * measurement.trueResidualNorm)
+                    : 0.0;
+            measurement.blocks.push_back(
+                {.stableId                             = std::string(block.stableId),
+                 .size                                 = block.size,
+                 .descriptorScale                      = block.scale,
+                 .rightHandSideNorm                    = blockRightHandSideNorm,
+                 .trueResidualNorm                     = blockResidualNorm,
+                 .blockRelativeResidual                = blockResidualNorm / blockDenominator,
+                 .scaledRightHandSideNorm              = blockRightHandSideNorm / block.scale,
+                 .scaledTrueResidualNorm               = blockResidualNorm / block.scale,
+                 .contributionToGlobalRelativeResidual = blockResidualNorm / denominator,
+                 .fractionOfGlobalSquaredResidualNorm  = globalResidualFraction}
+            );
+        }
+        return measurement;
+    }
+
+    LinearSolveMeasurement measureLinearSolve(
+        const mfem::IterativeSolver &iterativeSolver,
+        const mfem::Operator &jacobian,
+        const mfem::Vector &rightHandSide,
+        const mfem::Vector &solution,
+        const std::span<const operators::RootBlockDescriptor> residualBlocks,
+        const OperatorApplicationStatistics &jacobianStatistics,
+        const OperatorApplicationStatistics &inversePreconditionerStatistics,
+        const PreconditionerLifecycleStatistics &inversePreconditionerLifecycle,
+        const ResidualHistoryMonitor &monitor,
+        const double localSolveSeconds,
+        const MPI_Comm communicator,
+        const double denominatorFloor
+    ) {
+        if (!std::isfinite(localSolveSeconds) || localSolveSeconds < 0.0) {
+            throw std::invalid_argument("A linear-solve duration must be finite and nonnegative.");
+        }
+
+        const DirectResidualMeasurement directResidual =
+            measureDirectResidual(jacobian, rightHandSide, solution, residualBlocks, communicator, denominatorFloor);
+        const double reportedInitial = iterativeSolver.GetInitialNorm();
+        const double reportedFinal   = iterativeSolver.GetFinalNorm();
+        const double reportedReduction =
+            std::abs(reportedInitial) > denominatorFloor ? std::abs(reportedFinal) / std::abs(reportedInitial) : 0.0;
+        double digitsPerJacobianApplication = 0.0;
+        if (jacobianStatistics.applications > 0 && directResidual.relativeResidual >= 0.0 &&
+            std::isfinite(directResidual.relativeResidual)) {
+            digitsPerJacobianApplication = -std::log10(std::max(directResidual.relativeResidual, denominatorFloor)) /
+                                           static_cast<double>(jacobianStatistics.applications);
+        }
+
+        return {
+            .solverConverged                                 = iterativeSolver.GetConverged(),
+            .outerIterations                                 = iterativeSolver.GetNumIterations(),
+            .solverReportedInitialNorm                       = reportedInitial,
+            .solverReportedFinalNorm                         = reportedFinal,
+            .solverReportedResidualReduction                 = reportedReduction,
+            .trueResidualDigitsReducedPerJacobianApplication = digitsPerJacobianApplication,
+            .solveSecondsMaximumRank                         = maximum_rank_value(localSolveSeconds, communicator),
+            .jacobian              = maximum_rank_statistics(jacobianStatistics, communicator),
+            .inversePreconditioner = maximum_rank_statistics(inversePreconditionerStatistics, communicator),
+            .inversePreconditionerLifecycle =
+                maximum_rank_lifecycle_statistics(inversePreconditionerLifecycle, communicator),
+            .directResidual          = directResidual,
+            .reportedResidualHistory = monitor.GetHistory()
+        };
+    }
+
+    ArnoldiSpectralMeasurement measureArnoldiSpectrum(
+        const mfem::Operator &operation,
+        const mfem::Vector &initialDirection,
+        const MPI_Comm communicator,
+        const ArnoldiOptions &options
+    ) {
+        if (operation.Height() != operation.Width() || operation.Width() <= 0) {
+            throw std::invalid_argument("Arnoldi diagnostics require a nonempty square operator.");
+        }
+        if (initialDirection.Size() != operation.Width()) {
+            throw std::invalid_argument("The Arnoldi initial direction has the wrong size.");
+        }
+        if (options.krylovDimension <= 0 || !std::isfinite(options.breakdownRelativeTolerance) ||
+            options.breakdownRelativeTolerance < 0.0 || !std::isfinite(options.ritzConvergenceRelativeTolerance) ||
+            options.ritzConvergenceRelativeTolerance < 0.0) {
+            throw std::invalid_argument("Arnoldi diagnostic options are invalid.");
+        }
+        verify_finite_vector(initialDirection, "Arnoldi diagnostics received a non-finite initial direction.");
+
+        const Clock::time_point measurementStart = Clock::now();
+        OperatorApplicationStatistics localApplicationStatistics;
+
+        const double initialNorm = global_norm(initialDirection, communicator);
+        if (!std::isfinite(initialNorm) || initialNorm <= 0.0) {
+            throw std::invalid_argument("Arnoldi diagnostics require a nonzero initial direction.");
+        }
+
+        const int requestedDimension = std::min(options.krylovDimension, operation.Width());
+        Eigen::MatrixXd hessenberg   = Eigen::MatrixXd::Zero(requestedDimension + 1, requestedDimension);
+        std::vector<mfem::Vector> basis;
+        basis.reserve(static_cast<std::size_t>(requestedDimension + 1));
+        basis.emplace_back(initialDirection);
+        basis.back() /= initialNorm;
+
+        int achievedDimension{0};
+        bool invariantSubspaceFound{false};
+
+        for (int column = 0; column < requestedDimension; ++column) {
+            mfem::Vector candidate(operation.Height());
+            const Clock::time_point applicationStart = Clock::now();
+            operation.Mult(basis[static_cast<std::size_t>(column)], candidate);
+            const double applicationSeconds = seconds_between(applicationStart, Clock::now());
+            ++localApplicationStatistics.applications;
+            localApplicationStatistics.totalSeconds += applicationSeconds;
+            localApplicationStatistics.maximumSeconds =
+                std::max(localApplicationStatistics.maximumSeconds, applicationSeconds);
+            if (candidate.Size() != operation.Height()) {
+                throw std::runtime_error("The Arnoldi operator returned a vector with the wrong size.");
+            }
+            verify_finite_vector(candidate, "The Arnoldi operator produced a non-finite vector.");
+            const double unorthogonalizedNorm = global_norm(candidate, communicator);
+
+            const int passCount               = options.reorthogonalize ? 2 : 1;
+            for (int pass = 0; pass < passCount; ++pass) {
+                for (int row = 0; row <= column; ++row) {
+                    const double projection = global_dot(basis[static_cast<std::size_t>(row)], candidate, communicator);
+                    hessenberg(row, column) += projection;
+                    candidate.Add(-projection, basis[static_cast<std::size_t>(row)]);
+                }
+            }
+
+            const double nextNorm          = global_norm(candidate, communicator);
+            hessenberg(column + 1, column) = nextNorm;
+            achievedDimension              = column + 1;
+            const double breakdownScale    = std::max(unorthogonalizedNorm, 1.0);
+            if (nextNorm <= options.breakdownRelativeTolerance * breakdownScale) {
+                invariantSubspaceFound = true;
+                break;
+            }
+            if (column + 1 < requestedDimension) {
+                candidate /= nextNorm;
+                basis.push_back(std::move(candidate));
+            }
+        }
+
+        if (achievedDimension <= 0) {
+            throw std::runtime_error("Arnoldi diagnostics did not construct a Krylov projection.");
+        }
+
+        const Eigen::MatrixXd projected = copy_hessenberg(hessenberg, achievedDimension, achievedDimension);
+        const Eigen::MatrixXd projectedRectangular =
+            copy_hessenberg(hessenberg, achievedDimension + 1, achievedDimension);
+
+        Eigen::EigenSolver<Eigen::MatrixXd> eigenSolver(projected, true);
+        if (eigenSolver.info() != Eigen::Success) {
+            throw std::runtime_error("The projected Arnoldi eigenproblem did not converge.");
+        }
+        Eigen::JacobiSVD<Eigen::MatrixXd> singularValueDecomposition(projectedRectangular);
+        if (singularValueDecomposition.info() != Eigen::Success) {
+            throw std::runtime_error("The projected Arnoldi singular-value problem did not converge.");
+        }
+
+        ArnoldiSpectralMeasurement measurement;
+        measurement.requestedDimension     = requestedDimension;
+        measurement.achievedDimension      = achievedDimension;
+        measurement.invariantSubspaceFound = invariantSubspaceFound;
+        const OperatorApplicationStatistics globalApplicationStatistics =
+            maximum_rank_statistics(localApplicationStatistics, communicator);
+        measurement.operatorApplications                         = globalApplicationStatistics.applications;
+        measurement.operatorApplicationSecondsMaximumRank        = globalApplicationStatistics.totalSeconds;
+        measurement.operatorMaximumApplicationSecondsMaximumRank = globalApplicationStatistics.maximumSeconds;
+        measurement.ritzValues.reserve(static_cast<std::size_t>(achievedDimension));
+
+        const Eigen::VectorXd singularValues       = singularValueDecomposition.singularValues();
+        measurement.projectedLargestSingularValue  = singularValues(0);
+        measurement.projectedSmallestSingularValue = singularValues(singularValues.size() - 1);
+        measurement.projectedConditionProxy =
+            measurement.projectedSmallestSingularValue > 0.0
+                ? measurement.projectedLargestSingularValue / measurement.projectedSmallestSingularValue
+                : std::numeric_limits<double>::infinity();
+
+        const double finalSubdiagonal = hessenberg(achievedDimension, achievedDimension - 1);
+        std::complex<double> centroid{0.0, 0.0};
+        const auto eigenvalues  = eigenSolver.eigenvalues();
+        const auto eigenvectors = eigenSolver.eigenvectors();
+        for (int index = 0; index < achievedDimension; ++index) {
+            const std::complex<double> eigenvalue = eigenvalues(index);
+            const double eigenvectorNorm          = eigenvectors.col(index).norm();
+            const double residualEstimate =
+                eigenvectorNorm > 0.0
+                    ? std::abs(finalSubdiagonal * eigenvectors(achievedDimension - 1, index)) / eigenvectorNorm
+                    : std::numeric_limits<double>::infinity();
+            const double convergenceScale         = std::max(std::abs(eigenvalue), 1.0);
+            const double relativeResidualEstimate = residualEstimate / convergenceScale;
+            const bool converged = relativeResidualEstimate <= options.ritzConvergenceRelativeTolerance;
+
+            measurement.ritzValues.push_back(
+                {.realPart                 = eigenvalue.real(),
+                 .imaginaryPart            = eigenvalue.imag(),
+                 .magnitude                = std::abs(eigenvalue),
+                 .distanceFromOne          = std::abs(eigenvalue - std::complex<double>{1.0, 0.0}),
+                 .residualEstimate         = residualEstimate,
+                 .relativeResidualEstimate = relativeResidualEstimate,
+                 .converged                = converged}
+            );
+            centroid += eigenvalue;
+            measurement.convergedRitzValueCount += converged ? 1 : 0;
+            measurement.negativeRealPartCount += eigenvalue.real() < 0.0 ? 1 : 0;
+        }
+        centroid /= static_cast<double>(achievedDimension);
+        measurement.centroidRealPart      = centroid.real();
+        measurement.centroidImaginaryPart = centroid.imag();
+
+        measurement.minimumMagnitude      = std::numeric_limits<double>::infinity();
+        measurement.minimumRealPart       = std::numeric_limits<double>::infinity();
+        measurement.maximumRealPart       = -std::numeric_limits<double>::infinity();
+        double squaredDistanceFromOne{0.0};
+        double squaredClusterRadius{0.0};
+        for (const RitzValueMeasurement &ritz : measurement.ritzValues) {
+            const std::complex<double> value{ritz.realPart, ritz.imaginaryPart};
+            measurement.minimumMagnitude = std::min(measurement.minimumMagnitude, ritz.magnitude);
+            measurement.maximumMagnitude = std::max(measurement.maximumMagnitude, ritz.magnitude);
+            measurement.minimumRealPart  = std::min(measurement.minimumRealPart, ritz.realPart);
+            measurement.maximumRealPart  = std::max(measurement.maximumRealPart, ritz.realPart);
+            measurement.maximumAbsoluteImaginaryPart =
+                std::max(measurement.maximumAbsoluteImaginaryPart, std::abs(ritz.imaginaryPart));
+            squaredDistanceFromOne += ritz.distanceFromOne * ritz.distanceFromOne;
+            squaredClusterRadius += std::norm(value - centroid);
+
+            double pairDefect = std::numeric_limits<double>::infinity();
+            for (const RitzValueMeasurement &candidate : measurement.ritzValues) {
+                pairDefect = std::min(
+                    pairDefect,
+                    std::abs(std::complex<double>{candidate.realPart, candidate.imaginaryPart} - std::conj(value))
+                );
+            }
+            measurement.conjugatePairDefect = std::max(measurement.conjugatePairDefect, pairDefect);
+        }
+        measurement.rmsDistanceFromOne         = std::sqrt(squaredDistanceFromOne / achievedDimension);
+        measurement.rmsClusterRadius           = std::sqrt(squaredClusterRadius / achievedDimension);
+
+        const double projectedFrobeniusSquared = projected.squaredNorm();
+        if (projectedFrobeniusSquared > 0.0) {
+            const Eigen::MatrixXd normalityCommutator =
+                projected.transpose() * projected - projected * projected.transpose();
+            measurement.projectedDepartureFromNormality = normalityCommutator.norm() / projectedFrobeniusSquared;
+        }
+
+        const Eigen::MatrixXd hermitianPart = 0.5 * (projected + projected.transpose());
+        Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> fieldOfValuesSolver(hermitianPart);
+        if (fieldOfValuesSolver.info() != Eigen::Success) {
+            throw std::runtime_error("The projected field-of-values problem did not converge.");
+        }
+        measurement.projectedFieldOfValuesMinimumRealPart = fieldOfValuesSolver.eigenvalues().minCoeff();
+        measurement.projectedFieldOfValuesMaximumRealPart = fieldOfValuesSolver.eigenvalues().maxCoeff();
+        const double localMeasurementSeconds              = seconds_between(measurementStart, Clock::now());
+        const double localNonApplicationSeconds =
+            std::max(localMeasurementSeconds - localApplicationStatistics.totalSeconds, 0.0);
+        measurement.measurementSecondsMaximumRank    = maximum_rank_value(localMeasurementSeconds, communicator);
+        measurement.nonApplicationSecondsMaximumRank = maximum_rank_value(localNonApplicationSeconds, communicator);
+        return measurement;
+    }
+
+    std::vector<RitzValueMeasurement> selectRitzValues(
+        const ArnoldiSpectralMeasurement &measurement,
+        const RitzValueOrdering ordering,
+        const int count
+    ) {
+        if (count < 0) {
+            throw std::invalid_argument("The requested Ritz-value count must be nonnegative.");
+        }
+
+        std::vector<RitzValueMeasurement> selected;
+        selected.reserve(measurement.ritzValues.size());
+        for (const RitzValueMeasurement &value : measurement.ritzValues) {
+            if (value.converged) {
+                selected.push_back(value);
+            }
+        }
+
+        std::ranges::sort(selected, [ordering](const RitzValueMeasurement &left, const RitzValueMeasurement &right) {
+            switch (ordering) {
+            case RitzValueOrdering::closest_to_zero:
+                return left.magnitude < right.magnitude;
+            case RitzValueOrdering::farthest_from_one:
+                return left.distanceFromOne > right.distanceFromOne;
+            case RitzValueOrdering::smallest_real_part:
+                return left.realPart < right.realPart;
+            case RitzValueOrdering::largest_magnitude:
+                return left.magnitude > right.magnitude;
+            }
+            return false;
+        });
+        if (static_cast<int>(selected.size()) > count) {
+            selected.resize(static_cast<std::size_t>(count));
+        }
+        return selected;
+    }
+} // namespace mean_field::solver

libmeanfield/interface/dimensions/quantities.cppm

@@ -0,0 +1,312 @@
+module;
+
+#include <compare>
+#include <concepts>
+#include <string_view>
+#include <type_traits>
+
+export module mean_field:dimensions.quantities;
+
+export namespace mean_field::dimensions {
+    /*
+     * QuantityValue provides semantic strong typing for scalar physical
+     * values expressed in the unit system selected by a model. It does not
+     * perform dimensional algebra or unit conversion.
+     */
+    struct PhysicalQuantity { };
+
+    struct ThermodynamicQuantity : PhysicalQuantity { };
+
+    template <typename Candidate>
+    concept PhysicalQuantityType =
+        std::same_as<Candidate, std::remove_cv_t<Candidate>> && std::derived_from<Candidate, PhysicalQuantity>;
+
+    template <typename Candidate>
+    concept ThermodynamicQuantityType =
+        PhysicalQuantityType<Candidate> && std::derived_from<Candidate, ThermodynamicQuantity>;
+
+    namespace quantity {
+        struct Dimensionless final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "dimensionless";
+        };
+
+        struct Mass final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "mass";
+        };
+
+        struct Length final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "length";
+        };
+
+        struct Time final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "time";
+        };
+
+        struct Area final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "area";
+        };
+
+        struct Volume final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "volume";
+        };
+
+        struct Density final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "density";
+        };
+
+        struct SurfaceDensity final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "surface_density";
+        };
+
+        struct NumberDensity final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "number_density";
+        };
+
+        struct Pressure final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "pressure";
+        };
+
+        struct Temperature final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "temperature";
+        };
+
+        struct Entropy final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "entropy";
+        };
+
+        struct SpecificEntropy final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "specific_entropy";
+        };
+
+        struct ChemicalPotential final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "chemical_potential";
+        };
+
+        struct Energy final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "energy";
+        };
+
+        struct InternalEnergy final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "internal_energy";
+        };
+
+        struct SpecificEnergy final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "specific_energy";
+        };
+
+        struct SpecificInternalEnergy final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "specific_internal_energy";
+        };
+
+        struct SpecificEnthalpy final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "specific_enthalpy";
+        };
+
+        struct EnergyDensity final : ThermodynamicQuantity {
+            static constexpr std::string_view identifier = "energy_density";
+        };
+
+        struct GravitationalPotential final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "gravitational_potential";
+        };
+
+        struct Velocity final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "velocity";
+        };
+
+        struct Acceleration final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "acceleration";
+        };
+
+        struct Frequency final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "frequency";
+        };
+
+        struct AngularVelocity final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "angular_velocity";
+        };
+
+        struct Momentum final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "momentum";
+        };
+
+        struct AngularMomentum final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "angular_momentum";
+        };
+
+        struct MomentOfInertia final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "moment_of_inertia";
+        };
+
+        struct Force final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "force";
+        };
+
+        struct Torque final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "torque";
+        };
+
+        struct Power final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "power";
+        };
+
+        struct Luminosity final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "luminosity";
+        };
+
+        struct MassFlowRate final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "mass_flow_rate";
+        };
+
+        struct Opacity final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "opacity";
+        };
+
+        struct DynamicViscosity final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "dynamic_viscosity";
+        };
+
+        struct KinematicViscosity final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "kinematic_viscosity";
+        };
+
+        struct MagneticFluxDensity final : PhysicalQuantity {
+            static constexpr std::string_view identifier = "magnetic_flux_density";
+        };
+    } // namespace quantity
+
+    template <typename T>
+    concept Numeric = std::integral<T> || std::floating_point<T>;
+
+    template <PhysicalQuantityType Quantity> class QuantityValue final {
+    public:
+        explicit constexpr QuantityValue(const double value) noexcept : m_value(value) {
+        }
+
+        [[nodiscard]] constexpr double value() const noexcept {
+            return m_value;
+        }
+
+        [[nodiscard]] friend constexpr bool operator==(
+            const QuantityValue &,
+            const QuantityValue &
+        ) noexcept = default;
+
+        friend constexpr QuantityValue operator+(
+            const QuantityValue &lhs,
+            const QuantityValue &rhs
+        ) noexcept {
+            return QuantityValue{lhs.m_value + rhs.m_value};
+        }
+
+        friend constexpr QuantityValue operator-(
+            const QuantityValue &lhs,
+            const QuantityValue &rhs
+        ) noexcept {
+            return QuantityValue{lhs.m_value - rhs.m_value};
+        }
+
+        template <Numeric Scalar>
+        friend constexpr QuantityValue operator*(
+            const QuantityValue &lhs,
+            const Scalar rhs
+        ) noexcept {
+            return QuantityValue{lhs.m_value * static_cast<double>(rhs)};
+        }
+
+        template <Numeric Scalar>
+        friend constexpr QuantityValue operator*(
+            const Scalar lhs,
+            const QuantityValue &rhs
+        ) noexcept {
+            return QuantityValue{static_cast<double>(lhs) * rhs.m_value};
+        }
+
+        template <Numeric Scalar>
+        friend constexpr QuantityValue operator/(
+            const QuantityValue &lhs,
+            const Scalar rhs
+        ) noexcept {
+            return QuantityValue{lhs.m_value / static_cast<double>(rhs)};
+        }
+
+        template <Numeric Scalar>
+        friend constexpr std::partial_ordering operator<=>(
+            const QuantityValue &lhs,
+            const Scalar rhs
+        ) noexcept {
+            return lhs.m_value <=> static_cast<double>(rhs);
+        }
+
+        template <Numeric Scalar>
+        friend constexpr std::partial_ordering operator<=>(
+            const Scalar lhs,
+            const QuantityValue &rhs
+        ) noexcept {
+            return static_cast<double>(lhs) <=> rhs.m_value;
+        }
+
+        friend constexpr std::partial_ordering operator<=>(
+            const QuantityValue &lhs,
+            const QuantityValue &rhs
+        ) noexcept {
+            return lhs.m_value <=> rhs.m_value;
+        }
+
+    private:
+        double m_value;
+    };
+
+    template <typename Candidate> struct IsQuantityValue : std::false_type { };
+
+    template <PhysicalQuantityType Quantity> struct IsQuantityValue<QuantityValue<Quantity>> : std::true_type { };
+
+    template <typename Candidate>
+    concept QuantityValueType = IsQuantityValue<std::remove_cvref_t<Candidate>>::value;
+
+    template <typename Candidate> struct QuantityOf;
+
+    template <PhysicalQuantityType Quantity> struct QuantityOf<QuantityValue<Quantity>> {
+        using Type = Quantity;
+    };
+
+    template <QuantityValueType Value> using QuantityOfT = typename QuantityOf<std::remove_cvref_t<Value>>::Type;
+
+    using DimensionlessValue                             = QuantityValue<quantity::Dimensionless>;
+    using MassValue                                      = QuantityValue<quantity::Mass>;
+    using LengthValue                                    = QuantityValue<quantity::Length>;
+    using TimeValue                                      = QuantityValue<quantity::Time>;
+    using AreaValue                                      = QuantityValue<quantity::Area>;
+    using VolumeValue                                    = QuantityValue<quantity::Volume>;
+    using DensityValue                                   = QuantityValue<quantity::Density>;
+    using SurfaceDensityValue                            = QuantityValue<quantity::SurfaceDensity>;
+    using NumberDensityValue                             = QuantityValue<quantity::NumberDensity>;
+    using PressureValue                                  = QuantityValue<quantity::Pressure>;
+    using TemperatureValue                               = QuantityValue<quantity::Temperature>;
+    using EntropyValue                                   = QuantityValue<quantity::Entropy>;
+    using SpecificEntropyValue                           = QuantityValue<quantity::SpecificEntropy>;
+    using ChemicalPotentialValue                         = QuantityValue<quantity::ChemicalPotential>;
+    using EnergyValue                                    = QuantityValue<quantity::Energy>;
+    using InternalEnergyValue                            = QuantityValue<quantity::InternalEnergy>;
+    using SpecificEnergyValue                            = QuantityValue<quantity::SpecificEnergy>;
+    using SpecificInternalEnergyValue                    = QuantityValue<quantity::SpecificInternalEnergy>;
+    using SpecificEnthalpyValue                          = QuantityValue<quantity::SpecificEnthalpy>;
+    using EnergyDensityValue                             = QuantityValue<quantity::EnergyDensity>;
+    using GravitationalPotentialValue                    = QuantityValue<quantity::GravitationalPotential>;
+    using VelocityValue                                  = QuantityValue<quantity::Velocity>;
+    using AccelerationValue                              = QuantityValue<quantity::Acceleration>;
+    using FrequencyValue                                 = QuantityValue<quantity::Frequency>;
+    using AngularVelocityValue                           = QuantityValue<quantity::AngularVelocity>;
+    using MomentumValue                                  = QuantityValue<quantity::Momentum>;
+    using AngularMomentumValue                           = QuantityValue<quantity::AngularMomentum>;
+    using MomentOfInertiaValue                           = QuantityValue<quantity::MomentOfInertia>;
+    using ForceValue                                     = QuantityValue<quantity::Force>;
+    using TorqueValue                                    = QuantityValue<quantity::Torque>;
+    using PowerValue                                     = QuantityValue<quantity::Power>;
+    using LuminosityValue                                = QuantityValue<quantity::Luminosity>;
+    using MassFlowRateValue                              = QuantityValue<quantity::MassFlowRate>;
+    using OpacityValue                                   = QuantityValue<quantity::Opacity>;
+    using DynamicViscosityValue                          = QuantityValue<quantity::DynamicViscosity>;
+    using KinematicViscosityValue                        = QuantityValue<quantity::KinematicViscosity>;
+    using MagneticFluxDensityValue                       = QuantityValue<quantity::MagneticFluxDensity>;
+} // namespace mean_field::dimensions

libmeanfield/interface/eos/concepts.cppm

@@ -91,10 +91,10 @@ export namespace mean_field::eos {
     template <typename Candidate>
     concept BarotropicClosureEquationOfState =
         EquationOfStateModel<Candidate> && SupportsRelation<Candidate, DensityFromSpecificEnthalpy> &&
-        SupportsPartialDerivative<Candidate, DensityFromSpecificEnthalpy, quantity::SpecificEnthalpy>;
+        SupportsPartialDerivative<Candidate, DensityFromSpecificEnthalpy, dimensions::quantity::SpecificEnthalpy>;
 
     template <typename Candidate>
     concept PressureForceEquationOfState =
         EquationOfStateModel<Candidate> && SupportsRelation<Candidate, PressureFromSpecificEnthalpy> &&
-        SupportsPartialDerivative<Candidate, PressureFromSpecificEnthalpy, quantity::SpecificEnthalpy>;
+        SupportsPartialDerivative<Candidate, PressureFromSpecificEnthalpy, dimensions::quantity::SpecificEnthalpy>;
 } // namespace mean_field::eos

libmeanfield/interface/eos/polytropic.cppm

@@ -8,6 +8,11 @@ export import :eos.evaluation;
 export namespace mean_field::eos {
     class Polytrope final {
     public:
+        struct Parameters final {
+            double n;
+            double K;
+        };
+
         using Relations = RelationCatalog<
             PressureFromDensity,
             PressureFromSpecificEnthalpy,
@@ -15,6 +20,13 @@ export namespace mean_field::eos {
             SpecificEnthalpyFromPressure,
             DensityFromSpecificEnthalpy>;
 
+        explicit Polytrope(const Parameters parameters)
+            : Polytrope(
+                  parameters.n,
+                  parameters.K
+              ) {
+        }
+
         Polytrope(
             const double polytropic_index,
             const double polytropic_constant
@@ -56,125 +68,131 @@ export namespace mean_field::eos {
             return m_enthalpy_scale;
         }
 
-        [[nodiscard]] PressureValue evaluate(
+        [[nodiscard]] dimensions::PressureValue evaluate(
             PressureFromDensity,
-            const DensityValue density
+            const dimensions::DensityValue density
         ) const {
             validate_nonnegativity(density.value(), "density");
             if (density.value() == 0.0) {
-                return PressureValue{0.0};
+                return dimensions::PressureValue{0.0};
             }
 
-            return PressureValue{m_polytropic_constant * std::pow(density.value(), 1.0 + 1.0 / m_polytropic_index)};
+            return dimensions::PressureValue{
+                m_polytropic_constant * std::pow(density.value(), 1.0 + 1.0 / m_polytropic_index)
+            };
         }
 
-        [[nodiscard]] SpecificEnthalpyValue evaluate(
+        [[nodiscard]] dimensions::SpecificEnthalpyValue evaluate(
             SpecificEnthalpyFromDensity,
-            const DensityValue density
+            const dimensions::DensityValue density
         ) const {
             validate_nonnegativity(density.value(), "density");
             if (density.value() == 0.0) {
-                return SpecificEnthalpyValue{0.0};
+                return dimensions::SpecificEnthalpyValue{0.0};
             }
 
-            return SpecificEnthalpyValue{m_enthalpy_scale * std::pow(density.value(), 1.0 / m_polytropic_index)};
+            return dimensions::SpecificEnthalpyValue{
+                m_enthalpy_scale * std::pow(density.value(), 1.0 / m_polytropic_index)
+            };
         }
 
-        [[nodiscard]] DensityValue evaluate(
+        [[nodiscard]] dimensions::DensityValue evaluate(
             DensityFromSpecificEnthalpy,
-            const SpecificEnthalpyValue specificEnthalpy
+            const dimensions::SpecificEnthalpyValue specificEnthalpy
         ) const {
             validate_finite(specificEnthalpy.value(), "specific enthalpy");
 
             if (specificEnthalpy.value() <= 0.0) {
-                return DensityValue{0.0};
+                return dimensions::DensityValue{0.0};
             }
 
-            return DensityValue{std::pow(specificEnthalpy.value() / m_enthalpy_scale, m_polytropic_index)};
+            return dimensions::DensityValue{std::pow(specificEnthalpy.value() / m_enthalpy_scale, m_polytropic_index)};
         }
 
-        [[nodiscard]] PressureValue evaluate(
+        [[nodiscard]] dimensions::PressureValue evaluate(
             PressureFromSpecificEnthalpy,
-            const SpecificEnthalpyValue specificEnthalpy
+            const dimensions::SpecificEnthalpyValue specificEnthalpy
         ) const {
-            const DensityValue density = evaluate(DensityFromSpecificEnthalpy{}, specificEnthalpy);
+            const dimensions::DensityValue density = evaluate(DensityFromSpecificEnthalpy{}, specificEnthalpy);
 
             if (specificEnthalpy.value() <= 0.0) {
-                return PressureValue{0.0};
+                return dimensions::PressureValue{0.0};
             }
 
-            return PressureValue{density.value() * specificEnthalpy.value() / (m_polytropic_index + 1.0)};
+            return dimensions::PressureValue{density.value() * specificEnthalpy.value() / (m_polytropic_index + 1.0)};
         }
 
-        [[nodiscard]] SpecificEnthalpyValue evaluate(
+        [[nodiscard]] dimensions::SpecificEnthalpyValue evaluate(
             SpecificEnthalpyFromPressure,
-            const PressureValue pressure
+            const dimensions::PressureValue pressure
         ) const {
             validate_nonnegativity(pressure.value(), "pressure");
             if (pressure.value() == 0.0) {
-                return SpecificEnthalpyValue{0.0};
+                return dimensions::SpecificEnthalpyValue{0.0};
             }
 
             const double indexPlusOne = m_polytropic_index + 1.0;
 
-            return SpecificEnthalpyValue{
+            return dimensions::SpecificEnthalpyValue{
                 indexPlusOne * std::pow(m_polytropic_constant, m_polytropic_index / indexPlusOne) *
                 std::pow(pressure.value(), 1.0 / indexPlusOne)
             };
         }
 
         [[nodiscard]] PartialDerivative<
-            quantity::Density,
-            quantity::SpecificEnthalpy>
+            dimensions::quantity::Density,
+            dimensions::quantity::SpecificEnthalpy>
         partialDerivative(
             DensityFromSpecificEnthalpy,
-            WithRespectTo<quantity::SpecificEnthalpy>,
-            const SpecificEnthalpyValue specificEnthalpy
+            WithRespectTo<dimensions::quantity::SpecificEnthalpy>,
+            const dimensions::SpecificEnthalpyValue specificEnthalpy
         ) const {
             validate_finite(specificEnthalpy.value(), "specific enthalpy");
             if (specificEnthalpy.value() < 0.0) {
-                return PartialDerivative<quantity::Density, quantity::SpecificEnthalpy>{0.0};
+                return PartialDerivative<dimensions::quantity::Density, dimensions::quantity::SpecificEnthalpy>{0.0};
             }
 
             if (specificEnthalpy.value() == 0.0) {
-                return PartialDerivative<quantity::Density, quantity::SpecificEnthalpy>{
+                return PartialDerivative<dimensions::quantity::Density, dimensions::quantity::SpecificEnthalpy>{
                     m_polytropic_index == 1.0 ? 1.0 / m_enthalpy_scale : 0.0
                 };
             }
 
-            return PartialDerivative<quantity::Density, quantity::SpecificEnthalpy>{
+            return PartialDerivative<dimensions::quantity::Density, dimensions::quantity::SpecificEnthalpy>{
                 m_polytropic_index / m_enthalpy_scale *
                 std::pow(specificEnthalpy.value() / m_enthalpy_scale, m_polytropic_index - 1.0)
             };
         }
 
         [[nodiscard]] PartialDerivative<
-            quantity::Pressure,
-            quantity::SpecificEnthalpy>
+            dimensions::quantity::Pressure,
+            dimensions::quantity::SpecificEnthalpy>
         partialDerivative(
             PressureFromSpecificEnthalpy,
-            WithRespectTo<quantity::SpecificEnthalpy>,
-            const SpecificEnthalpyValue specificEnthalpy
+            WithRespectTo<dimensions::quantity::SpecificEnthalpy>,
+            const dimensions::SpecificEnthalpyValue specificEnthalpy
         ) const {
-            const DensityValue density = evaluate(DensityFromSpecificEnthalpy{}, specificEnthalpy);
+            const dimensions::DensityValue density = evaluate(DensityFromSpecificEnthalpy{}, specificEnthalpy);
 
-            return PartialDerivative<quantity::Pressure, quantity::SpecificEnthalpy>{density.value()};
+            return PartialDerivative<dimensions::quantity::Pressure, dimensions::quantity::SpecificEnthalpy>{
+                density.value()
+            };
         }
 
         [[nodiscard]] PartialDerivative<
-            quantity::Pressure,
-            quantity::Density>
+            dimensions::quantity::Pressure,
+            dimensions::quantity::Density>
         partialDerivative(
             PressureFromDensity,
-            WithRespectTo<quantity::Density>,
-            const DensityValue density
+            WithRespectTo<dimensions::quantity::Density>,
+            const dimensions::DensityValue density
         ) const {
             validate_nonnegativity(density.value(), "density");
             if (density.value() == 0.0) {
-                return PartialDerivative<quantity::Pressure, quantity::Density>{0.0};
+                return PartialDerivative<dimensions::quantity::Pressure, dimensions::quantity::Density>{0.0};
             }
 
-            return PartialDerivative<quantity::Pressure, quantity::Density>{
+            return PartialDerivative<dimensions::quantity::Pressure, dimensions::quantity::Density>{
                 m_polytropic_constant * (1.0 + 1.0 / m_polytropic_index) *
                 std::pow(density.value(), 1.0 / m_polytropic_index)
             };

libmeanfield/interface/eos/pressure_surface.cppm

@@ -13,10 +13,10 @@ export namespace mean_field::eos {
             ThermodynamicQuantityType InputQuantity,
             typename SurfaceState>
         [[nodiscard]] constexpr auto pressureSurfaceRelationInput(
-            const PressureValue targetPressure,
+            const dimensions::PressureValue targetPressure,
             const SurfaceState &state
         ) {
-            if constexpr (std::same_as<InputQuantity, quantity::Pressure>) {
+            if constexpr (std::same_as<InputQuantity, dimensions::quantity::Pressure>) {
                 return targetPressure;
             } else {
                 return state.value(InputQuantity{});
@@ -30,9 +30,9 @@ export namespace mean_field::eos {
             template <
                 typename EquationOfState,
                 typename SurfaceState>
-            [[nodiscard]] static QuantityValue<CarrierQuantity> requiredCarrierValue(
+            [[nodiscard]] static dimensions::QuantityValue<CarrierQuantity> requiredCarrierValue(
                 const EquationOfState &equationOfState,
-                const PressureValue targetPressure,
+                const dimensions::PressureValue targetPressure,
                 const SurfaceState &state
             ) {
                 return evaluate<CarrierQuantity>(
@@ -47,11 +47,11 @@ export namespace mean_field::eos {
                 typename SurfaceVariation>
             [[nodiscard]] static double inputJacobianContribution(
                 const EquationOfState &equationOfState,
-                const PressureValue targetPressure,
+                const dimensions::PressureValue targetPressure,
                 const SurfaceState &state,
                 const SurfaceVariation &variation
             ) {
-                if constexpr (std::same_as<InputQuantity, quantity::Pressure>) {
+                if constexpr (std::same_as<InputQuantity, dimensions::quantity::Pressure>) {
                     return 0.0;
                 } else {
                     const auto derivative = partialDerivative<CarrierQuantity, InputQuantity>(
@@ -67,7 +67,7 @@ export namespace mean_field::eos {
                 typename SurfaceVariation>
             [[nodiscard]] static double carrierCorrectionJacobianAction(
                 const EquationOfState &equationOfState,
-                const PressureValue targetPressure,
+                const dimensions::PressureValue targetPressure,
                 const SurfaceState &state,
                 const SurfaceVariation &variation
             ) {
@@ -92,18 +92,18 @@ export namespace mean_field::eos {
 
         ResolvedPressureSurfaceRelation(
             const EquationOfState &equationOfState,
-            const PressureValue targetPressure
+            const dimensions::PressureValue targetPressure
         ) noexcept
             : m_equationOfState(std::addressof(equationOfState)),
               m_targetPressure(targetPressure) {
         }
 
-        [[nodiscard]] PressureValue targetPressure() const noexcept {
+        [[nodiscard]] dimensions::PressureValue targetPressure() const noexcept {
             return m_targetPressure;
         }
 
         template <typename SurfaceState>
-        [[nodiscard]] QuantityValue<CarrierQuantity> requiredCarrierValue(const SurfaceState &state) const {
+        [[nodiscard]] dimensions::QuantityValue<CarrierQuantity> requiredCarrierValue(const SurfaceState &state) const {
             return detail::PressureSurfaceRelationOperations<RelationType>::requiredCarrierValue(
                 *m_equationOfState, m_targetPressure, state
             );
@@ -123,6 +123,6 @@ export namespace mean_field::eos {
 
     private:
         const EquationOfState *m_equationOfState;
-        PressureValue m_targetPressure;
+        dimensions::PressureValue m_targetPressure;
     };
 } // namespace mean_field::eos

libmeanfield/interface/eos/quantities.cppm

@@ -2,132 +2,42 @@ module;
 
 #include <compare>
 #include <concepts>
-#include <string_view>
-#include <type_traits>
 
 export module mean_field:eos.quantities;
+export import :dimensions.quantities;
 
 export namespace mean_field::eos {
-    struct ThermodynamicQuantity { };
+    // Compatibility names for the thermodynamic subset now owned by the
+    // general dimensions partition.
+    using ThermodynamicQuantity = dimensions::ThermodynamicQuantity;
 
     template <typename Candidate>
-    concept ThermodynamicQuantityType =
-        std::same_as<Candidate, std::remove_cv_t<Candidate>> && std::derived_from<Candidate, ThermodynamicQuantity>;
+    concept ThermodynamicQuantityType = dimensions::ThermodynamicQuantityType<Candidate>;
 
     namespace quantity {
-        struct Density final : ThermodynamicQuantity {
-            static constexpr std::string_view identifier = "density";
-        };
-
-        struct Pressure final : ThermodynamicQuantity {
-            static constexpr std::string_view identifier = "pressure";
-        };
-
-        struct SpecificEnthalpy final : ThermodynamicQuantity {
-            static constexpr std::string_view identifier = "specific_enthalpy";
-        };
+        using Density          = dimensions::quantity::Density;
+        using Pressure         = dimensions::quantity::Pressure;
+        using SpecificEnthalpy = dimensions::quantity::SpecificEnthalpy;
     } // namespace quantity
 
     template <typename T>
-    concept Numeric = std::integral<T> || std::floating_point<T>;
+    concept Numeric                                                   = dimensions::Numeric<T>;
 
-    template <ThermodynamicQuantityType Quantity> class QuantityValue final {
-    public:
-        explicit constexpr QuantityValue(const double value) noexcept : m_value(value) {
-        }
+    template <ThermodynamicQuantityType Quantity> using QuantityValue = dimensions::QuantityValue<Quantity>;
 
-        [[nodiscard]] constexpr double value() const noexcept {
-            return m_value;
-        }
+    using DensityValue                                                = dimensions::DensityValue;
+    using PressureValue                                               = dimensions::PressureValue;
+    using SpecificEnthalpyValue                                       = dimensions::SpecificEnthalpyValue;
 
-        [[nodiscard]] friend constexpr bool operator==(
-            const QuantityValue &,
-            const QuantityValue &
-        ) noexcept = default;
-
-        friend constexpr QuantityValue<Quantity> operator+(
-            const QuantityValue<Quantity> &lhs,
-            const QuantityValue<Quantity> &rhs
-        ) noexcept {
-            return QuantityValue<Quantity>{lhs.m_value + rhs.m_value};
-        }
-
-        friend constexpr QuantityValue<Quantity> operator-(
-            const QuantityValue<Quantity> &lhs,
-            const QuantityValue<Quantity> &rhs
-        ) noexcept {
-            return QuantityValue<Quantity>{lhs.m_value - rhs.m_value};
-        }
-
-        template <Numeric rhsT>
-        friend constexpr QuantityValue<Quantity> operator*(
-            const QuantityValue<Quantity> &lhs,
-            rhsT rhs
-        ) noexcept {
-            return QuantityValue<Quantity>{lhs.m_value * static_cast<double>(rhs)};
-        }
-
-        template <Numeric lhsT>
-        friend constexpr QuantityValue<Quantity> operator*(
-            lhsT lhs,
-            const QuantityValue<Quantity> &rhs
-        ) noexcept {
-            return QuantityValue<Quantity>{static_cast<double>(lhs) * rhs.m_value};
-        }
-
-        template <Numeric rhsT>
-        friend constexpr QuantityValue<Quantity> operator/(
-            const QuantityValue<Quantity> &lhs,
-            rhsT rhs
-        ) noexcept {
-            return QuantityValue<Quantity>{lhs.m_value / static_cast<double>(rhs)};
-        }
-
-        template <Numeric compT>
-        friend constexpr std::partial_ordering operator<=>(
-            const QuantityValue<Quantity> &lhs,
-            compT rhs
-        ) noexcept {
-            return lhs.m_value <=> static_cast<double>(rhs);
-        }
-
-        template <Numeric compT>
-        friend constexpr std::partial_ordering operator<=>(
-            compT lhs,
-            const QuantityValue<Quantity> &rhs
-        ) noexcept {
-            return static_cast<double>(lhs) <=> rhs.m_value;
-        }
-
-        friend constexpr std::partial_ordering operator<=>(
-            const QuantityValue<Quantity> &lhs,
-            const QuantityValue<Quantity> &rhs
-        ) noexcept {
-            return lhs.m_value <=> rhs.m_value;
-        }
-
-    private:
-        double m_value;
-    };
-
-    using DensityValue          = QuantityValue<quantity::Density>;
-    using PressureValue         = QuantityValue<quantity::Pressure>;
-    using SpecificEnthalpyValue = QuantityValue<quantity::SpecificEnthalpy>;
-
-    template <typename Candidate> struct IsQuantityValue : std::false_type { };
-
-    template <ThermodynamicQuantityType Quantity> struct IsQuantityValue<QuantityValue<Quantity>> : std::true_type { };
+    template <typename Candidate> using IsQuantityValue               = dimensions::IsQuantityValue<Candidate>;
 
     template <typename Candidate>
-    concept QuantityValueType = IsQuantityValue<std::remove_cvref_t<Candidate>>::value;
+    concept QuantityValueType =
+        dimensions::QuantityValueType<Candidate> && ThermodynamicQuantityType<dimensions::QuantityOfT<Candidate>>;
 
-    template <typename Candidate> struct QuantityOf;
+    template <typename Candidate> using QuantityOf       = dimensions::QuantityOf<Candidate>;
 
-    template <ThermodynamicQuantityType Quantity> struct QuantityOf<QuantityValue<Quantity>> {
-        using Type = Quantity;
-    };
-
-    template <QuantityValueType Value> using QuantityOfT = typename QuantityOf<std::remove_cvref_t<Value>>::Type;
+    template <QuantityValueType Value> using QuantityOfT = dimensions::QuantityOfT<Value>;
 
     template <ThermodynamicQuantityType OutputQuantity, ThermodynamicQuantityType InputQuantity>
     class PartialDerivative final {
@@ -163,7 +73,7 @@ export namespace mean_field::eos {
                 InputQuantity> &rhs
         ) noexcept;
 
-        template <Numeric rhsT>
+        template <Numeric Scalar>
         friend constexpr PartialDerivative<
             OutputQuantity,
             InputQuantity>
@@ -171,21 +81,21 @@ export namespace mean_field::eos {
             const PartialDerivative<
                 OutputQuantity,
                 InputQuantity> &,
-            rhsT
+            Scalar
         ) noexcept;
 
-        template <Numeric lhsT>
+        template <Numeric Scalar>
         friend constexpr PartialDerivative<
             OutputQuantity,
             InputQuantity>
         operator*(
-            lhsT,
+            Scalar,
             const PartialDerivative<
                 OutputQuantity,
                 InputQuantity> &
         ) noexcept;
 
-        template <Numeric rhsT>
+        template <Numeric Scalar>
         friend constexpr PartialDerivative<
             OutputQuantity,
             InputQuantity>
@@ -193,22 +103,22 @@ export namespace mean_field::eos {
             const PartialDerivative<
                 OutputQuantity,
                 InputQuantity> &,
-            rhsT
+            Scalar
         ) noexcept;
 
-        template <Numeric cmpT>
+        template <Numeric Scalar>
         friend constexpr std::partial_ordering operator<=>(
             const PartialDerivative<
                 OutputQuantity,
                 InputQuantity> &lhs,
-            cmpT rhs
+            Scalar rhs
         ) noexcept {
             return lhs.m_value <=> static_cast<double>(rhs);
         }
 
-        template <Numeric cmpT>
+        template <Numeric Scalar>
         friend constexpr std::partial_ordering operator<=>(
-            cmpT lhs,
+            Scalar lhs,
             const PartialDerivative<
                 OutputQuantity,
                 InputQuantity> &rhs

libmeanfield/interface/eos/relations.cppm

@@ -85,9 +85,11 @@ export namespace mean_field::eos {
     template <std::size_t Index, ThermodynamicRelationType RelationType>
     using RelationInputT      = typename detail::QuantityAt<Index, typename RelationType::InputQuantities>::Type;
 
-    using PressureFromDensity = Relation<quantity::Pressure, quantity::Density>;
-    using PressureFromSpecificEnthalpy = Relation<quantity::Pressure, quantity::SpecificEnthalpy>;
-    using SpecificEnthalpyFromDensity  = Relation<quantity::SpecificEnthalpy, quantity::Density>;
-    using SpecificEnthalpyFromPressure = Relation<quantity::SpecificEnthalpy, quantity::Pressure>;
-    using DensityFromSpecificEnthalpy  = Relation<quantity::Density, quantity::SpecificEnthalpy>;
+    using PressureFromDensity = Relation<dimensions::quantity::Pressure, dimensions::quantity::Density>;
+    using PressureFromSpecificEnthalpy =
+        Relation<dimensions::quantity::Pressure, dimensions::quantity::SpecificEnthalpy>;
+    using SpecificEnthalpyFromDensity = Relation<dimensions::quantity::SpecificEnthalpy, dimensions::quantity::Density>;
+    using SpecificEnthalpyFromPressure =
+        Relation<dimensions::quantity::SpecificEnthalpy, dimensions::quantity::Pressure>;
+    using DensityFromSpecificEnthalpy = Relation<dimensions::quantity::Density, dimensions::quantity::SpecificEnthalpy>;
 } // namespace mean_field::eos

libmeanfield/interface/equilibrium/stellar_discretization.cppm

@@ -0,0 +1,64 @@
+module;
+
+#include <memory>
+#include <stdexcept>
+
+export module mean_field:equilibrium.stellar_discretization;
+
+export import :fem;
+export import :mapping.domain_mapper;
+
+export namespace mean_field::equilibrium {
+    /*
+     * An explicit, non-owning view of the numerical discretization used by a
+     * stellar equilibrium problem. The referenced FEM and mapper must outlive
+     * every problem and structure that uses this view.
+     *
+     * Ownership cannot move here yet because FEM currently also contains
+     * mutable field workspaces. Separating those workspaces is a prerequisite
+     * for shared discretization ownership by solved Structure objects.
+     */
+    class StellarDiscretization final {
+    public:
+        explicit StellarDiscretization(fem::FEM &finiteElementModel)
+            : StellarDiscretization(
+                  finiteElementModel,
+                  RequireDomainMapper(finiteElementModel)
+              ) {
+        }
+
+        StellarDiscretization(
+            fem::FEM &finiteElementModel,
+            const mapping::DomainMapper &domainMapper
+        )
+            : m_finiteElementModel(std::addressof(finiteElementModel)),
+              m_domainMapper(std::addressof(domainMapper)) {
+            if (!finiteElementModel.okay()) {
+                throw std::invalid_argument("A stellar discretization requires a complete finite-element model.");
+            }
+        }
+
+        [[nodiscard]] fem::FEM &finiteElementModel() const noexcept {
+            return *m_finiteElementModel;
+        }
+
+        [[nodiscard]] const mapping::DomainMapper &domainMapper() const noexcept {
+            return *m_domainMapper;
+        }
+
+        [[nodiscard]] bool isCurrent() const noexcept {
+            return m_finiteElementModel != nullptr && m_domainMapper != nullptr && m_finiteElementModel->okay();
+        }
+
+    private:
+        [[nodiscard]] static const mapping::DomainMapper &RequireDomainMapper(const fem::FEM &finiteElementModel) {
+            if (finiteElementModel.domainMapperStateless == nullptr) {
+                throw std::invalid_argument("A stellar discretization requires a domain mapper.");
+            }
+            return *finiteElementModel.domainMapperStateless;
+        }
+
+        fem::FEM *m_finiteElementModel;
+        const mapping::DomainMapper *m_domainMapper;
+    };
+} // namespace mean_field::equilibrium

libmeanfield/interface/field/field_registry.cppm

@@ -208,6 +208,9 @@ export namespace mean_field::field {
         using FormList = TypeList<Form::MeshExtension, Form::GravityForce, Form::CentrifugalForce, Form::ErrorNorm>;
     };
 
+    // Current realization of MultiplierFor<FixedTotalMass>. This remains a
+    // barotrope-specific field representation: the specification compiler,
+    // rather than the universal state registry, decides when it is present.
     struct BarotropicConstant {
         static constexpr std::string_view name = "barotropic_constant";
 
@@ -226,6 +229,26 @@ export namespace mean_field::field {
         static_assert(constraintsAreValid);
     };
 
+    // Solver border generated by FixedCentralDensity. This is deliberately a
+    // non-spatial numerical coordinate rather than a physical stellar field.
+    struct CentralDensityBorder {
+        static constexpr std::string_view name = "central_density_border";
+
+        using Support                          = NonSpatialSupport;
+
+        struct Scalar final : GlobalScalarQ {
+            static constexpr std::string_view symbol = "lambda_rho_c";
+        };
+
+        using Quantities                          = TypeList<Scalar>;
+        using Constraints                         = TypeList<>;
+        using FormList                            = TypeList<>;
+
+        static constexpr bool constraintsAreValid = validate_constraints(Constraints{});
+
+        static_assert(constraintsAreValid);
+    };
+
     // =========================================================================
     // Specific enthalpy
     //

libmeanfield/interface/mapping/domain_mapper.cppm

@@ -78,6 +78,7 @@ export namespace mean_field::mapping {
             int m_dimension;
 
             mfem::Vector m_shape;
+            mfem::DenseMatrix m_reference_dshape;
             mfem::DenseMatrix m_mesh_dshape;
             mfem::Vector m_field_value;
             mfem::DenseMatrix m_field_jacobian;
@@ -178,7 +179,8 @@ export namespace mean_field::mapping {
             const mfem::IntegrationPoint &integration_point,
             Workspace &workspace,
             mfem::Vector &value,
-            mfem::DenseMatrix &jacobian
+            mfem::DenseMatrix &jacobian,
+            const mfem::DenseMatrix *inverse_mesh_jacobian
         ) const;
 
         [[nodiscard]] MappingStatus EvaluateCompactificationCoordinate(
@@ -186,7 +188,19 @@ export namespace mean_field::mapping {
             mfem::ElementTransformation &transformation,
             const mfem::IntegrationPoint &integration_point,
             Workspace &workspace,
-            CompactificationPointData &point_data
+            CompactificationPointData &point_data,
+            const mfem::DenseMatrix *inverse_mesh_jacobian
+        ) const;
+
+        [[nodiscard]] MappingStatus EvaluatePointVariationImpl(
+            const ElementMappingData &element_data,
+            const ElementDisplacementData &direction,
+            mfem::ElementTransformation &transformation,
+            const mfem::IntegrationPoint &integration_point,
+            const MappingPointContext &base_context,
+            Workspace &workspace,
+            MappingPointVariation &variation,
+            const mfem::DenseMatrix *inverse_mesh_jacobian
         ) const;
 
         [[nodiscard]] static mfem::ElementTransformation &SelectFaceElementTransformation(

libmeanfield/interface/mean_field.cppm

@@ -25,6 +25,7 @@ export import :integrators.viscosity;
 export import :quadrature.policy;
 export import :quadrature.mfem;
 export import :solver.fields;
+export import :solver.preconditioning_diagnostics;
 export import :utils.blocks;
 export import :operators.gravity_field;
 export import :operators.gravity_field_jacobian;
@@ -51,9 +52,14 @@ export import :operators.context.rotational_displacement_force;
 export import :operators.kernels.rotational_displacement_force;
 export import :operators.prepared_rotational_displacement_force;
 export import :operators.prepared_displacement_residual;
+export import :dimensions.quantities;
 export import :model.structure_profile;
 export import :model.structure.base;
 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_central_density;
 export import :eos.quantities;
 export import :eos.relations;
 export import :eos.concepts;
@@ -61,6 +67,7 @@ export import :eos.evaluation;
 export import :eos.pressure_surface;
 export import :eos.runtime;
 export import :eos.polytrope;
+export import :seed.lane_emden;
 export import :surface.constant;
 export import :surface.dependencies;
 export import :surface.compiled;
@@ -73,7 +80,15 @@ export import :deformation.vacuum_extension;
 export import :deformation.radial_extensions;
 export import :deformation.domain_deformation;
 export import :model.stellar;
+export import :operators.root_manifest;
+export import :operators.prepared_constraint;
 export import :operators.prepared_mass_normalization;
+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 :equilibrium.stellar_discretization;
+export import :operators.stellar_equilibrium_problem;
+export import :seed.stellar_equilibrium_projection;
+export import :operators.stellar_equilibrium_system;

libmeanfield/interface/models/compiled_fixed_central_density.cppm

@@ -0,0 +1,95 @@
+module;
+
+#include <concepts>
+#include <type_traits>
+
+export module mean_field:model.compiled_fixed_central_density;
+
+export import :eos.polytrope;
+export import :field.registry;
+export import :model.compiled_fixed_mass;
+export import :utils.blocks;
+
+export namespace mean_field::models {
+    struct CentralDensityLayoutRequest final {
+        using SpecificationType     = FixedCentralDensity;
+        using GeneratedValueType    = BorderFor<FixedCentralDensity>;
+        using GeneratedResidualType = ResidualFor<FixedCentralDensity>;
+        using ValueBlockType        = utils::blocks::fixed_central_density::central_value::value;
+        using ResidualBlockType     = utils::blocks::fixed_central_density::central_value::residual;
+        using TermType              = utils::blocks::fixed_central_density::central_value;
+        using StateValueBlockTypes  = ModelTypeList<utils::blocks::enthalpy::specific::value>;
+
+        static constexpr ConstraintRowInjection rowInjection = ConstraintRowInjection::solver_border;
+        static constexpr std::size_t valueArity              = GeneratedValueType::scalarArity;
+        static constexpr std::size_t residualArity           = GeneratedResidualType::scalarArity;
+
+        template <typename Form> [[nodiscard]] static consteval auto valueBlock() {
+            return utils::blocks::get_value_block<Form>(TermType{});
+        }
+
+        template <typename Form> [[nodiscard]] static consteval auto residualBlock() {
+            return utils::blocks::get_residual_block<Form>(TermType{});
+        }
+    };
+
+    class CompiledFixedCentralDensity final {
+    public:
+        using SpecificationType = FixedCentralDensity;
+        using LayoutRequest     = CentralDensityLayoutRequest;
+        using BorderType        = typename LayoutRequest::GeneratedValueType;
+        using ResidualType      = typename LayoutRequest::GeneratedResidualType;
+        using CarrierField      = field::Enthalpy;
+        using BorderField       = field::CentralDensityBorder;
+
+        CompiledFixedCentralDensity(
+            const FixedCentralDensity specification,
+            const eos::Polytrope &equationOfState
+        )
+            : m_specification(specification),
+              m_equationOfState(equationOfState),
+              m_targetEnthalpy(
+                  eos::evaluate<eos::quantity::SpecificEnthalpy>(
+                      m_equationOfState,
+                      m_specification.targetDensity()
+                  )
+              ) {
+        }
+
+        [[nodiscard]] const FixedCentralDensity &specification() const noexcept {
+            return m_specification;
+        }
+
+        [[nodiscard]] dimensions::DensityValue targetDensity() const noexcept {
+            return m_specification.targetDensity();
+        }
+
+        [[nodiscard]] dimensions::SpecificEnthalpyValue targetEnthalpy() const noexcept {
+            return m_targetEnthalpy;
+        }
+
+        [[nodiscard]] dimensions::DensityValue
+        densityFromEnthalpy(const dimensions::SpecificEnthalpyValue enthalpy) const {
+            return eos::evaluate<eos::quantity::Density>(m_equationOfState, enthalpy);
+        }
+
+        [[nodiscard]] static consteval LayoutRequest layoutRequest() noexcept {
+            return {};
+        }
+
+    private:
+        FixedCentralDensity m_specification;
+        eos::Polytrope m_equationOfState;
+        dimensions::SpecificEnthalpyValue m_targetEnthalpy;
+    };
+
+    [[nodiscard]] inline CompiledFixedCentralDensity compileConstraint(
+        const FixedCentralDensity specification,
+        const eos::Polytrope &equationOfState
+    ) {
+        return {specification, equationOfState};
+    }
+
+    static_assert(ConstraintLayoutRequestType<CentralDensityLayoutRequest>);
+    static_assert(CompiledConstraint<CompiledFixedCentralDensity>);
+} // namespace mean_field::models

libmeanfield/interface/models/compiled_fixed_mass.cppm

@@ -0,0 +1,115 @@
+module;
+
+#include <concepts>
+#include <type_traits>
+
+export module mean_field:model.compiled_fixed_mass;
+
+export import :field.registry;
+export import :model.specifications;
+export import :utils.blocks;
+
+export namespace mean_field::models {
+    enum class ConstraintRowInjection { append, solver_border };
+
+    template <
+        ModelSpecification Specification,
+        typename GeneratedValue,
+        typename GeneratedResidual,
+        typename ValueBlock,
+        typename ResidualBlock,
+        typename Term,
+        typename... StateValueBlocks>
+    struct ConstraintLayoutRequest final {
+        using SpecificationType                              = Specification;
+        using GeneratedValueType                             = GeneratedValue;
+        using GeneratedResidualType                          = GeneratedResidual;
+        using ValueBlockType                                 = ValueBlock;
+        using ResidualBlockType                              = ResidualBlock;
+        using TermType                                       = Term;
+        using StateValueBlockTypes                           = ModelTypeList<StateValueBlocks...>;
+
+        static constexpr ConstraintRowInjection rowInjection = ConstraintRowInjection::append;
+        static constexpr std::size_t valueArity              = GeneratedValue::scalarArity;
+        static constexpr std::size_t residualArity           = GeneratedResidual::scalarArity;
+
+        template <typename Form> [[nodiscard]] static consteval auto valueBlock() {
+            return utils::blocks::get_value_block<Form>(Term{});
+        }
+
+        template <typename Form> [[nodiscard]] static consteval auto residualBlock() {
+            return utils::blocks::get_residual_block<Form>(Term{});
+        }
+    };
+
+    template <typename Candidate>
+    concept ConstraintLayoutRequestType = requires {
+        typename std::remove_cvref_t<Candidate>::SpecificationType;
+        typename std::remove_cvref_t<Candidate>::GeneratedValueType;
+        typename std::remove_cvref_t<Candidate>::GeneratedResidualType;
+        typename std::remove_cvref_t<Candidate>::ValueBlockType;
+        typename std::remove_cvref_t<Candidate>::ResidualBlockType;
+        typename std::remove_cvref_t<Candidate>::StateValueBlockTypes;
+        requires ModelSpecification<typename std::remove_cvref_t<Candidate>::SpecificationType>;
+        requires std::remove_cvref_t<Candidate>::valueArity == std::remove_cvref_t<Candidate>::residualArity;
+    };
+
+    using FixedMassLayoutRequest = ConstraintLayoutRequest<
+        FixedTotalMass,
+        MultiplierFor<FixedTotalMass>,
+        ResidualFor<FixedTotalMass>,
+        utils::blocks::fixed_total_mass::mass_normalization::value,
+        utils::blocks::fixed_total_mass::mass_normalization::residual,
+        utils::blocks::fixed_total_mass::mass_normalization,
+        utils::blocks::density::mass::value,
+        utils::blocks::displacement::geometry::value>;
+
+    class CompiledFixedMass final {
+    public:
+        using SpecificationType = FixedTotalMass;
+        using LayoutRequest     = FixedMassLayoutRequest;
+        using MultiplierType    = typename LayoutRequest::GeneratedValueType;
+        using ResidualType      = typename LayoutRequest::GeneratedResidualType;
+
+        // In the current barotropic formulation, the multiplier generated by
+        // FixedTotalMass is realized by the historical scalar C field.
+        using MultiplierField   = field::BarotropicConstant;
+
+        explicit CompiledFixedMass(const FixedTotalMass specification) noexcept : m_specification(specification) {
+        }
+
+        [[nodiscard]] const FixedTotalMass &specification() const noexcept {
+            return m_specification;
+        }
+
+        [[nodiscard]] dimensions::MassValue targetMass() const noexcept {
+            return m_specification.targetMass();
+        }
+
+        [[nodiscard]] static consteval LayoutRequest layoutRequest() noexcept {
+            return {};
+        }
+
+    private:
+        FixedTotalMass m_specification;
+    };
+
+    template <typename Candidate>
+    concept CompiledConstraint = requires(const std::remove_cvref_t<Candidate> &constraint) {
+        typename std::remove_cvref_t<Candidate>::SpecificationType;
+        typename std::remove_cvref_t<Candidate>::LayoutRequest;
+        requires ModelSpecification<typename std::remove_cvref_t<Candidate>::SpecificationType>;
+        requires ConstraintLayoutRequestType<typename std::remove_cvref_t<Candidate>::LayoutRequest>;
+        {
+            constraint.specification()
+        } noexcept -> std::same_as<const typename std::remove_cvref_t<Candidate>::SpecificationType &>;
+        { constraint.layoutRequest() } noexcept -> std::same_as<typename std::remove_cvref_t<Candidate>::LayoutRequest>;
+    };
+
+    [[nodiscard]] inline CompiledFixedMass compileConstraint(const FixedTotalMass specification) noexcept {
+        return CompiledFixedMass{specification};
+    }
+
+    static_assert(ConstraintLayoutRequestType<FixedMassLayoutRequest>);
+    static_assert(CompiledConstraint<CompiledFixedMass>);
+} // namespace mean_field::models

libmeanfield/interface/models/specifications.cppm

@@ -0,0 +1,481 @@
+module;
+
+#include <array>
+#include <cmath>
+#include <compare>
+#include <concepts>
+#include <cstddef>
+#include <format>
+#include <span>
+#include <stdexcept>
+#include <string_view>
+#include <tuple>
+#include <type_traits>
+#include <utility>
+
+export module mean_field:model.specifications;
+
+export import :eos.polytrope;
+export import :surface.constant;
+
+export namespace mean_field::models {
+    enum class SpecificationRole {
+        constitutive_law,
+        boundary_condition,
+        invariant,
+        phase_condition,
+        gauge_choice,
+        rotation_law
+    };
+
+    struct SpecificationKey final {
+        SpecificationRole role;
+        std::size_t ordinal;
+
+        constexpr auto operator<=>(const SpecificationKey &) const = default;
+    };
+
+    struct SpecificationDescriptor final {
+        std::string_view name;
+        SpecificationRole role;
+        SpecificationKey key;
+        std::size_t generatedValueArity;
+        std::size_t generatedResidualArity;
+
+        constexpr bool operator==(const SpecificationDescriptor &) const = default;
+    };
+
+    enum class EquilibriumSystemCompilation {
+        complete_equilibrium_system,
+        equation_contributions_only,
+
+        // Transitional spellings retained while internal solver code is
+        // migrated to physics-facing equilibrium-system terminology.
+        isolated_root = complete_equilibrium_system,
+        assembly_only = equation_contributions_only
+    };
+
+    using ModelCompilationClass = EquilibriumSystemCompilation;
+
+    struct RuntimeSpecificationDescriptor final {
+        SpecificationDescriptor specification;
+        std::size_t canonicalIndex;
+        bool hasRootCompiler;
+
+        constexpr bool operator==(const RuntimeSpecificationDescriptor &) const = default;
+    };
+
+    template <typename Candidate> struct SpecificationTraits;
+
+    template <typename Candidate>
+    concept ModelSpecification = requires {
+        typename std::remove_cvref_t<Candidate>::Parameters;
+        { SpecificationTraits<std::remove_cvref_t<Candidate>>::name } -> std::convertible_to<std::string_view>;
+        { SpecificationTraits<std::remove_cvref_t<Candidate>>::role } -> std::convertible_to<SpecificationRole>;
+        { SpecificationTraits<std::remove_cvref_t<Candidate>>::key } -> std::convertible_to<SpecificationKey>;
+    } && std::constructible_from<std::remove_cvref_t<Candidate>, typename std::remove_cvref_t<Candidate>::Parameters>;
+
+    class FixedTotalMass final {
+    public:
+        struct Parameters final {
+            dimensions::MassValue Mtotal;
+        };
+
+        using TargetValue = dimensions::MassValue;
+
+        explicit FixedTotalMass(const Parameters parameters) : FixedTotalMass(parameters.Mtotal) {
+        }
+
+        explicit FixedTotalMass(const TargetValue targetMass) : m_targetMass(targetMass) {
+            if (!std::isfinite(targetMass.value()) || targetMass.value() <= 0.0) {
+                throw std::invalid_argument(
+                    std::format(
+                        "The fixed total mass must be finite and positive. Instead M = {} was provided.",
+                        targetMass.value()
+                    )
+                );
+            }
+        }
+
+        [[nodiscard]] TargetValue targetMass() const noexcept {
+            return m_targetMass;
+        }
+
+    private:
+        TargetValue m_targetMass;
+    };
+
+    class FixedCentralDensity final {
+    public:
+        struct Parameters final {
+            dimensions::DensityValue RhoC;
+        };
+
+        using TargetValue = dimensions::DensityValue;
+
+        explicit FixedCentralDensity(const Parameters parameters) : FixedCentralDensity(parameters.RhoC) {
+        }
+
+        explicit FixedCentralDensity(const TargetValue targetDensity) : m_targetDensity(targetDensity) {
+            if (!std::isfinite(targetDensity.value()) || targetDensity.value() <= 0.0) {
+                throw std::invalid_argument(
+                    std::format(
+                        "The fixed central density must be finite and positive. Instead rho_c = {} was provided.",
+                        targetDensity.value()
+                    )
+                );
+            }
+        }
+
+        [[nodiscard]] TargetValue targetDensity() const noexcept {
+            return m_targetDensity;
+        }
+
+    private:
+        TargetValue m_targetDensity;
+    };
+
+    template <> struct SpecificationTraits<eos::Polytrope> {
+        static constexpr std::string_view name  = "Polytrope";
+        static constexpr SpecificationRole role = SpecificationRole::constitutive_law;
+        static constexpr SpecificationKey key{role, 0};
+    };
+
+    template <> struct SpecificationTraits<surface::ConstantPressureSurface> {
+        static constexpr std::string_view name  = "IsobaricSurface";
+        static constexpr SpecificationRole role = SpecificationRole::boundary_condition;
+        static constexpr SpecificationKey key{role, 0};
+    };
+
+    template <> struct SpecificationTraits<FixedTotalMass> {
+        static constexpr std::string_view name  = "FixedTotalMass";
+        static constexpr SpecificationRole role = SpecificationRole::invariant;
+        static constexpr SpecificationKey key{role, 0};
+    };
+
+    template <> struct SpecificationTraits<FixedCentralDensity> {
+        static constexpr std::string_view name  = "FixedCentralDensity";
+        static constexpr SpecificationRole role = SpecificationRole::phase_condition;
+        static constexpr SpecificationKey key{role, 0};
+    };
+
+    template <typename... Types> struct ModelTypeList final {
+        static constexpr std::size_t size = sizeof...(Types);
+    };
+
+    template <typename Query, typename List> struct ModelTypeListContains;
+
+    template <typename Query, typename... Types>
+    struct ModelTypeListContains<Query, ModelTypeList<Types...>>
+        : std::bool_constant<(std::same_as<Query, Types> || ...)> { };
+
+    template <typename Query, typename List>
+    inline constexpr bool modelTypeListContains = ModelTypeListContains<Query, List>::value;
+
+    template <ModelSpecification Specification> struct ResidualFor final {
+        using SpecificationType                  = Specification;
+
+        static constexpr std::size_t scalarArity = 1;
+    };
+
+    template <ModelSpecification Specification> struct MultiplierFor final {
+        using SpecificationType                  = Specification;
+
+        static constexpr std::size_t scalarArity = 1;
+    };
+
+    template <ModelSpecification Specification> struct BorderFor final {
+        using SpecificationType                  = Specification;
+
+        static constexpr std::size_t scalarArity = 1;
+    };
+
+    template <ModelSpecification Specification> struct SpecificationContribution {
+        using GeneratedValues                 = ModelTypeList<>;
+        using GeneratedResiduals              = ModelTypeList<>;
+
+        static constexpr bool isDefined       = false;
+        static constexpr bool hasRootCompiler = false;
+    };
+
+    template <> struct SpecificationContribution<eos::Polytrope> {
+        using GeneratedValues                 = ModelTypeList<>;
+        using GeneratedResiduals              = ModelTypeList<>;
+
+        static constexpr bool isDefined       = true;
+        static constexpr bool hasRootCompiler = true;
+    };
+
+    template <> struct SpecificationContribution<surface::ConstantPressureSurface> {
+        using GeneratedValues                 = ModelTypeList<>;
+        using GeneratedResiduals              = ModelTypeList<>;
+
+        static constexpr bool isDefined       = true;
+        static constexpr bool hasRootCompiler = true;
+    };
+
+    template <> struct SpecificationContribution<FixedTotalMass> {
+        using GeneratedValues                 = ModelTypeList<MultiplierFor<FixedTotalMass>>;
+        using GeneratedResiduals              = ModelTypeList<ResidualFor<FixedTotalMass>>;
+
+        static constexpr bool isDefined       = true;
+        static constexpr bool hasRootCompiler = true;
+    };
+
+    template <> struct SpecificationContribution<FixedCentralDensity> {
+        using GeneratedValues                 = ModelTypeList<BorderFor<FixedCentralDensity>>;
+        using GeneratedResiduals              = ModelTypeList<ResidualFor<FixedCentralDensity>>;
+
+        static constexpr bool isDefined       = true;
+        static constexpr bool hasRootCompiler = true;
+    };
+
+    template <typename Candidate>
+    concept ResolvedModelSpecification =
+        ModelSpecification<Candidate> && SpecificationContribution<std::remove_cvref_t<Candidate>>::isDefined;
+
+    namespace detail {
+        template <typename... Specifications> struct SpecificationSetStorage final {
+            static constexpr std::size_t size = sizeof...(Specifications);
+        };
+
+        template <typename... Lists> struct ConcatenateModelTypeLists;
+
+        template <> struct ConcatenateModelTypeLists<> {
+            using Type = ModelTypeList<>;
+        };
+
+        template <typename... Types> struct ConcatenateModelTypeLists<ModelTypeList<Types...>> {
+            using Type = ModelTypeList<Types...>;
+        };
+
+        template <typename... First, typename... Second, typename... Remaining>
+        struct ConcatenateModelTypeLists<ModelTypeList<First...>, ModelTypeList<Second...>, Remaining...> {
+            using Type = typename ConcatenateModelTypeLists<ModelTypeList<First..., Second...>, Remaining...>::Type;
+        };
+
+        template <ModelSpecification Specification, typename Set> struct InsertSpecification;
+
+        template <ModelSpecification Specification>
+        struct InsertSpecification<Specification, SpecificationSetStorage<>> {
+            using Type = SpecificationSetStorage<Specification>;
+        };
+
+        template <ModelSpecification Specification, ModelSpecification Head, ModelSpecification... Tail>
+        struct InsertSpecification<Specification, SpecificationSetStorage<Head, Tail...>> {
+        private:
+            using InsertedTail = typename InsertSpecification<Specification, SpecificationSetStorage<Tail...>>::Type;
+
+            template <typename First, typename Rest> struct PrependSpecification;
+
+            template <typename First, ModelSpecification... Rest>
+            struct PrependSpecification<First, SpecificationSetStorage<Rest...>> {
+                using Type = SpecificationSetStorage<First, Rest...>;
+            };
+
+        public:
+            using Type = std::conditional_t<
+                (SpecificationTraits<Specification>::key < SpecificationTraits<Head>::key),
+                SpecificationSetStorage<Specification, Head, Tail...>,
+                typename PrependSpecification<Head, InsertedTail>::Type>;
+        };
+
+        template <typename Set, ModelSpecification... Specifications> struct CanonicalizeSpecifications;
+
+        template <typename Set> struct CanonicalizeSpecifications<Set> {
+            using Type = Set;
+        };
+
+        template <typename Set, ModelSpecification Head, ModelSpecification... Tail>
+        struct CanonicalizeSpecifications<Set, Head, Tail...> {
+            using Inserted = typename InsertSpecification<Head, Set>::Type;
+            using Type     = typename CanonicalizeSpecifications<Inserted, Tail...>::Type;
+        };
+
+        template <ModelSpecification... Specifications>
+        using CanonicalSpecificationSet =
+            typename CanonicalizeSpecifications<SpecificationSetStorage<>, Specifications...>::Type;
+
+        template <
+            ModelSpecification Head,
+            ModelSpecification... Tail>
+        consteval bool specificationKeyIsUnique() {
+            return ((SpecificationTraits<Head>::key != SpecificationTraits<Tail>::key) && ...);
+        }
+
+        template <ModelSpecification... Specifications> struct SpecificationKeysAreUnique;
+
+        template <> struct SpecificationKeysAreUnique<> : std::true_type { };
+
+        template <ModelSpecification Head, ModelSpecification... Tail>
+        struct SpecificationKeysAreUnique<Head, Tail...>
+            : std::bool_constant<
+                  specificationKeyIsUnique<Head, Tail...>() && SpecificationKeysAreUnique<Tail...>::value> { };
+
+        template <SpecificationRole Role, ModelSpecification... Specifications>
+        inline constexpr std::size_t specificationRoleCount =
+            (std::size_t{0} + ... + (SpecificationTraits<Specifications>::role == Role ? 1 : 0));
+
+        template <typename List> struct ModelTypeListScalarArity;
+
+        template <typename... Types>
+        struct ModelTypeListScalarArity<ModelTypeList<Types...>>
+            : std::integral_constant<std::size_t, (std::size_t{0} + ... + Types::scalarArity)> { };
+
+        template <typename Query, typename... Types>
+        inline constexpr bool isOneOf = (std::same_as<Query, Types> || ...);
+
+        template <typename Query, typename... Types>
+        inline constexpr std::size_t typeCount =
+            (std::size_t{0} + ... +
+             (std::same_as<Query, std::remove_cvref_t<Types>> ? std::size_t{1} : std::size_t{0}));
+
+        template <typename CanonicalSet, typename... Arguments> struct ArgumentsMatchCanonicalSpecifications;
+
+        template <ModelSpecification... CanonicalSpecifications, typename... Arguments>
+        struct ArgumentsMatchCanonicalSpecifications<SpecificationSetStorage<CanonicalSpecifications...>, Arguments...>
+            : std::bool_constant<
+                  sizeof...(CanonicalSpecifications) == sizeof...(Arguments) &&
+                  (isOneOf<std::remove_cvref_t<Arguments>, CanonicalSpecifications...> && ...) &&
+                  ((typeCount<CanonicalSpecifications, Arguments...> == 1) && ...)> { };
+    } // namespace detail
+
+    template <ModelSpecification... Specifications>
+    inline constexpr bool specificationKeysAreUnique = detail::SpecificationKeysAreUnique<Specifications...>::value;
+
+    template <typename... Specifications>
+    concept ValidModelSpecificationPack =
+        (ResolvedModelSpecification<Specifications> && ...) && specificationKeysAreUnique<Specifications...> &&
+        detail::specificationRoleCount<SpecificationRole::constitutive_law, Specifications...> == 1;
+
+    template <ModelSpecification... Specifications>
+        requires specificationKeysAreUnique<Specifications...>
+    using SpecificationSet = detail::CanonicalSpecificationSet<Specifications...>;
+
+    template <typename SpecificationSet> struct SpecificationOperatorSignature;
+
+    template <ModelSpecification... Specifications>
+    struct SpecificationOperatorSignature<detail::SpecificationSetStorage<Specifications...>> final {
+        using GeneratedValues = typename detail::ConcatenateModelTypeLists<
+            typename SpecificationContribution<Specifications>::GeneratedValues...>::Type;
+
+        using GeneratedResiduals = typename detail::ConcatenateModelTypeLists<
+            typename SpecificationContribution<Specifications>::GeneratedResiduals...>::Type;
+
+        static constexpr std::size_t generatedValueArity = detail::ModelTypeListScalarArity<GeneratedValues>::value;
+
+        static constexpr std::size_t generatedResidualArity =
+            detail::ModelTypeListScalarArity<GeneratedResiduals>::value;
+
+        static constexpr bool symbolicallySquare = generatedValueArity == generatedResidualArity;
+    };
+
+    template <ResolvedModelSpecification Specification>
+    [[nodiscard]] consteval SpecificationDescriptor specificationDescriptor() {
+        using Contribution = SpecificationContribution<Specification>;
+
+        return {
+            .name                   = SpecificationTraits<Specification>::name,
+            .role                   = SpecificationTraits<Specification>::role,
+            .key                    = SpecificationTraits<Specification>::key,
+            .generatedValueArity    = detail::ModelTypeListScalarArity<typename Contribution::GeneratedValues>::value,
+            .generatedResidualArity = detail::ModelTypeListScalarArity<typename Contribution::GeneratedResiduals>::value
+        };
+    }
+
+    namespace detail {
+        template <typename Specifications> class SpecifiedModel;
+
+        template <ModelSpecification... Specifications>
+        class SpecifiedModel<SpecificationSetStorage<Specifications...>> final {
+        public:
+            using SpecificationTypes                 = SpecificationSetStorage<Specifications...>;
+            using OperatorSignature                  = SpecificationOperatorSignature<SpecificationTypes>;
+
+            static constexpr bool symbolicallySquare = OperatorSignature::symbolicallySquare;
+            static constexpr bool hasCompleteRootCompiler =
+                (SpecificationContribution<Specifications>::hasRootCompiler && ...);
+            static constexpr EquilibriumSystemCompilation compilationClass =
+                symbolicallySquare && hasCompleteRootCompiler
+                    ? EquilibriumSystemCompilation::complete_equilibrium_system
+                    : EquilibriumSystemCompilation::equation_contributions_only;
+
+            template <typename... Arguments>
+                requires ArgumentsMatchCanonicalSpecifications<
+                    SpecificationTypes,
+                    Arguments...>::value
+            explicit SpecifiedModel(Arguments &&...arguments)
+                : m_specifications(
+                      std::get<Specifications>(
+                          std::tuple<std::remove_cvref_t<Arguments>...>{std::forward<Arguments>(arguments)...}
+                      )...
+                  ) {
+            }
+
+            template <ModelSpecification Specification>
+                requires isOneOf<
+                    Specification,
+                    Specifications...>
+            [[nodiscard]] const Specification &specification() const noexcept {
+                return std::get<Specification>(m_specifications);
+            }
+
+            template <ModelSpecification Specification>
+            static constexpr bool containsSpecification = isOneOf<Specification, Specifications...>;
+
+            [[nodiscard]] static constexpr std::span<const RuntimeSpecificationDescriptor>
+            runtimeSpecificationDescriptors() noexcept {
+                return runtimeDescriptors;
+            }
+
+        private:
+            inline static constexpr std::array<RuntimeSpecificationDescriptor, sizeof...(Specifications)>
+                runtimeDescriptors = [] {
+                    std::array<RuntimeSpecificationDescriptor, sizeof...(Specifications)> descriptors{};
+                    std::size_t index = 0;
+                    ((descriptors[index] =
+                          {.specification   = specificationDescriptor<Specifications>(),
+                           .canonicalIndex  = index,
+                           .hasRootCompiler = SpecificationContribution<Specifications>::hasRootCompiler},
+                      ++index),
+                     ...);
+                    return descriptors;
+                }();
+
+            std::tuple<Specifications...> m_specifications;
+        };
+    } // namespace detail
+
+    template <ModelSpecification... Specifications>
+        requires ValidModelSpecificationPack<Specifications...> &&
+                     SpecificationOperatorSignature<SpecificationSet<Specifications...>>::symbolicallySquare
+    using Model = detail::SpecifiedModel<SpecificationSet<Specifications...>>;
+
+    template <typename Candidate>
+    concept SpecifiedModelType = requires {
+        typename std::remove_cvref_t<Candidate>::SpecificationTypes;
+        typename std::remove_cvref_t<Candidate>::OperatorSignature;
+        requires std::remove_cvref_t<Candidate>::symbolicallySquare;
+        { std::remove_cvref_t<Candidate>::compilationClass } -> std::convertible_to<ModelCompilationClass>;
+        {
+            std::remove_cvref_t<Candidate>::runtimeSpecificationDescriptors()
+        } -> std::same_as<std::span<const RuntimeSpecificationDescriptor>>;
+    };
+
+    static_assert(ModelSpecification<eos::Polytrope>);
+    static_assert(ModelSpecification<surface::ConstantPressureSurface>);
+    static_assert(ModelSpecification<FixedTotalMass>);
+    static_assert(ModelSpecification<FixedCentralDensity>);
+    static_assert(ResolvedModelSpecification<eos::Polytrope>);
+    static_assert(ResolvedModelSpecification<surface::ConstantPressureSurface>);
+    static_assert(ResolvedModelSpecification<FixedTotalMass>);
+    static_assert(ResolvedModelSpecification<FixedCentralDensity>);
+} // namespace mean_field::models
+
+export namespace mean_field::integral {
+    using FixedTotalMass = models::FixedTotalMass;
+}
+
+export namespace mean_field::constraint {
+    using FixedCentralDensity = models::FixedCentralDensity;
+}

libmeanfield/interface/models/structure/polytropic.cppm

@@ -1,13 +1,12 @@
 module;
 
-#include <vector>
-
 #include <mfem.hpp>
 
 export module mean_field:model.structure.polytropic;
 
 export import :eos.polytrope;
 export import :model.structure.base;
+export import :seed.lane_emden;
 
 import :utils.misc;
 
@@ -28,40 +27,6 @@ export namespace mean_field::models::structure {
         void validate() const;
 
     private:
-        struct LaneEmdenPoint {
-            double coordinate{0.0};
-            double value{0.0};
-            double derivative{0.0};
-        };
-
-        struct LaneEmdenDerivative {
-            double value{0.0};
-            double derivative{0.0};
-        };
-
-        static void validateSeedRequest(const StructureSeedRequest &request);
-
-        [[nodiscard]] static LaneEmdenDerivative evaluateLaneEmdenRhs(
-            double coordinate,
-            double value,
-            double derivative,
-            double polytropicIndex
-        );
-
-        [[nodiscard]] static LaneEmdenPoint takeLaneEmdenStep(
-            const LaneEmdenPoint &point,
-            double step,
-            double polytropicIndex
-        );
-
-        [[nodiscard]] static std::vector<LaneEmdenPoint> solveLaneEmden(double polytropicIndex);
-
-        [[nodiscard]] static double interpolateLaneEmdenValue(
-            const std::vector<LaneEmdenPoint> &solution,
-            double coordinate,
-            std::size_t &lowerIndex
-        );
-
         eos::Polytrope m_equationOfState;
         double m_targetMass;
     };

libmeanfield/interface/models/typed_stellar_model.cppm

@@ -0,0 +1,66 @@
+module;
+
+#include <concepts>
+#include <cstddef>
+#include <span>
+#include <type_traits>
+#include <utility>
+
+export module mean_field:model.typed_stellar;
+
+export import :model.specifications;
+
+export namespace mean_field::model {
+    template <typename SpecificationSet> class StellarModel;
+
+    template <models::ModelSpecification... CanonicalSpecifications>
+    class StellarModel<models::detail::SpecificationSetStorage<CanonicalSpecifications...>> final {
+    public:
+        using SpecificationTypes = models::detail::SpecificationSetStorage<CanonicalSpecifications...>;
+        using OperatorSignature  = models::SpecificationOperatorSignature<SpecificationTypes>;
+        using Storage            = models::Model<CanonicalSpecifications...>;
+
+        static constexpr std::size_t specificationCount                        = sizeof...(CanonicalSpecifications);
+        static constexpr bool symbolicallySquare                               = Storage::symbolicallySquare;
+        static constexpr bool hasCompleteEquilibriumCompiler                   = Storage::hasCompleteRootCompiler;
+        static constexpr models::EquilibriumSystemCompilation compilationClass = Storage::compilationClass;
+
+        template <typename... Arguments>
+            requires std::constructible_from<
+                Storage,
+                Arguments...>
+        explicit StellarModel(Arguments &&...arguments) : m_specifications(std::forward<Arguments>(arguments)...) {
+        }
+
+        template <models::ModelSpecification Specification>
+            requires Storage::template
+        containsSpecification<Specification> [[nodiscard]] const Specification &specification() const noexcept {
+            return m_specifications.template specification<Specification>();
+        }
+
+        template <models::ModelSpecification Specification>
+        static constexpr bool containsSpecification = Storage::template containsSpecification<Specification>;
+
+        [[nodiscard]] static constexpr std::span<const models::RuntimeSpecificationDescriptor>
+        runtimeSpecificationDescriptors() noexcept {
+            return Storage::runtimeSpecificationDescriptors();
+        }
+
+    private:
+        Storage m_specifications;
+    };
+
+    template <models::ResolvedModelSpecification... Specifications>
+        requires models::ValidModelSpecificationPack<std::remove_cvref_t<Specifications>...>
+    StellarModel(Specifications &&...)
+        -> StellarModel<models::SpecificationSet<std::remove_cvref_t<Specifications>...>>;
+
+    namespace detail {
+        template <typename Candidate> struct IsStellarModel : std::false_type { };
+
+        template <typename SpecificationSet> struct IsStellarModel<StellarModel<SpecificationSet>> : std::true_type { };
+    } // namespace detail
+
+    template <typename Candidate>
+    concept StellarModelType = detail::IsStellarModel<std::remove_cvref_t<Candidate>>::value;
+} // namespace mean_field::model

libmeanfield/interface/operators/prepared_barotropic_closure_operator.cppm

@@ -84,15 +84,24 @@ export namespace mean_field::operators {
             mfem::Vector &actionTrue
         ) const;
 
+        void ApplyDisplacementActionFull(
+            const mfem::Vector &displacementVariationTrue,
+            mfem::Vector &actionTrue
+        ) const;
+
         struct ElementPAData {
+            int elementId{-1};
             mfem::Array<int> densityDofs;
             mfem::Array<int> enthalpyDofs;
+            mfem::Array<int> displacementDofs;
 
             mfem::DofTransformation *densityDofTransformation{nullptr};
             mfem::DofTransformation *enthalpyDofTransformation{nullptr};
+            mfem::DofTransformation *displacementDofTransformation{nullptr};
 
             mfem::DenseMatrix densityBasis;
             mfem::DenseMatrix enthalpyBasis;
+            mfem::DenseMatrix inverseElementJacobians;
 
             mfem::Vector weightedResidual;
             mfem::Vector quadratureWeights;
@@ -122,6 +131,14 @@ export namespace mean_field::operators {
         mutable mfem::Vector m_fullDisplacementAction;
         mutable mfem::Vector m_fullResidual;
 
+        mutable mfem::Vector m_displacementVariationLocal;
+        mutable mfem::Vector m_localDisplacementAction;
+        mutable mfem::Vector m_elementDisplacementVariation;
+        mutable mfem::Vector m_quadratureDisplacementAction;
+        mutable mfem::Vector m_elementDisplacementAction;
+        mutable mfem::DenseMatrix m_referenceDShape;
+        mutable mfem::DenseMatrix m_referenceDisplacementJacobian;
+
         std::uint64_t m_preparationCount{0};
         bool m_isPrepared{false};
     };

libmeanfield/interface/operators/prepared_central_density.cppm

@@ -0,0 +1,264 @@
+module;
+
+#include <algorithm>
+#include <cmath>
+#include <compare>
+#include <cstdint>
+#include <optional>
+#include <utility>
+
+#include <mfem.hpp>
+
+export module mean_field:operators.prepared_central_density;
+
+export import :field.mfem;
+export import :model.compiled_fixed_central_density;
+
+export namespace mean_field::operators {
+    struct CentralDensityDependencyStamp final {
+        std::uint64_t identity{0};
+        std::uint64_t revision{0};
+
+        constexpr auto operator<=>(const CentralDensityDependencyStamp &) const = default;
+    };
+
+    struct CentralDensityDependencies final {
+        CentralDensityDependencyStamp enthalpy;
+
+        constexpr auto operator<=>(const CentralDensityDependencies &) const = default;
+    };
+
+    struct PreparedCentralDensityReport final {
+        bool refreshedCentralEnthalpy{false};
+        bool refreshedBorder{false};
+        bool assembledResidual{false};
+
+        [[nodiscard]] bool DidAnyWork() const noexcept {
+            return refreshedCentralEnthalpy || refreshedBorder || assembledResidual;
+        }
+
+        constexpr auto operator<=>(const PreparedCentralDensityReport &) const = default;
+    };
+
+    struct CentralDensityConstraintReport final {
+        double targetDensity;
+        double achievedDensity;
+        double targetEnthalpy;
+        double achievedEnthalpy;
+        double enthalpyResidual;
+        double scaledResidual;
+    };
+
+    struct CentralDensityJacobianInput final {
+        const mfem::Vector &enthalpyVariation;
+        double borderVariation;
+    };
+
+    struct CentralDensityJacobianOutput final {
+        mfem::Vector &enthalpyAction;
+        mfem::Vector &phaseAction;
+    };
+
+    struct CentralDensityJacobianTransposeInput final {
+        const mfem::Vector &enthalpyResidualDual;
+        double phaseResidualDual;
+    };
+
+    struct CentralDensityJacobianTransposeOutput final {
+        mfem::Vector &enthalpyDual;
+        mfem::Vector &borderDual;
+    };
+
+    /*
+     * Bordered central-density phase condition
+     *
+     *   R_c(h) = h(0) - h(rho_c,target),
+     *   R_h    <- R_h + lambda_c e_c.
+     *
+     * The point functional e_c selects the unique scalar H1 vertex at the
+     * computational origin. Its coordinate transpose supplies the border
+     * column, so this contribution is algebraically symmetric before any
+     * independent scaling is applied by a solver.
+     */
+    class PreparedCentralDensityConstraint final {
+    public:
+        PreparedCentralDensityConstraint(
+            field::FieldPointDofMap centerDof,
+            const MPI_Comm communicator
+        )
+            : m_centerDof(std::move(centerDof)),
+              m_communicator(communicator) {
+        }
+
+        PreparedCentralDensityReport Prepare(
+            const models::CompiledFixedCentralDensity &constraint,
+            const mfem::Vector &enthalpy,
+            const double border,
+            const CentralDensityDependencies &dependencies
+        ) {
+            MFEM_VERIFY(
+                enthalpy.Size() == m_centerDof.field_size(),
+                "The central-density phase received an enthalpy vector with the wrong size."
+            );
+            MFEM_VERIFY(std::isfinite(border), "The central-density phase received a non-finite border value.");
+
+            const bool wasPrepared = m_isPrepared;
+            PreparedCentralDensityReport report;
+
+            if (!wasPrepared || dependencies.enthalpy != m_preparedDependencies.enthalpy) {
+                double localCentralEnthalpy = 0.0;
+                for (const int reducedDof : m_centerDof.reduced_dofs()) {
+                    const double value = enthalpy(reducedDof);
+                    MFEM_VERIFY(std::isfinite(value), "The central enthalpy is non-finite.");
+                    localCentralEnthalpy += value;
+                }
+                m_centralEnthalpy               = GlobalSum(localCentralEnthalpy);
+                report.refreshedCentralEnthalpy = true;
+            }
+
+            if (!wasPrepared || border != m_border) {
+                m_border               = border;
+                report.refreshedBorder = true;
+            }
+
+            const bool targetChanged =
+                !m_constraint.has_value() || constraint.targetDensity() != m_constraint->targetDensity();
+            if (targetChanged) {
+                m_constraint = constraint;
+            }
+
+            if (report.refreshedCentralEnthalpy || report.refreshedBorder || targetChanged) {
+                m_cachedPhaseResidual    = m_centralEnthalpy - m_constraint->targetEnthalpy().value();
+                report.assembledResidual = true;
+            }
+
+            m_preparedDependencies = dependencies;
+            m_isPrepared           = true;
+            ++m_preparationCount;
+            return report;
+        }
+
+        void AddResidual(
+            mfem::Vector &enthalpyResidual,
+            mfem::Vector &phaseResidual
+        ) const {
+            VerifyPrepared();
+            VerifyOutputSizes(enthalpyResidual, phaseResidual);
+            for (const int reducedDof : m_centerDof.reduced_dofs()) {
+                enthalpyResidual(reducedDof) += m_border;
+            }
+            phaseResidual(0) = m_cachedPhaseResidual;
+        }
+
+        void ApplyJacobian(
+            const CentralDensityJacobianInput &input,
+            CentralDensityJacobianOutput output
+        ) const {
+            VerifyPrepared();
+            MFEM_VERIFY(
+                input.enthalpyVariation.Size() == m_centerDof.field_size(),
+                "The central-density Jacobian received an enthalpy direction with the wrong size."
+            );
+            VerifyOutputSizes(output.enthalpyAction, output.phaseAction);
+
+            double localPhaseAction = 0.0;
+            for (const int reducedDof : m_centerDof.reduced_dofs()) {
+                output.enthalpyAction(reducedDof) += input.borderVariation;
+                localPhaseAction += input.enthalpyVariation(reducedDof);
+            }
+            output.phaseAction(0) = GlobalSum(localPhaseAction);
+            ++m_jacobianApplicationCount;
+        }
+
+        void ApplyJacobianTranspose(
+            const CentralDensityJacobianTransposeInput &input,
+            CentralDensityJacobianTransposeOutput output
+        ) const {
+            VerifyPrepared();
+            MFEM_VERIFY(
+                input.enthalpyResidualDual.Size() == m_centerDof.field_size(),
+                "The central-density transpose received an enthalpy residual dual with the wrong size."
+            );
+            MFEM_VERIFY(
+                output.enthalpyDual.Size() == m_centerDof.field_size() && output.borderDual.Size() == 1,
+                "The central-density transpose received output vectors with the wrong size."
+            );
+
+            double localBorderDual = 0.0;
+            for (const int reducedDof : m_centerDof.reduced_dofs()) {
+                output.enthalpyDual(reducedDof) += input.phaseResidualDual;
+                localBorderDual += input.enthalpyResidualDual(reducedDof);
+            }
+            output.borderDual(0) += GlobalSum(localBorderDual);
+            ++m_transposeApplicationCount;
+        }
+
+        [[nodiscard]] CentralDensityConstraintReport GetConstraintReport() const {
+            VerifyPrepared();
+            const double targetEnthalpy = m_constraint->targetEnthalpy().value();
+            const double scale          = std::max(std::abs(targetEnthalpy), 1.0e-300);
+            return {
+                .targetDensity = m_constraint->targetDensity().value(),
+                .achievedDensity =
+                    m_constraint->densityFromEnthalpy(dimensions::SpecificEnthalpyValue{m_centralEnthalpy}).value(),
+                .targetEnthalpy   = targetEnthalpy,
+                .achievedEnthalpy = m_centralEnthalpy,
+                .enthalpyResidual = m_cachedPhaseResidual,
+                .scaledResidual   = m_cachedPhaseResidual / scale
+            };
+        }
+
+        [[nodiscard]] bool IsPrepared() const noexcept {
+            return m_isPrepared;
+        }
+
+        [[nodiscard]] const field::FieldPointDofMap &GetCenterDof() const noexcept {
+            return m_centerDof;
+        }
+
+        [[nodiscard]] std::uint64_t GetPreparationCount() const noexcept {
+            return m_preparationCount;
+        }
+
+        [[nodiscard]] std::uint64_t GetJacobianApplicationCount() const noexcept {
+            return m_jacobianApplicationCount;
+        }
+
+        [[nodiscard]] std::uint64_t GetTransposeApplicationCount() const noexcept {
+            return m_transposeApplicationCount;
+        }
+
+    private:
+        [[nodiscard]] double GlobalSum(const double localValue) const {
+            double globalValue = 0.0;
+            MPI_Allreduce(&localValue, &globalValue, 1, MPI_DOUBLE, MPI_SUM, m_communicator);
+            return globalValue;
+        }
+
+        void VerifyOutputSizes(
+            const mfem::Vector &enthalpyOutput,
+            const mfem::Vector &phaseOutput
+        ) const {
+            MFEM_VERIFY(
+                enthalpyOutput.Size() == m_centerDof.field_size() && phaseOutput.Size() == 1,
+                "The central-density phase received output vectors with the wrong size."
+            );
+        }
+
+        void VerifyPrepared() const {
+            MFEM_VERIFY(m_isPrepared, "The central-density phase must be prepared before application.");
+        }
+
+        field::FieldPointDofMap m_centerDof;
+        MPI_Comm m_communicator;
+        std::optional<models::CompiledFixedCentralDensity> m_constraint;
+        CentralDensityDependencies m_preparedDependencies;
+        double m_centralEnthalpy{0.0};
+        double m_border{0.0};
+        double m_cachedPhaseResidual{0.0};
+        std::uint64_t m_preparationCount{0};
+        mutable std::uint64_t m_jacobianApplicationCount{0};
+        mutable std::uint64_t m_transposeApplicationCount{0};
+        bool m_isPrepared{false};
+    };
+} // namespace mean_field::operators

libmeanfield/interface/operators/prepared_central_density_stellar_equilibrium.cppm

@@ -0,0 +1,117 @@
+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_constraint.cppm

@@ -0,0 +1,37 @@
+module;
+
+#include <concepts>
+#include <type_traits>
+
+#include <mfem.hpp>
+
+export module mean_field:operators.prepared_constraint;
+
+export import :model.compiled_fixed_mass;
+
+export namespace mean_field::operators {
+    template <typename Candidate>
+    concept PreparedConstraint = requires(
+        std::remove_cvref_t<Candidate> &prepared,
+        const std::remove_cvref_t<Candidate> &constPrepared,
+        const typename std::remove_cvref_t<Candidate>::CompiledConstraintType &constraint,
+        const typename std::remove_cvref_t<Candidate>::Dependencies &dependencies,
+        const typename std::remove_cvref_t<Candidate>::JacobianInput &jacobianInput,
+        typename std::remove_cvref_t<Candidate>::JacobianTransposeOutput transposeOutput,
+        const mfem::Vector &residualDual,
+        mfem::Vector &result
+    ) {
+        typename std::remove_cvref_t<Candidate>::SpecificationType;
+        typename std::remove_cvref_t<Candidate>::CompiledConstraintType;
+        typename std::remove_cvref_t<Candidate>::Dependencies;
+        typename std::remove_cvref_t<Candidate>::Report;
+        typename std::remove_cvref_t<Candidate>::JacobianInput;
+        typename std::remove_cvref_t<Candidate>::JacobianTransposeOutput;
+        requires models::CompiledConstraint<typename std::remove_cvref_t<Candidate>::CompiledConstraintType>;
+        { prepared.Prepare(constraint, dependencies) } -> std::same_as<typename std::remove_cvref_t<Candidate>::Report>;
+        { constPrepared.BuildResidual(result) } -> std::same_as<void>;
+        { constPrepared.ApplyJacobian(jacobianInput, result) } -> std::same_as<void>;
+        { constPrepared.ApplyJacobianTranspose(residualDual, transposeOutput) } -> std::same_as<void>;
+        { constPrepared.IsPrepared() } noexcept -> std::same_as<bool>;
+    };
+} // namespace mean_field::operators

libmeanfield/interface/operators/prepared_gravity_displacement_force.cppm

@@ -2,6 +2,7 @@ module;
 
 #include <compare>
 #include <cstdint>
+#include <vector>
 
 #include <mfem.hpp>
 
@@ -105,6 +106,29 @@ export namespace mean_field::operators {
 
     private:
         void VerifyPrepared() const;
+        void PrepareElementData();
+        void ApplyPreparedCompleteJacobianActionTrue(
+            const mfem::Vector &densityVariationTrue,
+            const mfem::Vector &displacementVariationTrue,
+            const mfem::Vector &gravityGradientVariationTrue,
+            mfem::Vector &actionTrue
+        ) const;
+
+        struct ElementPAData {
+            int elementId{-1};
+            mfem::Array<int> densityDofs;
+            mfem::Array<int> gravityGradientDofs;
+            mfem::Array<int> displacementDofs;
+            mfem::DofTransformation *densityDofTransformation{nullptr};
+            mfem::DofTransformation *gravityGradientDofTransformation{nullptr};
+            mfem::DofTransformation *displacementDofTransformation{nullptr};
+            const mfem::IntegrationRule *integrationRule{nullptr};
+            mfem::DenseMatrix mappingJacobians;
+            mfem::DenseMatrix inverseMeshJacobians;
+            mfem::DenseMatrix baseGravityReferenceValues;
+            mfem::Vector baseDensityValues;
+            mfem::Vector referenceWeights;
+        };
 
         const fem::FEM &m_fem;
         const mapping::DomainMapper &m_domainMapper;
@@ -112,11 +136,34 @@ export namespace mean_field::operators {
 
         context::gravity_field::GravityFieldRevisions m_preparedRevisions;
         mfem::Vector m_cachedResidual;
+        std::vector<ElementPAData> m_elements;
 
         mutable mfem::Vector m_densityVariationTrue;
         mutable mfem::Vector m_gravityGradientVariationTrue;
         mutable mfem::Vector m_displacementVariationTrue;
         mutable mfem::Vector m_actionTrue;
+        mutable mfem::Vector m_densityVariationLocal;
+        mutable mfem::Vector m_gravityGradientVariationLocal;
+        mutable mfem::Vector m_displacementVariationLocal;
+        mutable mfem::Vector m_localAction;
+        mutable mfem::Vector m_elementDensityVariation;
+        mutable mfem::Vector m_elementGravityGradientVariation;
+        mutable mfem::Vector m_elementDisplacementVariation;
+        mutable mfem::Vector m_elementAction;
+        mutable mfem::Vector m_densityShape;
+        mutable mfem::Vector m_displacementShape;
+        mutable mfem::Vector m_baseGravityReferenceValue;
+        mutable mfem::Vector m_gravityVariationReferenceValue;
+        mutable mfem::Vector m_mappedBaseGravity;
+        mutable mfem::Vector m_mappedGravityVariation;
+        mutable mfem::Vector m_mappedGeometryVariation;
+        mutable mfem::Vector m_forceValue;
+        mutable mfem::DenseMatrix m_gravityGradientShape;
+        mutable mfem::DenseMatrix m_referenceDisplacementDShape;
+        mutable mfem::DenseMatrix m_referenceDisplacementJacobian;
+        mutable mfem::DenseMatrix m_displacementJacobianVariation;
+        mutable mfem::DenseMatrix m_mappingJacobian;
+        mutable mfem::DenseMatrix m_inverseMeshJacobian;
 
         std::uint64_t m_residualPreparationCount{0};
         mutable std::uint64_t m_residualApplicationCount{0};

libmeanfield/interface/operators/prepared_gravity_source.cppm

@@ -22,6 +22,11 @@ export namespace mean_field::operators {
             const mfem::Vector &density,
             mfem::Vector &action
         ) const override;
+        void MultDisplacementVariationTrue(
+            const mfem::Vector &densityTrue,
+            const mfem::Vector &displacementVariationTrue,
+            mfem::Vector &actionVariationTrue
+        ) const;
 
         [[nodiscard]] bool IsPrepared() const noexcept;
         [[nodiscard]] std::uint64_t GetPreparationCount() const noexcept;
@@ -41,13 +46,18 @@ export namespace mean_field::operators {
 
             mfem::Array<int> density_dofs;
             mfem::Array<int> potential_dofs;
+            mfem::Array<int> displacement_dofs;
 
             mfem::DofTransformation *density_dof_transformation{nullptr};
             mfem::DofTransformation *potential_dof_transformation{nullptr};
+            mfem::DofTransformation *displacement_dof_transformation{nullptr};
+
+            const mfem::IntegrationRule *integration_rule{nullptr};
 
             // Rows are quadrature points; columns are element DOFs.
             mfem::DenseMatrix density_basis;
             mfem::DenseMatrix potential_basis;
+            mfem::DenseMatrix inverse_element_jacobians;
 
             // Contains quadrature weight, mesh Jacobian, mapped Jacobian,
             // and 4*pi*G.
@@ -67,6 +77,15 @@ export namespace mean_field::operators {
         mutable mfem::Vector m_density_true;
         mutable mfem::Vector m_potential_true;
         mutable mfem::Vector m_action_true;
+        mutable mfem::Vector m_density_local;
+        mutable mfem::Vector m_displacement_variation_local;
+        mutable mfem::Vector m_local_variation_action;
+        mutable mfem::Vector m_element_density;
+        mutable mfem::Vector m_element_displacement_variation;
+        mutable mfem::Vector m_quadrature_variation_action;
+        mutable mfem::Vector m_element_variation_action;
+        mutable mfem::DenseMatrix m_reference_displacement_dshape;
+        mutable mfem::DenseMatrix m_reference_displacement_jacobian;
         mfem::Vector m_displacement_true;
 
         std::uint64_t m_preparation_count{0};

libmeanfield/interface/operators/prepared_hdiv_mass.cppm

@@ -2,6 +2,7 @@ module;
 #include <cstdint>
 #include <memory>
 #include <mfem.hpp>
+#include <vector>
 
 export module mean_field:operators.prepared_hdiv_mass;
 export import :fem;
@@ -21,6 +22,11 @@ export namespace mean_field::operators {
             const mfem::Vector &gravity_gradient,
             mfem::Vector &action
         ) const override;
+        void MultDisplacementVariationTrue(
+            const mfem::Vector &gravityGradientTrue,
+            const mfem::Vector &displacementVariationTrue,
+            mfem::Vector &actionVariationTrue
+        ) const;
         void AssembleDiagonal(mfem::Vector &diagonal) const override;
         void AssembleTrueDiagonal(mfem::Vector &diagonal) const;
 
@@ -31,6 +37,21 @@ export namespace mean_field::operators {
         [[nodiscard]] const field::FieldDofMap &GetDisplacementMap() const noexcept;
 
     private:
+        struct ElementVariationData {
+            int elementId{-1};
+            mfem::Array<int> gravityGradientDofs;
+            mfem::Array<int> displacementDofs;
+            mfem::Array<int> compactificationDofs;
+            mfem::DofTransformation *gravityGradientDofTransformation{nullptr};
+            mfem::DofTransformation *displacementDofTransformation{nullptr};
+            mfem::Vector baseDisplacement;
+            mfem::Vector compactification;
+            const mfem::IntegrationRule *integrationRule{nullptr};
+            mfem::DenseMatrix frozenMappingData;
+        };
+
+        void PrepareVariationData();
+
         const fem::FEM &m_fem;
         const mapping::DomainMapper &m_domain_mapper;
 
@@ -48,6 +69,21 @@ export namespace mean_field::operators {
         mutable mfem::Vector m_action_true;
         mutable mfem::Vector m_domain_action_true;
         mfem::Vector m_displacement_true;
+        std::vector<ElementVariationData> m_variationElements;
+
+        mutable mapping::DomainMapper::Workspace m_variationWorkspace;
+        mutable mapping::VolumeMappingContext m_baseMappingContext;
+        mutable mapping::VolumeMappingVariation m_mappingVariation;
+        mutable mfem::Vector m_gravityGradientLocal;
+        mutable mfem::Vector m_displacementVariationLocal;
+        mutable mfem::Vector m_localVariationAction;
+        mutable mfem::Vector m_elementGravityGradient;
+        mutable mfem::Vector m_elementDisplacementVariation;
+        mutable mfem::Vector m_elementVariationAction;
+        mutable mfem::Vector m_gravityGradientValue;
+        mutable mfem::Vector m_massTensorVariationAction;
+        mutable mfem::DenseMatrix m_gravityGradientShape;
+        mutable mfem::DenseMatrix m_massTensorVariation;
         std::uint64_t m_preparation_count{0};
         bool m_is_prepared{false};
     };

libmeanfield/interface/operators/prepared_mass_normalization.cppm

@@ -9,7 +9,9 @@ export module mean_field:operators.prepared_mass_normalization;
 
 export import :fem;
 export import :mapping.domain_mapper;
+export import :model.compiled_fixed_mass;
 export import :operators.context.gravity_field;
+export import :operators.prepared_constraint;
 export import :utils.blocks;
 
 export namespace mean_field::operators {
@@ -33,6 +35,16 @@ export namespace mean_field::operators {
         double targetMass{0.0};
     };
 
+    struct FixedMassJacobianInput final {
+        const mfem::Vector &densityVariation;
+        const mfem::Vector &displacementVariation;
+    };
+
+    struct FixedMassJacobianTransposeOutput final {
+        mfem::Vector &densityDual;
+        mfem::Vector &displacementDual;
+    };
+
     struct PreparedMassNormalizationReport final {
         bool rebuiltStaticPlan{false};
         bool refreshedGeometry{false};
@@ -43,12 +55,15 @@ export namespace mean_field::operators {
         [[nodiscard]] bool DidAnyWork() const noexcept {
             return rebuiltStaticPlan || refreshedGeometry || refreshedDensity || updatedTargetMass || assembledResidual;
         }
+
+        constexpr auto operator<=>(const PreparedMassNormalizationReport &) const = default;
     };
 
     struct PreparedMassNormalizationActionStatistics final {
         std::uint64_t densityApplications{0};
         std::uint64_t displacementApplications{0};
         std::uint64_t completeApplications{0};
+        std::uint64_t transposeApplications{0};
 
         constexpr auto operator<=>(const PreparedMassNormalizationActionStatistics &) const = default;
     };
@@ -66,6 +81,13 @@ export namespace mean_field::operators {
      */
     class PreparedMassNormalizationOperator final {
     public:
+        using SpecificationType       = models::FixedTotalMass;
+        using CompiledConstraintType  = models::CompiledFixedMass;
+        using Dependencies            = MassNormalizationDependencies;
+        using Report                  = PreparedMassNormalizationReport;
+        using JacobianInput           = FixedMassJacobianInput;
+        using JacobianTransposeOutput = FixedMassJacobianTransposeOutput;
+
         PreparedMassNormalizationOperator(
             const fem::FEM &f,
             const mapping::DomainMapper &domainMapper,
@@ -82,6 +104,11 @@ export namespace mean_field::operators {
             const MassNormalizationDependencies &dependencies
         );
 
+        PreparedMassNormalizationReport Prepare(
+            const models::CompiledFixedMass &constraint,
+            const MassNormalizationDependencies &dependencies
+        );
+
         void BuildResidual(mfem::Vector &residual) const;
 
         void ApplyDensityJacobianAction(
@@ -100,6 +127,22 @@ export namespace mean_field::operators {
             mfem::Vector &action
         ) const;
 
+        void ApplyJacobian(
+            const FixedMassJacobianInput &input,
+            mfem::Vector &action
+        ) const;
+
+        void ApplyCompleteJacobianTransposeAction(
+            double residualDual,
+            mfem::Vector &densityDual,
+            mfem::Vector &displacementDual
+        ) const;
+
+        void ApplyJacobianTranspose(
+            const mfem::Vector &residualDual,
+            FixedMassJacobianTransposeOutput output
+        ) const;
+
         [[nodiscard]] bool IsPrepared() const noexcept;
         [[nodiscard]] double GetCurrentMass() const;
         [[nodiscard]] double GetTargetMass() const;
@@ -145,6 +188,16 @@ export namespace mean_field::operators {
 
         [[nodiscard]] double EvaluateDisplacementActionLocal(const mfem::Vector &displacementVariation) const;
 
+        void AssembleDensityTransposeAction(
+            double residualDual,
+            mfem::Vector &densityDual
+        ) const;
+
+        void AssembleDisplacementTransposeAction(
+            double residualDual,
+            mfem::Vector &displacementDual
+        ) const;
+
         [[nodiscard]] double GlobalSum(double localValue) const;
 
         const fem::FEM &m_fem;
@@ -167,6 +220,10 @@ export namespace mean_field::operators {
         bool m_isPrepared{false};
     };
 
+    using PreparedFixedMass = PreparedMassNormalizationOperator;
+
+    static_assert(PreparedConstraint<PreparedFixedMass>);
+
     using MassNormalizationLayout = utils::blocks::form_layout<utils::blocks::barotropic_equilibrium_form>;
 
     class PreparedMassNormalizationJacobianOperator final : public mfem::Operator {
@@ -181,6 +238,11 @@ export namespace mean_field::operators {
             mfem::Vector &action
         ) const override;
 
+        void MultTranspose(
+            const mfem::Vector &residualDual,
+            mfem::Vector &stateDual
+        ) const override;
+
         [[nodiscard]] const MassNormalizationLayout &GetLayout() const noexcept;
 
     private:

libmeanfield/interface/operators/prepared_rotation_displacement_force.cppm

@@ -3,6 +3,7 @@ module;
 #include <compare>
 #include <cstdint>
 #include <optional>
+#include <vector>
 
 #include <mfem.hpp>
 
@@ -103,6 +104,25 @@ export namespace mean_field::operators {
 
     private:
         void VerifyPrepared() const;
+        void PrepareElementData();
+        void ApplyPreparedCompleteJacobianActionTrue(
+            const mfem::Vector &densityVariationTrue,
+            const mfem::Vector &displacementVariationTrue,
+            mfem::Vector &actionTrue
+        ) const;
+
+        struct ElementPAData {
+            int elementId{-1};
+            mfem::Array<int> densityDofs;
+            mfem::Array<int> displacementDofs;
+            mfem::DofTransformation *densityDofTransformation{nullptr};
+            mfem::DofTransformation *displacementDofTransformation{nullptr};
+            const mfem::IntegrationRule *integrationRule{nullptr};
+            mfem::DenseMatrix inverseElementJacobians;
+            mfem::DenseMatrix centrifugalAccelerations;
+            mfem::Vector baseDensityValues;
+            mfem::Vector quadratureWeights;
+        };
 
         const fem::FEM &m_fem;
         const mapping::DomainMapper &m_domainMapper;
@@ -111,9 +131,24 @@ export namespace mean_field::operators {
 
         std::optional<physics::RigidRotation> m_rotation;
         mfem::Vector m_cachedResidual;
+        std::vector<ElementPAData> m_elements;
         mutable mfem::Vector m_densityVariationTrue;
         mutable mfem::Vector m_displacementVariationTrue;
         mutable mfem::Vector m_actionTrue;
+        mutable mfem::Vector m_densityVariationLocal;
+        mutable mfem::Vector m_displacementVariationLocal;
+        mutable mfem::Vector m_localAction;
+        mutable mfem::Vector m_elementDensityVariation;
+        mutable mfem::Vector m_elementDisplacementVariation;
+        mutable mfem::Vector m_elementAction;
+        mutable mfem::Vector m_densityShape;
+        mutable mfem::Vector m_displacementShape;
+        mutable mfem::Vector m_physicalPositionVariation;
+        mutable mfem::Vector m_centrifugalAcceleration;
+        mutable mfem::Vector m_centrifugalAccelerationVariation;
+        mutable mfem::Vector m_weightedForce;
+        mutable mfem::DenseMatrix m_referenceDisplacementDShape;
+        mutable mfem::DenseMatrix m_referenceDisplacementJacobian;
 
         context::rotational_displacement_force::RotationalDisplacementForceDependencies m_preparedDependencies;
 

libmeanfield/interface/operators/prepared_stellar_equilibrium.cppm

@@ -15,6 +15,7 @@ export import :fem;
 export import :field.mfem;
 export import :mapping.domain_mapper;
 export import :model.stellar;
+export import :model.typed_stellar;
 export import :operators.context.gravity_field;
 export import :operators.gravity_field;
 export import :operators.gravity_field_jacobian;
@@ -23,6 +24,7 @@ export import :operators.prepared_displacement_residual;
 export import :operators.prepared_hydrostatic_equilibrium;
 export import :operators.prepared_mass_normalization;
 export import :operators.prepared_surface_constraint;
+export import :operators.root_manifest;
 export import :physics.rigid_rotation;
 export import :utils.blocks;
 
@@ -82,6 +84,16 @@ export namespace mean_field::operators {
     using StellarEquilibriumLayout =
         utils::blocks::form_layout<utils::blocks::surface_deformed_stellar_equilibrium_form>;
 
+    using StellarEquilibriumSpecificationModel =
+        model::StellarModel<models::SpecificationSet<eos::Polytrope, models::FixedTotalMass, surface::Isobaric>>;
+
+    using StellarEquilibriumSystemManifest = EquilibriumSystemManifest<
+        StellarEquilibriumSpecificationModel,
+        utils::blocks::surface_deformed_stellar_equilibrium_form,
+        utils::blocks::surface_deformed_stellar_equilibrium_jacobian_form>;
+
+    using StellarEquilibriumRootManifest = StellarEquilibriumSystemManifest;
+
     class PreparedStellarEquilibriumOperator final : public mfem::Operator {
     public:
         template <models::StellarModelType Model>
@@ -101,12 +113,27 @@ export namespace mean_field::operators {
                   f,
                   domainMapper,
                   stellarModel.equationOfState(),
-                  stellarModel.targetMass(),
+                  models::compileConstraint(models::FixedTotalMass{dimensions::MassValue{stellarModel.targetMass()}}),
                   PressureSurfaceConstraintView{stellarModel.compiledSurfaceConstraint()},
                   deformation::PreparedDomainDeformationRuntime{stellarModel.compileDomainDeformation(f)}
               ) {
         }
 
+        /*
+         * Authoritative construction path for a compiled equilibrium system.
+         * The caller owns the EOS and compiled surface constraint for this
+         * operator's lifetime; the remaining compiled contributions are
+         * transferred into the operator.
+         */
+        PreparedStellarEquilibriumOperator(
+            fem::FEM &f,
+            const mapping::DomainMapper &domainMapper,
+            const eos::Polytrope &equationOfState,
+            models::CompiledFixedMass fixedMassConstraint,
+            PressureSurfaceConstraintView surfaceConstraint,
+            deformation::PreparedDomainDeformationRuntime domainDeformation
+        );
+
         PreparedStellarEquilibriumOperator(const PreparedStellarEquilibriumOperator &)            = delete;
         PreparedStellarEquilibriumOperator &operator=(const PreparedStellarEquilibriumOperator &) = delete;
         PreparedStellarEquilibriumOperator(PreparedStellarEquilibriumOperator &&)                 = delete;
@@ -128,6 +155,12 @@ export namespace mean_field::operators {
         [[nodiscard]] bool IsPrepared() const noexcept;
         [[nodiscard]] double GetTargetMass() const noexcept;
         [[nodiscard]] const StellarEquilibriumLayout &GetLayout() const noexcept;
+        [[nodiscard]] const StellarEquilibriumRootManifest &GetRootManifest() const noexcept;
+        [[nodiscard]] RootStateView<utils::blocks::surface_deformed_stellar_equilibrium_form>
+        GetRootStateView(const mfem::Vector &state) const;
+        [[nodiscard]] ResidualView<utils::blocks::surface_deformed_stellar_equilibrium_form>
+        GetResidualView(mfem::Vector &residual) const;
+        [[nodiscard]] RootConstraintReport GetFixedMassReport() const;
         [[nodiscard]] const StellarEquilibriumDependencies &GetDependencies() const;
         [[nodiscard]] const PreparedStellarEquilibriumStatistics &GetStatistics() const noexcept;
 
@@ -160,16 +193,7 @@ export namespace mean_field::operators {
             fem::FEM &f,
             const mapping::DomainMapper &domainMapper,
             const eos::Polytrope &equationOfState,
-            double targetMass,
-            PressureSurfaceConstraintView surfaceConstraint,
-            deformation::PreparedDomainDeformationRuntime domainDeformation
-        );
-
-        PreparedStellarEquilibriumOperator(
-            fem::FEM &f,
-            const mapping::DomainMapper &domainMapper,
-            const eos::Polytrope &equationOfState,
-            double targetMass,
+            models::CompiledFixedMass fixedMassConstraint,
             PressureSurfaceConstraintView surfaceConstraint,
             ConstructionData constructionData
         );
@@ -177,7 +201,7 @@ export namespace mean_field::operators {
         void AssembleResidual();
         void VerifyPrepared() const;
 
-        StellarEquilibriumLayout m_layout;
+        StellarEquilibriumRootManifest m_rootManifest;
         mfem::Array<int> m_gravityStateOffsets;
 
         context::gravity_field::GravityFieldLinearizationContext m_gravityContext;
@@ -198,7 +222,7 @@ export namespace mean_field::operators {
         mfem::Vector m_generatedVolumeDisplacement;
         mfem::Vector m_fullMechanicalResidual;
         mfem::Vector m_cachedResidual;
-        double m_targetMass{0.0};
+        models::CompiledFixedMass m_fixedMassConstraint;
 
         mutable PreparedStellarEquilibriumStatistics m_statistics;
         bool m_isPrepared{false};

libmeanfield/interface/operators/prepared_surface_constraint.cppm

@@ -14,10 +14,10 @@ export import :field.mfem;
 export import :surface.compiled;
 
 namespace mean_field::operators::detail {
-    template <eos::ThermodynamicQuantityType Quantity> struct SingleQuantitySurfaceState final {
-        eos::QuantityValue<Quantity> quantityValue;
+    template <dimensions::ThermodynamicQuantityType Quantity> struct SingleQuantitySurfaceState final {
+        dimensions::QuantityValue<Quantity> quantityValue;
 
-        [[nodiscard]] eos::QuantityValue<Quantity> value(Quantity) const noexcept {
+        [[nodiscard]] dimensions::QuantityValue<Quantity> value(Quantity) const noexcept {
             return quantityValue;
         }
     };
@@ -37,7 +37,7 @@ export namespace mean_field::operators {
             typename std::remove_cvref_t<Candidate>::CarrierQuantity;
             typename std::remove_cvref_t<Candidate>::CarrierField;
             typename std::remove_cvref_t<Candidate>::SurfaceDependencies;
-        } && std::same_as<typename std::remove_cvref_t<Candidate>::PhysicalQuantity, eos::quantity::Pressure> &&
+        } && std::same_as<typename std::remove_cvref_t<Candidate>::PhysicalQuantity, dimensions::quantity::Pressure> &&
         std::same_as<
             typename std::remove_cvref_t<Candidate>::SurfaceDependencies::RowField,
             typename std::remove_cvref_t<Candidate>::CarrierField> &&
@@ -112,7 +112,7 @@ export namespace mean_field::operators {
 
             for (int surfaceIndex = 0; surfaceIndex < surfaceRows.size(); ++surfaceIndex) {
                 const detail::SingleQuantitySurfaceState<CarrierQuantity> state{
-                    eos::QuantityValue<CarrierQuantity>{surfaceState(surfaceIndex)}
+                    dimensions::QuantityValue<CarrierQuantity>{surfaceState(surfaceIndex)}
                 };
                 rowResidual(surfaceRows.reduced_dofs()[surfaceIndex]) =
                     static_cast<const Constraint *>(constraint)->residual(state);
@@ -132,10 +132,10 @@ export namespace mean_field::operators {
             for (int surfaceIndex = 0; surfaceIndex < surfaceRows.size(); ++surfaceIndex) {
                 const int reducedDof = surfaceRows.reduced_dofs()[surfaceIndex];
                 const detail::SingleQuantitySurfaceState<CarrierQuantity> state{
-                    eos::QuantityValue<CarrierQuantity>{surfaceState(surfaceIndex)}
+                    dimensions::QuantityValue<CarrierQuantity>{surfaceState(surfaceIndex)}
                 };
                 const detail::SingleQuantitySurfaceState<CarrierQuantity> variation{
-                    eos::QuantityValue<CarrierQuantity>{stateVariation(reducedDof)}
+                    dimensions::QuantityValue<CarrierQuantity>{stateVariation(reducedDof)}
                 };
                 rowAction(reducedDof) = static_cast<const Constraint *>(constraint)->jacobianAction(state, variation);
             }

libmeanfield/interface/operators/root_manifest.cppm

@@ -0,0 +1,601 @@
+module;
+
+#include <algorithm>
+#include <array>
+#include <cmath>
+#include <concepts>
+#include <cstddef>
+#include <optional>
+#include <span>
+#include <stdexcept>
+#include <string_view>
+#include <type_traits>
+
+#include <mfem.hpp>
+
+export module mean_field:operators.root_manifest;
+
+export import :model.compiled_fixed_mass;
+export import :model.compiled_fixed_central_density;
+export import :model.specifications;
+export import :utils.blocks;
+
+export namespace mean_field::operators {
+    enum class RootBlockKind { value, residual };
+    enum class RootBlockProvenance { physical_operator, model_specification };
+    enum class RootRowInjection { physical_equation, append_global, replace_carrier_rows };
+    enum class RootColumnPolicy { physical_state, existing_physical_multiplier, solver_border, no_column };
+    enum class RootScalePolicy { unscaled, target_relative };
+
+    struct RootBlockDescriptor final {
+        std::string_view stableId;
+        std::string_view symbol;
+        RootBlockKind kind;
+        RootBlockProvenance provenance;
+        std::string_view source;
+        RootRowInjection rowInjection;
+        RootColumnPolicy columnPolicy;
+        RootScalePolicy scalePolicy;
+        int canonicalIndex;
+        int offset;
+        int size;
+        double scale;
+    };
+
+    struct RootRowReplacementDescriptor final {
+        std::string_view stableId;
+        std::string_view sourceSpecification;
+        models::SpecificationRole role;
+        int carrierResidualBlock;
+        int replacedRowCount;
+    };
+
+    struct RootConstraintDescriptor final {
+        std::string_view stableId;
+        models::SpecificationRole role;
+        RootRowInjection rowInjection;
+        RootColumnPolicy columnPolicy;
+        int valueBlock;
+        int residualBlock;
+        int rowArity;
+        int columnArity;
+        double target;
+        std::optional<double> carrierTarget;
+        std::string_view targetUnits;
+        std::string_view residualUnits;
+        double residualScale;
+    };
+
+    struct CentralDensityManifestInput final {
+        double targetDensity;
+        double targetEnthalpy;
+        int centerDofCount;
+    };
+
+    struct RootConstraintReport final {
+        RootConstraintDescriptor descriptor;
+        double achieved;
+        double dimensionalResidual;
+        double scaledResidual;
+    };
+
+    namespace detail {
+        struct StaticRootBlockDescriptor final {
+            std::string_view stableId;
+            std::string_view symbol;
+            RootBlockProvenance provenance;
+            std::string_view source;
+            RootRowInjection rowInjection;
+            RootColumnPolicy columnPolicy;
+            RootScalePolicy scalePolicy;
+        };
+
+        template <typename Block> struct RootBlockTraits;
+
+#define MEAN_FIELD_PHYSICAL_VALUE_BLOCK(BlockType, StableId, Symbol)                                                   \
+    template <> struct RootBlockTraits<BlockType> {                                                                    \
+        static constexpr StaticRootBlockDescriptor descriptor{                                                         \
+            StableId,                                                                                                  \
+            Symbol,                                                                                                    \
+            RootBlockProvenance::physical_operator,                                                                    \
+            "stellar_equilibrium",                                                                                     \
+            RootRowInjection::physical_equation,                                                                       \
+            RootColumnPolicy::physical_state,                                                                          \
+            RootScalePolicy::unscaled                                                                                  \
+        };                                                                                                             \
+    }
+
+#define MEAN_FIELD_PHYSICAL_RESIDUAL_BLOCK(BlockType, StableId, Symbol)                                                \
+    template <> struct RootBlockTraits<BlockType> {                                                                    \
+        static constexpr StaticRootBlockDescriptor descriptor{                                                         \
+            StableId,                                                                                                  \
+            Symbol,                                                                                                    \
+            RootBlockProvenance::physical_operator,                                                                    \
+            "stellar_equilibrium",                                                                                     \
+            RootRowInjection::physical_equation,                                                                       \
+            RootColumnPolicy::no_column,                                                                               \
+            RootScalePolicy::unscaled                                                                                  \
+        };                                                                                                             \
+    }
+
+        MEAN_FIELD_PHYSICAL_VALUE_BLOCK(
+            utils::blocks::density::mass::value,
+            "density",
+            "rho"
+        );
+        MEAN_FIELD_PHYSICAL_VALUE_BLOCK(
+            utils::blocks::displacement::geometry::value,
+            "volume_displacement",
+            "d"
+        );
+        MEAN_FIELD_PHYSICAL_VALUE_BLOCK(
+            utils::blocks::surface_deformation::parameters::value,
+            "surface_deformation",
+            "q"
+        );
+        MEAN_FIELD_PHYSICAL_VALUE_BLOCK(
+            utils::blocks::gravity::gradient::value,
+            "gravity_gradient",
+            "g"
+        );
+        MEAN_FIELD_PHYSICAL_VALUE_BLOCK(
+            utils::blocks::gravity::poisson::value,
+            "gravity_potential",
+            "Phi"
+        );
+        MEAN_FIELD_PHYSICAL_VALUE_BLOCK(
+            utils::blocks::enthalpy::specific::value,
+            "specific_enthalpy",
+            "h"
+        );
+
+        MEAN_FIELD_PHYSICAL_RESIDUAL_BLOCK(
+            utils::blocks::gravity::gradient::residual,
+            "gravity_gradient_relation",
+            "R_g"
+        );
+        MEAN_FIELD_PHYSICAL_RESIDUAL_BLOCK(
+            utils::blocks::gravity::poisson::residual,
+            "poisson_balance",
+            "R_Phi"
+        );
+        MEAN_FIELD_PHYSICAL_RESIDUAL_BLOCK(
+            utils::blocks::density::mass::residual,
+            "barotropic_closure",
+            "R_rho"
+        );
+        MEAN_FIELD_PHYSICAL_RESIDUAL_BLOCK(
+            utils::blocks::displacement::geometry::residual,
+            "mechanical_balance",
+            "R_d"
+        );
+        MEAN_FIELD_PHYSICAL_RESIDUAL_BLOCK(
+            utils::blocks::surface_deformation::shape_equilibrium::residual,
+            "surface_shape_balance",
+            "R_q"
+        );
+        MEAN_FIELD_PHYSICAL_RESIDUAL_BLOCK(
+            utils::blocks::enthalpy::specific::residual,
+            "hydrostatic_balance",
+            "R_h"
+        );
+
+#undef MEAN_FIELD_PHYSICAL_VALUE_BLOCK
+#undef MEAN_FIELD_PHYSICAL_RESIDUAL_BLOCK
+
+        template <> struct RootBlockTraits<utils::blocks::fixed_total_mass::mass_normalization::value> {
+            static constexpr StaticRootBlockDescriptor descriptor{
+                "fixed_total_mass.multiplier",
+                "C",
+                RootBlockProvenance::model_specification,
+                "FixedTotalMass",
+                RootRowInjection::physical_equation,
+                RootColumnPolicy::existing_physical_multiplier,
+                RootScalePolicy::unscaled
+            };
+        };
+
+        template <> struct RootBlockTraits<utils::blocks::fixed_total_mass::mass_normalization::residual> {
+            static constexpr StaticRootBlockDescriptor descriptor{
+                "fixed_total_mass.residual",
+                "R_M",
+                RootBlockProvenance::model_specification,
+                "FixedTotalMass",
+                RootRowInjection::append_global,
+                RootColumnPolicy::no_column,
+                RootScalePolicy::target_relative
+            };
+        };
+
+        template <> struct RootBlockTraits<utils::blocks::fixed_central_density::central_value::value> {
+            static constexpr StaticRootBlockDescriptor descriptor{
+                "fixed_central_density.border",
+                "lambda_rho_c",
+                RootBlockProvenance::model_specification,
+                "FixedCentralDensity",
+                RootRowInjection::physical_equation,
+                RootColumnPolicy::solver_border,
+                RootScalePolicy::unscaled
+            };
+        };
+
+        template <> struct RootBlockTraits<utils::blocks::fixed_central_density::central_value::residual> {
+            static constexpr StaticRootBlockDescriptor descriptor{
+                "fixed_central_density.residual",         "R_rho_c",
+                RootBlockProvenance::model_specification, "FixedCentralDensity",
+                RootRowInjection::append_global,          RootColumnPolicy::no_column,
+                RootScalePolicy::target_relative
+            };
+        };
+
+        template <typename Block>
+        [[nodiscard]] constexpr double blockScale(
+            const double fixedMassScale,
+            const double centralDensityScale
+        ) noexcept {
+            if constexpr (std::same_as<Block, utils::blocks::fixed_total_mass::mass_normalization::residual>) {
+                return fixedMassScale;
+            } else if constexpr (std::same_as<Block, utils::blocks::fixed_central_density::central_value::residual>) {
+                return centralDensityScale;
+            } else {
+                return 1.0;
+            }
+        }
+
+        template <
+            RootBlockKind Kind,
+            typename... Blocks>
+        [[nodiscard]] std::array<
+            RootBlockDescriptor,
+            sizeof...(Blocks)>
+        makeBlockDescriptors(
+            const mfem::Array<int> &offsets,
+            const double fixedMassScale,
+            const double centralDensityScale,
+            utils::blocks::type_list<Blocks...>
+        ) {
+            std::array<RootBlockDescriptor, sizeof...(Blocks)> descriptors{};
+            int index = 0;
+            ((descriptors[index] =
+                  {.stableId       = RootBlockTraits<Blocks>::descriptor.stableId,
+                   .symbol         = RootBlockTraits<Blocks>::descriptor.symbol,
+                   .kind           = Kind,
+                   .provenance     = RootBlockTraits<Blocks>::descriptor.provenance,
+                   .source         = RootBlockTraits<Blocks>::descriptor.source,
+                   .rowInjection   = RootBlockTraits<Blocks>::descriptor.rowInjection,
+                   .columnPolicy   = RootBlockTraits<Blocks>::descriptor.columnPolicy,
+                   .scalePolicy    = RootBlockTraits<Blocks>::descriptor.scalePolicy,
+                   .canonicalIndex = index,
+                   .offset         = offsets[index],
+                   .size           = offsets[index + 1] - offsets[index],
+                   .scale          = blockScale<Blocks>(fixedMassScale, centralDensityScale)},
+              ++index),
+             ...);
+            return descriptors;
+        }
+
+        template <models::SpecifiedModelType Model>
+        inline constexpr bool hasCentralDensity = Model::template containsSpecification<models::FixedCentralDensity>;
+
+        template <models::SpecifiedModelType Model>
+        inline constexpr std::size_t rootConstraintCount = 2 + (hasCentralDensity<Model> ? 1 : 0);
+
+        template <
+            models::SpecifiedModelType Model,
+            typename Form>
+        [[nodiscard]] std::array<
+            RootConstraintDescriptor,
+            rootConstraintCount<Model>>
+        makeConstraintDescriptors(
+            const double targetMass,
+            const double targetSurfacePressure,
+            const double fixedMassScale,
+            const std::optional<CentralDensityManifestInput> centralDensity
+        ) {
+            std::array<RootConstraintDescriptor, rootConstraintCount<Model>> descriptors{};
+            descriptors[0] = {
+                .stableId      = "FixedTotalMass",
+                .role          = models::SpecificationRole::invariant,
+                .rowInjection  = RootRowInjection::append_global,
+                .columnPolicy  = RootColumnPolicy::existing_physical_multiplier,
+                .valueBlock    = models::FixedMassLayoutRequest::valueBlock<Form>().index,
+                .residualBlock = models::FixedMassLayoutRequest::residualBlock<Form>().index,
+                .rowArity      = 1,
+                .columnArity   = 1,
+                .target        = targetMass,
+                .carrierTarget = targetMass,
+                .targetUnits   = "mass",
+                .residualUnits = "mass",
+                .residualScale = fixedMassScale
+            };
+            descriptors[1] = {
+                .stableId     = "IsobaricSurface",
+                .role         = models::SpecificationRole::boundary_condition,
+                .rowInjection = RootRowInjection::replace_carrier_rows,
+                .columnPolicy = RootColumnPolicy::no_column,
+                .valueBlock   = -1,
+                .residualBlock =
+                    utils::blocks::get_residual_block<Form>(utils::blocks::enthalpy_field.specific_term).index,
+                .rowArity      = 0,
+                .columnArity   = 0,
+                .target        = targetSurfacePressure,
+                .carrierTarget = std::nullopt,
+                .targetUnits   = "pressure",
+                .residualUnits = "specific_enthalpy",
+                .residualScale = 1.0
+            };
+
+            if constexpr (hasCentralDensity<Model>) {
+                if (!centralDensity.has_value()) {
+                    throw std::invalid_argument(
+                        "A model containing FixedCentralDensity requires central-density manifest metadata."
+                    );
+                }
+                descriptors[2] = {
+                    .stableId      = "FixedCentralDensity",
+                    .role          = models::SpecificationRole::phase_condition,
+                    .rowInjection  = RootRowInjection::append_global,
+                    .columnPolicy  = RootColumnPolicy::solver_border,
+                    .valueBlock    = models::CentralDensityLayoutRequest::valueBlock<Form>().index,
+                    .residualBlock = models::CentralDensityLayoutRequest::residualBlock<Form>().index,
+                    .rowArity      = 1,
+                    .columnArity   = 1,
+                    .target        = centralDensity->targetDensity,
+                    .carrierTarget = centralDensity->targetEnthalpy,
+                    .targetUnits   = "density",
+                    .residualUnits = "specific_enthalpy",
+                    .residualScale = std::max(std::abs(centralDensity->targetEnthalpy), 1.0e-300)
+                };
+            } else if (centralDensity.has_value()) {
+                throw std::invalid_argument(
+                    "Central-density manifest metadata was provided to a model without FixedCentralDensity."
+                );
+            }
+            return descriptors;
+        }
+    } // namespace detail
+
+    template <typename Form> class RootStateView final {
+    public:
+        RootStateView(
+            const mfem::Vector &state,
+            const utils::blocks::form_layout<Form> &layout
+        )
+            : m_state(state),
+              m_layout(layout) {
+            if (state.Size() != layout.value_offsets().Last()) {
+                throw std::invalid_argument("RootStateView received a vector with the wrong size.");
+            }
+        }
+
+        template <typename Term> [[nodiscard]] mfem::Vector block(const Term &term) const {
+            constexpr auto valueBlock = utils::blocks::get_value_block<Form>(term);
+            return mfem::Vector(
+                const_cast<mfem::real_t *>(m_state.GetData()) + m_layout.offset(valueBlock), m_layout.size(valueBlock)
+            );
+        }
+
+        [[nodiscard]] const mfem::Vector &vector() const noexcept {
+            return m_state;
+        }
+
+    private:
+        const mfem::Vector &m_state;
+        const utils::blocks::form_layout<Form> &m_layout;
+    };
+
+    template <typename Form> class ResidualView final {
+    public:
+        ResidualView(
+            mfem::Vector &residual,
+            const utils::blocks::form_layout<Form> &layout
+        )
+            : m_residual(residual),
+              m_layout(layout) {
+            if (residual.Size() != layout.residual_offsets().Last()) {
+                throw std::invalid_argument("ResidualView received a vector with the wrong size.");
+            }
+        }
+
+        template <typename Term> [[nodiscard]] mfem::Vector block(const Term &term) const {
+            constexpr auto residualBlock = utils::blocks::get_residual_block<Form>(term);
+            return mfem::Vector(m_residual.GetData() + m_layout.offset(residualBlock), m_layout.size(residualBlock));
+        }
+
+        template <typename Term>
+        void assign(
+            const Term &term,
+            const mfem::Vector &source
+        ) const {
+            mfem::Vector destination = block(term);
+            if (destination.Size() != source.Size()) {
+                throw std::invalid_argument("ResidualView block assignment has the wrong size.");
+            }
+            destination = source;
+        }
+
+        [[nodiscard]] mfem::Vector &vector() const noexcept {
+            return m_residual;
+        }
+
+    private:
+        mfem::Vector &m_residual;
+        const utils::blocks::form_layout<Form> &m_layout;
+    };
+
+    template <models::SpecifiedModelType Model, typename Form, typename JacobianForm>
+        requires utils::blocks::valid_jacobian_form<Form, JacobianForm>
+    class CompiledRootManifest final {
+    public:
+        using ModelType                                                 = Model;
+        using FormType                                                  = Form;
+        using JacobianType                                              = JacobianForm;
+        using Layout                                                    = utils::blocks::form_layout<Form>;
+        using StateView                                                 = RootStateView<Form>;
+        using DirectionView                                             = RootStateView<Form>;
+        using RootResidualView                                          = ResidualView<Form>;
+
+        static constexpr models::ModelCompilationClass compilationClass = Model::compilationClass;
+        static constexpr bool symbolicallySquare                        = Model::symbolicallySquare;
+
+        CompiledRootManifest(
+            const std::array<
+                int,
+                Form::value_block_count> &valueSizes,
+            const std::array<
+                int,
+                Form::residual_block_count> &residualSizes,
+            const double targetMass,
+            const double targetSurfacePressure,
+            const int replacedSurfaceRowCount,
+            const std::optional<CentralDensityManifestInput> centralDensity = std::nullopt
+        )
+            : m_layout(
+                  valueSizes,
+                  residualSizes
+              ),
+              m_fixedMassScale(
+                  std::max(
+                      std::abs(targetMass),
+                      1.0e-300
+                  )
+              ),
+              m_centralDensityScale(
+                  centralDensity.has_value() ? std::max(
+                                                   std::abs(centralDensity->targetEnthalpy),
+                                                   1.0e-300
+                                               )
+                                             : 1.0
+              ),
+              m_valueBlocks(
+                  detail::makeBlockDescriptors<RootBlockKind::value>(
+                      m_layout.value_offsets(),
+                      m_fixedMassScale,
+                      m_centralDensityScale,
+                      typename Form::value_blocks{}
+                  )
+              ),
+              m_residualBlocks(
+                  detail::makeBlockDescriptors<RootBlockKind::residual>(
+                      m_layout.residual_offsets(),
+                      m_fixedMassScale,
+                      m_centralDensityScale,
+                      typename Form::residual_blocks{}
+                  )
+              ),
+              m_replacements{RootRowReplacementDescriptor{
+                  .stableId            = "isobaric_surface.replacement",
+                  .sourceSpecification = "IsobaricSurface",
+                  .role                = models::SpecificationRole::boundary_condition,
+                  .carrierResidualBlock =
+                      utils::blocks::get_residual_block<Form>(utils::blocks::enthalpy_field.specific_term).index,
+                  .replacedRowCount = replacedSurfaceRowCount
+              }},
+              m_constraints(
+                  detail::makeConstraintDescriptors<
+                      Model,
+                      Form>(
+                      targetMass,
+                      targetSurfacePressure,
+                      m_fixedMassScale,
+                      centralDensity
+                  )
+              ) {
+            if (replacedSurfaceRowCount < 0) {
+                throw std::invalid_argument(
+                    "An equilibrium-system manifest cannot contain a negative replacement-row count."
+                );
+            }
+            if (centralDensity.has_value() && centralDensity->centerDofCount < 0) {
+                throw std::invalid_argument(
+                    "An equilibrium-system manifest cannot contain a negative central-DOF count."
+                );
+            }
+            if constexpr (compilationClass == models::EquilibriumSystemCompilation::complete_equilibrium_system) {
+                if (m_layout.value_offsets().Last() != m_layout.residual_offsets().Last()) {
+                    throw std::invalid_argument(
+                        "A complete equilibrium system must have equal state and equation dimensions."
+                    );
+                }
+            }
+        }
+
+        [[nodiscard]] const Layout &layout() const noexcept {
+            return m_layout;
+        }
+
+        [[nodiscard]] StateView stateView(const mfem::Vector &state) const {
+            return {state, m_layout};
+        }
+
+        [[nodiscard]] DirectionView directionView(const mfem::Vector &direction) const {
+            return {direction, m_layout};
+        }
+
+        [[nodiscard]] RootResidualView residualView(mfem::Vector &residual) const {
+            return {residual, m_layout};
+        }
+
+        [[nodiscard]] std::span<const RootBlockDescriptor> valueBlocks() const noexcept {
+            return m_valueBlocks;
+        }
+
+        [[nodiscard]] std::span<const RootBlockDescriptor> residualBlocks() const noexcept {
+            return m_residualBlocks;
+        }
+
+        [[nodiscard]] std::span<const RootRowReplacementDescriptor> rowReplacements() const noexcept {
+            return m_replacements;
+        }
+
+        [[nodiscard]] std::span<const RootConstraintDescriptor> constraints() const noexcept {
+            return m_constraints;
+        }
+
+        [[nodiscard]] static constexpr std::span<const models::RuntimeSpecificationDescriptor>
+        specificationDescriptors() noexcept {
+            return Model::runtimeSpecificationDescriptors();
+        }
+
+        [[nodiscard]] RootConstraintReport fixedMassReport(const double achievedMass) const {
+            const RootConstraintDescriptor &descriptor = m_constraints[0];
+            const double residual                      = achievedMass - descriptor.target;
+            return {
+                .descriptor          = descriptor,
+                .achieved            = achievedMass,
+                .dimensionalResidual = residual,
+                .scaledResidual      = residual / descriptor.residualScale
+            };
+        }
+
+    private:
+        Layout m_layout;
+        double m_fixedMassScale;
+        double m_centralDensityScale;
+        std::array<RootBlockDescriptor, Form::value_block_count> m_valueBlocks;
+        std::array<RootBlockDescriptor, Form::residual_block_count> m_residualBlocks;
+        std::array<RootRowReplacementDescriptor, 1> m_replacements;
+        std::array<RootConstraintDescriptor, detail::rootConstraintCount<Model>> m_constraints;
+    };
+
+    // Physics-facing names for the public equilibrium-system boundary. The
+    // root-oriented names remain available while existing solver consumers
+    // migrate, but new APIs should expose these aliases.
+    using EquilibriumBlockKind                             = RootBlockKind;
+    using EquilibriumBlockProvenance                       = RootBlockProvenance;
+    using EquilibriumEquationInjection                     = RootRowInjection;
+    using EquilibriumGeneratedVariablePolicy               = RootColumnPolicy;
+    using EquilibriumScalePolicy                           = RootScalePolicy;
+    using EquilibriumBlockDescriptor                       = RootBlockDescriptor;
+    using EquilibriumEquationReplacementDescriptor         = RootRowReplacementDescriptor;
+    using EquilibriumSpecificationDescriptor               = RootConstraintDescriptor;
+    using EquilibriumSpecificationReport                   = RootConstraintReport;
+
+    template <typename Form> using EquilibriumStateView    = RootStateView<Form>;
+
+    template <typename Form> using EquilibriumResidualView = ResidualView<Form>;
+
+    template <models::SpecifiedModelType Model, typename Form, typename JacobianForm>
+        requires utils::blocks::valid_jacobian_form<Form, JacobianForm>
+    using EquilibriumSystemManifest = CompiledRootManifest<Model, Form, JacobianForm>;
+} // namespace mean_field::operators

libmeanfield/interface/operators/stellar_equilibrium_problem.cppm

@@ -0,0 +1,210 @@
+module;
+
+#include <concepts>
+#include <cstddef>
+#include <type_traits>
+#include <utility>
+
+#include <mfem.hpp>
+
+export module mean_field:operators.stellar_equilibrium_problem;
+
+export import :deformation.domain_deformation;
+export import :equilibrium.stellar_discretization;
+export import :model.typed_stellar;
+export import :operators.prepared_central_density_stellar_equilibrium;
+export import :surface.compiler;
+
+export namespace mean_field::equilibrium {
+    template <typename Candidate>
+    concept StellarEquilibriumModel = model::StellarModelType<Candidate> && requires {
+        requires std::remove_cvref_t<Candidate>::template containsSpecification<eos::Polytrope>;
+        requires std::remove_cvref_t<Candidate>::template containsSpecification<surface::Isobaric>;
+        requires std::remove_cvref_t<Candidate>::template containsSpecification<models::FixedTotalMass>;
+        requires std::remove_cvref_t<Candidate>::specificationCount ==
+                     3 + static_cast<std::size_t>(
+                             std::remove_cvref_t<Candidate>::template containsSpecification<models::FixedCentralDensity>
+                         );
+    };
+
+    template <StellarEquilibriumModel Model> class StellarEquilibriumProblem final {
+    public:
+        using ModelType = std::remove_cvref_t<Model>;
+
+        static constexpr bool hasFixedCentralDensity =
+            ModelType::template containsSpecification<models::FixedCentralDensity>;
+        static constexpr bool symbolicallySquare = ModelType::symbolicallySquare;
+
+        using PreparedOperatorType               = std::conditional_t<
+            hasFixedCentralDensity,
+            operators::PreparedCentralDensityStellarEquilibriumOperator,
+            operators::PreparedStellarEquilibriumOperator>;
+        using CompiledSurfaceConstraintType =
+            surface::CompiledPressureSurfaceConstraintT<surface::BarotropicSurfaceFormulation, eos::Polytrope>;
+
+        StellarEquilibriumProblem(
+            ModelType stellarModel,
+            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
+        )
+            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()),
+                  models::compileConstraint(
+                      m_stellarModel.template specification<models::FixedCentralDensity>(),
+                      m_stellarModel.template specification<eos::Polytrope>()
+                  )
+              ) {
+            VerifyProblem();
+        }
+
+        StellarEquilibriumProblem(const StellarEquilibriumProblem &)            = delete;
+        StellarEquilibriumProblem &operator=(const StellarEquilibriumProblem &) = delete;
+        StellarEquilibriumProblem(StellarEquilibriumProblem &&)                 = delete;
+        StellarEquilibriumProblem &operator=(StellarEquilibriumProblem &&)      = delete;
+
+        [[nodiscard]] const ModelType &GetStellarModel() const noexcept {
+            return m_stellarModel;
+        }
+
+        [[nodiscard]] const StellarDiscretization &GetDiscretization() const noexcept {
+            return m_discretization;
+        }
+
+        [[nodiscard]] const CompiledSurfaceConstraintType &GetCompiledSurfaceConstraint() const noexcept {
+            return m_compiledSurfaceConstraint;
+        }
+
+        [[nodiscard]] PreparedOperatorType &GetPreparedOperator() noexcept {
+            return m_preparedOperator;
+        }
+
+        [[nodiscard]] const PreparedOperatorType &GetPreparedOperator() const noexcept {
+            return m_preparedOperator;
+        }
+
+        [[nodiscard]] const auto &GetManifest() const noexcept {
+            return m_preparedOperator.GetRootManifest();
+        }
+
+        [[nodiscard]] const field::FieldBoundaryDofMap &GetPressureSurfaceRows() const noexcept {
+            if constexpr (hasFixedCentralDensity) {
+                return m_preparedOperator.GetPhysicalOperator().GetSurfaceConstraintOperator().GetSurfaceRows();
+            } else {
+                return m_preparedOperator.GetSurfaceConstraintOperator().GetSurfaceRows();
+            }
+        }
+
+        [[nodiscard]] int StateSize() const noexcept {
+            return m_preparedOperator.Width();
+        }
+
+        [[nodiscard]] int EquationSize() const noexcept {
+            return m_preparedOperator.Height();
+        }
+
+        [[nodiscard]] const mfem::Operator &GetLinearizationOperator() const noexcept {
+            return m_preparedOperator;
+        }
+
+        [[nodiscard]] auto Prepare(
+            const mfem::Vector &state,
+            const operators::StellarEquilibriumDependencies &dependencies,
+            const physics::RigidRotation &rotation
+        ) {
+            return m_preparedOperator.Prepare(state, dependencies, rotation);
+        }
+
+        void BuildResidual(mfem::Vector &residual) const {
+            m_preparedOperator.BuildResidual(residual);
+        }
+
+        void ApplyLinearization(
+            const mfem::Vector &direction,
+            mfem::Vector &action
+        ) const {
+            m_preparedOperator.Mult(direction, action);
+        }
+
+    private:
+        [[nodiscard]] static CompiledSurfaceConstraintType CompileSurfaceConstraint(const ModelType &stellarModel) {
+            return surface::compilePressureSurfaceConstraint<surface::BarotropicSurfaceFormulation>(
+                stellarModel.template specification<surface::Isobaric>(),
+                stellarModel.template specification<eos::Polytrope>()
+            );
+        }
+
+        [[nodiscard]] static deformation::PreparedDomainDeformationRuntime
+        CompileDefaultDomainDeformation(fem::FEM &finiteElementModel) {
+            MFEM_VERIFY(
+                finiteElementModel.mesh != nullptr,
+                "Default stellar domain-deformation compilation requires a physical mesh."
+            );
+            mfem::Vector referenceCenter(finiteElementModel.mesh->SpaceDimension());
+            referenceCenter = 0.0;
+
+            return deformation::PreparedDomainDeformationRuntime{deformation::compileDomainDeformation(
+                deformation::NodalRadialSurface{std::move(referenceCenter)},
+                deformation::PowerLawRadialInteriorExtension{}, deformation::FixedInfinityRadialVacuumExtension{},
+                finiteElementModel
+            )};
+        }
+
+        void VerifyProblem() const {
+            MFEM_VERIFY(symbolicallySquare, "A stellar equilibrium problem must be symbolically square.");
+            MFEM_VERIFY(
+                StateSize() == EquationSize(),
+                "The discretized stellar equilibrium problem has unequal state and equation dimensions."
+            );
+            MFEM_VERIFY(m_discretization.isCurrent(), "The stellar equilibrium problem has a stale discretization.");
+        }
+
+        ModelType m_stellarModel;
+        StellarDiscretization m_discretization;
+        CompiledSurfaceConstraintType m_compiledSurfaceConstraint;
+        PreparedOperatorType m_preparedOperator;
+    };
+
+    template <StellarEquilibriumModel Model>
+    [[nodiscard]] auto discretize(
+        Model &&stellarModel,
+        const StellarDiscretization discretization
+    ) {
+        using ModelType = std::remove_cvref_t<Model>;
+        return StellarEquilibriumProblem<ModelType>{std::forward<Model>(stellarModel), discretization};
+    }
+
+    template <StellarEquilibriumModel Model>
+    [[nodiscard]] auto discretize(
+        Model &&stellarModel,
+        fem::FEM &finiteElementModel
+    ) {
+        return discretize(std::forward<Model>(stellarModel), StellarDiscretization{finiteElementModel});
+    }
+} // namespace mean_field::equilibrium

libmeanfield/interface/operators/stellar_equilibrium_system.cppm

@@ -0,0 +1,26 @@
+module;
+
+#include <type_traits>
+#include <utility>
+
+export module mean_field:operators.stellar_equilibrium_system;
+
+export import :operators.stellar_equilibrium_problem;
+
+export namespace mean_field::equilibrium {
+    // Transitional source-compatible names. New code should use
+    // StellarEquilibriumProblem and equilibrium::discretize.
+    template <typename Candidate>
+    concept CurrentlySupportedStellarModel                                  = StellarEquilibriumModel<Candidate>;
+
+    template <StellarEquilibriumModel Model> using StellarEquilibriumSystem = StellarEquilibriumProblem<Model>;
+
+    template <StellarEquilibriumModel Model>
+    [[nodiscard]] auto makeStellarEquilibriumSystem(
+        fem::FEM &finiteElementModel,
+        const mapping::DomainMapper &domainMapper,
+        Model &&stellarModel
+    ) {
+        return discretize(std::forward<Model>(stellarModel), StellarDiscretization{finiteElementModel, domainMapper});
+    }
+} // namespace mean_field::equilibrium

libmeanfield/interface/physics/gravity.cppm

@@ -6,6 +6,12 @@ export module mean_field:physics.gravity;
 export import :fem;
 
 export namespace mean_field::physics {
+    struct GravitySolveOptions final {
+        double relativeTolerance{1.0e-12};
+        double absoluteTolerance{1.0e-15};
+        int maximumIterations{1000};
+    };
+
     struct GravitySolution {
         mfem::ParGridFunction gradPhi;
         mfem::ParGridFunction phi;
@@ -16,6 +22,13 @@ export namespace mean_field::physics {
         }
     };
 
+    GravitySolution solve_gravity_field(
+        fem::FEM &f,
+        const GravitySolveOptions &options,
+        const mfem::GridFunction &rho,
+        const mfem::GridFunction &displacement
+    );
+
     GravitySolution solve_gravity_field(
         fem::FEM &f,
         const utils::Args &args,

libmeanfield/interface/seed/lane_emden.cppm

@@ -0,0 +1,120 @@
+module;
+
+#include <cmath>
+#include <concepts>
+#include <optional>
+#include <stdexcept>
+#include <type_traits>
+#include <vector>
+
+#include <mfem.hpp>
+
+export module mean_field:seed.lane_emden;
+
+export import :dimensions.quantities;
+export import :eos.polytrope;
+export import :model.typed_stellar;
+
+export namespace mean_field::seed {
+    struct DimensionlessLaneEmdenSolution final {
+        mfem::Vector coordinate;
+        mfem::Vector theta;
+        mfem::Vector thetaDerivative;
+        std::optional<double> firstZeroCoordinate;
+    };
+
+    /*
+     * Integrate the dimensionless Lane-Emden equation from the regular center
+     * to either the first zero of theta or coordinateLimit, whichever occurs
+     * first. This numerical kernel also supports the n = 0 and n = 5 analytic
+     * benchmark cases even though they do not both define admissible seeds for
+     * the current Polytrope EOS and finite stellar domain.
+     */
+    [[nodiscard]] DimensionlessLaneEmdenSolution integrateLaneEmden(
+        double polytropicIndex,
+        double coordinateLimit,
+        double integrationStep = 1.0e-3
+    );
+
+    struct RadialProfile final {
+        mfem::Vector radius;
+        mfem::Vector density;
+        mfem::Vector specificEnthalpy;
+
+        dimensions::LengthValue stellarRadius;
+        dimensions::DensityValue centralDensity;
+        dimensions::SpecificEnthalpyValue centralSpecificEnthalpy;
+    };
+
+    class LaneEmden final {
+    public:
+        struct Parameters final {
+            std::optional<dimensions::DensityValue> centralDensity{std::nullopt};
+            int radialSampleCount{512};
+        };
+
+        LaneEmden()
+            : m_centralDensity(std::nullopt),
+              m_radialSampleCount(512) {
+        }
+
+        explicit LaneEmden(const Parameters parameters)
+            : m_centralDensity(parameters.centralDensity),
+              m_radialSampleCount(parameters.radialSampleCount) {
+            if (m_centralDensity.has_value() &&
+                (!std::isfinite(m_centralDensity->value()) || m_centralDensity->value() <= 0.0)) {
+                throw std::invalid_argument("A Lane-Emden seed central density must be finite and positive.");
+            }
+            if (m_radialSampleCount < 2) {
+                throw std::invalid_argument("A Lane-Emden seed requires at least two radial samples.");
+            }
+        }
+
+        [[nodiscard]] const std::optional<dimensions::DensityValue> &centralDensity() const noexcept {
+            return m_centralDensity;
+        }
+
+        [[nodiscard]] int radialSampleCount() const noexcept {
+            return m_radialSampleCount;
+        }
+
+    private:
+        std::optional<dimensions::DensityValue> m_centralDensity;
+        int m_radialSampleCount;
+    };
+
+    [[nodiscard]] RadialProfile generateLaneEmdenProfile(
+        const eos::Polytrope &equationOfState,
+        dimensions::DensityValue centralDensity,
+        int radialSampleCount
+    );
+
+    template <model::StellarModelType Model>
+        requires std::remove_cvref_t<Model>::template
+    containsSpecification<eos::Polytrope> [[nodiscard]] RadialProfile generateRadialProfile(
+        const Model &stellarModel,
+        const LaneEmden &strategy
+    ) {
+        std::optional<dimensions::DensityValue> centralDensity = strategy.centralDensity();
+
+        if (!centralDensity.has_value()) {
+            if constexpr (std::remove_cvref_t<Model>::template containsSpecification<models::FixedCentralDensity>) {
+                centralDensity = stellarModel.template specification<models::FixedCentralDensity>().targetDensity();
+            } else {
+                throw std::invalid_argument(
+                    "Lane-Emden seed generation requires either FixedCentralDensity or an explicit seed-only central "
+                    "density."
+                );
+            }
+        }
+
+        return generateLaneEmdenProfile(
+            stellarModel.template specification<eos::Polytrope>(), *centralDensity, strategy.radialSampleCount()
+        );
+    }
+
+    template <typename Strategy, typename Model>
+    concept RadialSeedStrategyFor = requires(const Model &stellarModel, const Strategy &strategy) {
+        { generateRadialProfile(stellarModel, strategy) } -> std::same_as<RadialProfile>;
+    };
+} // namespace mean_field::seed

libmeanfield/interface/seed/stellar_equilibrium_projection.cppm

@@ -0,0 +1,136 @@
+module;
+
+#include <concepts>
+#include <stdexcept>
+#include <type_traits>
+#include <utility>
+
+#include <mfem.hpp>
+
+export module mean_field:seed.stellar_equilibrium_projection;
+
+export import :operators.stellar_equilibrium_problem;
+export import :physics.gravity;
+export import :seed.lane_emden;
+
+export namespace mean_field::seed {
+    struct StellarEquilibriumProjectionOptions final {
+        physics::GravitySolveOptions gravity{};
+        double surfaceRadiusRelativeTolerance{5.0e-4};
+    };
+
+    template <equilibrium::StellarEquilibriumModel Model> struct ProjectedEquilibriumState final {
+        using ModelType = std::remove_cvref_t<Model>;
+
+        mfem::Vector values;
+    };
+
+    namespace detail {
+        struct ProjectedRadialFields final {
+            mfem::Vector density;
+            mfem::Vector gravityGradient;
+            mfem::Vector gravityPotential;
+            mfem::Vector specificEnthalpy;
+            double bernoulliConstant;
+        };
+
+        [[nodiscard]] ProjectedRadialFields projectRadialFields(
+            const equilibrium::StellarDiscretization &discretization,
+            const RadialProfile &profile,
+            dimensions::MassValue targetMass,
+            dimensions::PressureValue targetSurfacePressure,
+            const StellarEquilibriumProjectionOptions &options
+        );
+
+        inline void assignProjectedBlock(
+            mfem::Vector destination,
+            const mfem::Vector &source,
+            const char *name
+        ) {
+            if (destination.Size() != source.Size()) {
+                throw std::invalid_argument(name);
+            }
+            destination = source;
+        }
+    } // namespace detail
+
+    template <equilibrium::StellarEquilibriumModel Model>
+    [[nodiscard]] ProjectedEquilibriumState<Model> projectRadialProfile(
+        const equilibrium::StellarEquilibriumProblem<Model> &problem,
+        const RadialProfile &profile,
+        const StellarEquilibriumProjectionOptions &options = {}
+    ) {
+        const detail::ProjectedRadialFields fields = detail::projectRadialFields(
+            problem.GetDiscretization(), profile,
+            problem.GetStellarModel().template specification<models::FixedTotalMass>().targetMass(),
+            problem.GetStellarModel().template specification<surface::Isobaric>().targetPressure(), options
+        );
+
+        mfem::Vector values(problem.StateSize());
+        values               = 0.0;
+        const auto stateView = problem.GetManifest().stateView(values);
+
+        detail::assignProjectedBlock(
+            stateView.block(utils::blocks::density_field.mass_term), fields.density,
+            "The projected density does not match the compiled equilibrium-state block."
+        );
+        stateView.block(utils::blocks::surface_deformation_field.parameters_term) = 0.0;
+        detail::assignProjectedBlock(
+            stateView.block(utils::blocks::gravity_field.gradient_term), fields.gravityGradient,
+            "The projected gravity gradient does not match the compiled equilibrium-state block."
+        );
+        detail::assignProjectedBlock(
+            stateView.block(utils::blocks::gravity_field.poisson_term), fields.gravityPotential,
+            "The projected gravity potential does not match the compiled equilibrium-state block."
+        );
+        detail::assignProjectedBlock(
+            stateView.block(utils::blocks::enthalpy_field.specific_term), fields.specificEnthalpy,
+            "The projected specific enthalpy does not match the compiled equilibrium-state block."
+        );
+
+        /*
+         * Projection of a continuous spherical profile onto a faceted
+         * reference mesh generally leaves a small trace error on the physical
+         * surface. The pressure condition replaces these carrier rows in the
+         * compiled equilibrium problem, so impose its required carrier value
+         * 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);
+        const dimensions::SpecificEnthalpyValue requiredSurfaceEnthalpy =
+            eos::evaluate<dimensions::quantity::SpecificEnthalpy>(
+                problem.GetStellarModel().template specification<eos::Polytrope>(),
+                problem.GetStellarModel().template specification<surface::Isobaric>().targetPressure()
+            );
+        for (const int surfaceRow : problem.GetPressureSurfaceRows().reduced_dofs()) {
+            enthalpy(surfaceRow) = requiredSurfaceEnthalpy.value();
+        }
+
+        mfem::Vector fixedMassCoordinate =
+            stateView.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term);
+        if (fixedMassCoordinate.Size() != 1) {
+            throw std::invalid_argument("FixedTotalMass must generate exactly one equilibrium-state coordinate.");
+        }
+        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,
+        typename Strategy>
+        requires RadialSeedStrategyFor<
+            Strategy,
+            typename equilibrium::StellarEquilibriumProblem<Model>::ModelType>
+    [[nodiscard]] ProjectedEquilibriumState<Model> makeProjectedEquilibriumState(
+        const equilibrium::StellarEquilibriumProblem<Model> &problem,
+        const Strategy &strategy,
+        const StellarEquilibriumProjectionOptions &options = {}
+    ) {
+        return projectRadialProfile(problem, generateRadialProfile(problem.GetStellarModel(), strategy), options);
+    }
+} // namespace mean_field::seed

libmeanfield/interface/solver/preconditioning_diagnostics.cppm

@@ -0,0 +1,259 @@
+module;
+
+#include <cstdint>
+#include <span>
+#include <string>
+#include <string_view>
+#include <vector>
+
+#include <mfem.hpp>
+#include <mpi.h>
+
+export module mean_field:solver.preconditioning_diagnostics;
+
+export import :operators.root_manifest;
+
+export namespace mean_field::solver {
+    struct OperatorApplicationStatistics final {
+        std::uint64_t applications{0};
+        double totalSeconds{0.0};
+        double maximumSeconds{0.0};
+    };
+
+    struct PreconditionerLifecycleStatistics final {
+        std::uint64_t setups{0};
+        std::uint64_t refreshes{0};
+        double setupSeconds{0.0};
+        double refreshSeconds{0.0};
+    };
+
+    /*
+     * A non-owning measurement wrapper. Statistics are local to an MPI rank;
+     * cross-rank wall-clock reductions are performed when a solve report is
+     * assembled. Krylov application is sequential, so counters intentionally
+     * do not impose atomic overhead.
+     */
+    class InstrumentedOperator final : public mfem::Operator {
+    public:
+        explicit InstrumentedOperator(const mfem::Operator &operation);
+
+        void Mult(
+            const mfem::Vector &input,
+            mfem::Vector &output
+        ) const override;
+
+        void ResetStatistics() const noexcept;
+        [[nodiscard]] const OperatorApplicationStatistics &GetStatistics() const noexcept;
+        [[nodiscard]] const mfem::Operator &GetOperation() const noexcept;
+
+    private:
+        const mfem::Operator *m_operation;
+        mutable OperatorApplicationStatistics m_statistics;
+    };
+
+    class InstrumentedPreconditioner final : public mfem::Solver {
+    public:
+        explicit InstrumentedPreconditioner(mfem::Solver &preconditioner);
+
+        void SetOperator(const mfem::Operator &operation) override;
+
+        void Mult(
+            const mfem::Vector &input,
+            mfem::Vector &output
+        ) const override;
+
+        void ResetStatistics() const noexcept;
+        [[nodiscard]] const OperatorApplicationStatistics &GetStatistics() const noexcept;
+        [[nodiscard]] const PreconditionerLifecycleStatistics &GetLifecycleStatistics() const noexcept;
+        [[nodiscard]] const mfem::Solver &GetPreconditioner() const noexcept;
+
+    private:
+        mfem::Solver *m_preconditioner;
+        mutable OperatorApplicationStatistics m_statistics;
+        PreconditionerLifecycleStatistics m_lifecycleStatistics;
+    };
+
+    class IdentityPreconditioner final : public mfem::Solver {
+    public:
+        explicit IdentityPreconditioner(int size);
+
+        void SetOperator(const mfem::Operator &operation) override;
+
+        void Mult(
+            const mfem::Vector &input,
+            mfem::Vector &output
+        ) const override;
+    };
+
+    /*
+     * If the supplied solver applies M^{-1}, this operator represents the
+     * fixed right-preconditioned product J M^{-1}. It is deliberately
+     * independent of the Krylov implementation used in production.
+     */
+    class FixedRightPreconditionedOperator final : public mfem::Operator {
+    public:
+        FixedRightPreconditionedOperator(
+            const mfem::Operator &jacobian,
+            const mfem::Solver &inversePreconditioner
+        );
+
+        void Mult(
+            const mfem::Vector &input,
+            mfem::Vector &output
+        ) const override;
+
+        [[nodiscard]] const mfem::Operator &GetJacobian() const noexcept;
+        [[nodiscard]] const mfem::Solver &GetInversePreconditioner() const noexcept;
+
+    private:
+        const mfem::Operator *m_jacobian;
+        const mfem::Solver *m_inversePreconditioner;
+        mutable mfem::Vector m_preconditionedDirection;
+    };
+
+    struct IterationResidualMeasurement final {
+        int iteration;
+        double reportedNorm;
+        bool final;
+    };
+
+    class ResidualHistoryMonitor final : public mfem::IterativeSolverMonitor {
+    public:
+        void Reset() override;
+
+        void MonitorResidual(
+            int iteration,
+            double norm,
+            const mfem::Vector &residual,
+            bool final
+        ) override;
+
+        [[nodiscard]] const std::vector<IterationResidualMeasurement> &GetHistory() const noexcept;
+
+    private:
+        std::vector<IterationResidualMeasurement> m_history;
+    };
+
+    struct ResidualBlockMeasurement final {
+        std::string stableId;
+        int size{0};
+        double descriptorScale{1.0};
+        double rightHandSideNorm{0.0};
+        double trueResidualNorm{0.0};
+        double blockRelativeResidual{0.0};
+        double scaledRightHandSideNorm{0.0};
+        double scaledTrueResidualNorm{0.0};
+        double contributionToGlobalRelativeResidual{0.0};
+        double fractionOfGlobalSquaredResidualNorm{0.0};
+    };
+
+    struct DirectResidualMeasurement final {
+        double rightHandSideNorm{0.0};
+        double trueResidualNorm{0.0};
+        double relativeResidual{0.0};
+        std::vector<ResidualBlockMeasurement> blocks;
+    };
+
+    [[nodiscard]] DirectResidualMeasurement measureDirectResidual(
+        const mfem::Operator &jacobian,
+        const mfem::Vector &rightHandSide,
+        const mfem::Vector &solution,
+        std::span<const operators::RootBlockDescriptor> residualBlocks,
+        MPI_Comm communicator,
+        double denominatorFloor = 1.0e-300
+    );
+
+    struct LinearSolveMeasurement final {
+        bool solverConverged{false};
+        int outerIterations{0};
+        double solverReportedInitialNorm{0.0};
+        double solverReportedFinalNorm{0.0};
+        double solverReportedResidualReduction{0.0};
+        double trueResidualDigitsReducedPerJacobianApplication{0.0};
+        double solveSecondsMaximumRank{0.0};
+        OperatorApplicationStatistics jacobian;
+        OperatorApplicationStatistics inversePreconditioner;
+        PreconditionerLifecycleStatistics inversePreconditionerLifecycle;
+        DirectResidualMeasurement directResidual;
+        std::vector<IterationResidualMeasurement> reportedResidualHistory;
+    };
+
+    [[nodiscard]] LinearSolveMeasurement measureLinearSolve(
+        const mfem::IterativeSolver &iterativeSolver,
+        const mfem::Operator &jacobian,
+        const mfem::Vector &rightHandSide,
+        const mfem::Vector &solution,
+        std::span<const operators::RootBlockDescriptor> residualBlocks,
+        const OperatorApplicationStatistics &jacobianStatistics,
+        const OperatorApplicationStatistics &inversePreconditionerStatistics,
+        const PreconditionerLifecycleStatistics &inversePreconditionerLifecycle,
+        const ResidualHistoryMonitor &monitor,
+        double localSolveSeconds,
+        MPI_Comm communicator,
+        double denominatorFloor = 1.0e-300
+    );
+
+    struct ArnoldiOptions final {
+        int krylovDimension{40};
+        double breakdownRelativeTolerance{1.0e-13};
+        double ritzConvergenceRelativeTolerance{1.0e-8};
+        bool reorthogonalize{true};
+    };
+
+    struct RitzValueMeasurement final {
+        double realPart{0.0};
+        double imaginaryPart{0.0};
+        double magnitude{0.0};
+        double distanceFromOne{0.0};
+        double residualEstimate{0.0};
+        double relativeResidualEstimate{0.0};
+        bool converged{false};
+    };
+
+    enum class RitzValueOrdering { closest_to_zero, farthest_from_one, smallest_real_part, largest_magnitude };
+
+    struct ArnoldiSpectralMeasurement final {
+        int requestedDimension{0};
+        int achievedDimension{0};
+        bool invariantSubspaceFound{false};
+        std::uint64_t operatorApplications{0};
+        double operatorApplicationSecondsMaximumRank{0.0};
+        double operatorMaximumApplicationSecondsMaximumRank{0.0};
+        double measurementSecondsMaximumRank{0.0};
+        double nonApplicationSecondsMaximumRank{0.0};
+        int convergedRitzValueCount{0};
+        int negativeRealPartCount{0};
+
+        double projectedLargestSingularValue{0.0};
+        double projectedSmallestSingularValue{0.0};
+        double projectedConditionProxy{0.0};
+        double centroidRealPart{0.0};
+        double centroidImaginaryPart{0.0};
+        double rmsDistanceFromOne{0.0};
+        double rmsClusterRadius{0.0};
+        double minimumMagnitude{0.0};
+        double maximumMagnitude{0.0};
+        double minimumRealPart{0.0};
+        double maximumRealPart{0.0};
+        double maximumAbsoluteImaginaryPart{0.0};
+        double conjugatePairDefect{0.0};
+        double projectedDepartureFromNormality{0.0};
+        double projectedFieldOfValuesMinimumRealPart{0.0};
+        double projectedFieldOfValuesMaximumRealPart{0.0};
+
+        std::vector<RitzValueMeasurement> ritzValues;
+    };
+
+    [[nodiscard]] ArnoldiSpectralMeasurement measureArnoldiSpectrum(
+        const mfem::Operator &operation,
+        const mfem::Vector &initialDirection,
+        MPI_Comm communicator,
+        const ArnoldiOptions &options = {}
+    );
+
+    [[nodiscard]] std::vector<RitzValueMeasurement> selectRitzValues(
+        const ArnoldiSpectralMeasurement &measurement,
+        RitzValueOrdering ordering,
+        int count
+    );
+} // namespace mean_field::solver

libmeanfield/interface/surface/compiled.cppm

@@ -15,7 +15,7 @@ export namespace mean_field::surface {
     class CompiledPressureSurfaceConstraint final {
     public:
         using PhysicalCondition   = ConstantPressureSurface;
-        using PhysicalQuantity    = eos::quantity::Pressure;
+        using PhysicalQuantity    = dimensions::quantity::Pressure;
         using CarrierQuantity     = typename Formulation::CarrierQuantity;
         using CarrierField        = typename Formulation::CarrierField;
         using Relation            = SelectedRelation;
@@ -32,7 +32,7 @@ export namespace mean_field::surface {
               ) {
         }
 
-        [[nodiscard]] eos::PressureValue targetPressure() const noexcept {
+        [[nodiscard]] dimensions::PressureValue targetPressure() const noexcept {
             return m_condition.targetPressure();
         }
 

libmeanfield/interface/surface/constant.cppm

@@ -7,7 +7,7 @@ module;
 
 export module mean_field:surface.constant;
 
-export import :eos.quantities;
+export import :dimensions.quantities;
 
 export namespace mean_field::surface {
     struct PressureSurfaceDescriptor final {
@@ -22,8 +22,15 @@ export namespace mean_field::surface {
      */
     class ConstantPressureSurface final {
     public:
-        using PhysicalQuantity = eos::quantity::Pressure;
-        using TargetValue      = eos::PressureValue;
+        struct Parameters final {
+            dimensions::PressureValue Psurf;
+        };
+
+        using PhysicalQuantity = dimensions::quantity::Pressure;
+        using TargetValue      = dimensions::PressureValue;
+
+        explicit ConstantPressureSurface(const Parameters parameters) : ConstantPressureSurface(parameters.Psurf) {
+        }
 
         explicit ConstantPressureSurface(const TargetValue targetPressure) : m_targetPressure(targetPressure) {
             if (!std::isfinite(targetPressure.value())) {

libmeanfield/interface/utils/blocks.cppm

@@ -6,6 +6,8 @@ module;
 #include <type_traits>
 export module mean_field:utils.blocks;
 
+export import :model.specifications;
+
 export namespace mean_field::utils::blocks {
     inline constexpr int dynamic_block_size = -1;
 
@@ -19,6 +21,18 @@ export namespace mean_field::utils::blocks {
         static constexpr int static_block_size = dynamic_block_size;
     };
 
+    template <typename GeneratedValue> struct generated_value_block final : value_block_base {
+        using GeneratedType                    = GeneratedValue;
+
+        static constexpr int static_block_size = static_cast<int>(GeneratedValue::scalarArity);
+    };
+
+    template <typename GeneratedResidual> struct generated_residual_block final : residual_block_base {
+        using GeneratedType                    = GeneratedResidual;
+
+        static constexpr int static_block_size = static_cast<int>(GeneratedResidual::scalarArity);
+    };
+
     struct term { };
     struct field { };
 
@@ -96,25 +110,44 @@ export namespace mean_field::utils::blocks {
         static inline constexpr specific specific_term{};
     };
 
-    struct barotropic_constant final : field {
-        struct mass_normalization final : term {
-            struct value final : value_block_base {
-                static constexpr int static_block_size = 1;
-            };
+    struct fixed_total_mass final : field {
+        using SpecificationType = models::FixedTotalMass;
+        using MultiplierType    = models::MultiplierFor<SpecificationType>;
+        using ResidualType      = models::ResidualFor<SpecificationType>;
 
-            struct residual final : residual_block_base {
-                static constexpr int static_block_size = 1;
-            };
+        struct mass_normalization final : term {
+            using value    = generated_value_block<MultiplierType>;
+            using residual = generated_residual_block<ResidualType>;
         };
 
         static inline constexpr mass_normalization mass_normalization_term{};
     };
 
+    struct fixed_central_density final : field {
+        using SpecificationType = models::FixedCentralDensity;
+        using BorderType        = models::BorderFor<SpecificationType>;
+        using ResidualType      = models::ResidualFor<SpecificationType>;
+
+        struct central_value final : term {
+            using value    = generated_value_block<BorderType>;
+            using residual = generated_residual_block<ResidualType>;
+        };
+
+        static inline constexpr central_value central_value_term{};
+    };
+
+    // Compatibility name for the current barotropic formulation. The scalar
+    // is generated by FixedTotalMass; its realization in this formulation is
+    // the historical C coordinate.
+    using barotropic_constant = fixed_total_mass;
+
     inline constexpr density density_field{};
     inline constexpr displacement displacement_field{};
     inline constexpr surface_deformation surface_deformation_field{};
     inline constexpr gravity gravity_field{};
     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 barotropic_constant barotropic_constant_field{};
 
     template <typename... Types> struct type_list {
@@ -489,6 +522,61 @@ export namespace mean_field::utils::blocks {
             density::mass::value,
             surface_deformation::parameters::value>>;
 
+    // Bordered n=3 family closure. The original stellar coordinates remain a
+    // contiguous prefix and the phase border and row are appended last.
+    using central_density_bordered_stellar_equilibrium_form = block_form<
+        type_list<
+            density::mass::value,
+            surface_deformation::parameters::value,
+            gravity::gradient::value,
+            gravity::poisson::value,
+            enthalpy::specific::value,
+            barotropic_constant::mass_normalization::value,
+            fixed_central_density::central_value::value>,
+        type_list<
+            gravity::gradient::residual,
+            gravity::poisson::residual,
+            density::mass::residual,
+            surface_deformation::shape_equilibrium::residual,
+            enthalpy::specific::residual,
+            barotropic_constant::mass_normalization::residual,
+            fixed_central_density::central_value::residual>>;
+
+    using central_density_bordered_stellar_equilibrium_jacobian_form = type_list<
+        block_row<
+            gravity::gradient::residual,
+            gravity::gradient::value,
+            gravity::poisson::value,
+            surface_deformation::parameters::value>,
+        block_row<
+            gravity::poisson::residual,
+            gravity::gradient::value,
+            density::mass::value,
+            surface_deformation::parameters::value>,
+        block_row<
+            density::mass::residual,
+            density::mass::value,
+            enthalpy::specific::value,
+            surface_deformation::parameters::value>,
+        block_row<
+            surface_deformation::shape_equilibrium::residual,
+            density::mass::value,
+            surface_deformation::parameters::value,
+            gravity::gradient::value,
+            enthalpy::specific::value>,
+        block_row<
+            enthalpy::specific::residual,
+            enthalpy::specific::value,
+            gravity::poisson::value,
+            surface_deformation::parameters::value,
+            barotropic_constant::mass_normalization::value,
+            fixed_central_density::central_value::value>,
+        block_row<
+            barotropic_constant::mass_normalization::residual,
+            density::mass::value,
+            surface_deformation::parameters::value>,
+        block_row<fixed_central_density::central_value::residual, enthalpy::specific::value>>;
+
     // Columns: [d, h]
     // Rows:    [R_d]
     using pressure_force_form = block_form<
@@ -509,4 +597,8 @@ export namespace mean_field::utils::blocks {
     static_assert(valid_jacobian_form<
                   surface_deformed_stellar_equilibrium_form,
                   surface_deformed_stellar_equilibrium_jacobian_form>);
+
+    static_assert(valid_jacobian_form<
+                  central_density_bordered_stellar_equilibrium_form,
+                  central_density_bordered_stellar_equilibrium_jacobian_form>);
 } // namespace mean_field::utils::blocks

tests/models/model_specifications.cpp

@@ -0,0 +1,196 @@
+#include <concepts>
+#include <limits>
+#include <stdexcept>
+#include <type_traits>
+
+#include <catch2/catch_test_macros.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    struct NotAModelSpecification final { };
+
+    using PolytropicMassSpecifications = mean_field::models::
+        SpecificationSet<mean_field::eos::Polytrope, mean_field::models::FixedTotalMass, mean_field::surface::Isobaric>;
+
+    using PermutedPolytropicMassSpecifications = mean_field::models::
+        SpecificationSet<mean_field::surface::Isobaric, mean_field::models::FixedTotalMass, mean_field::eos::Polytrope>;
+
+    using CentralDensityPolytropicMassSpecifications = mean_field::models::SpecificationSet<
+        mean_field::models::FixedCentralDensity,
+        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>;
+} // namespace
+
+TEST_CASE(
+    "Model Specifications Form Canonical Compile-Time Model Types",
+    tags::model_specification_type_contract
+) {
+    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::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::FixedCentralDensity>);
+
+    STATIC_CHECK(
+        mean_field::models::ValidModelSpecificationPack<
+            mean_field::eos::Polytrope, mean_field::models::FixedTotalMass, mean_field::surface::Isobaric>
+    );
+
+    STATIC_CHECK_FALSE(
+        mean_field::models::ValidModelSpecificationPack<
+            mean_field::eos::Polytrope, mean_field::models::FixedTotalMass, mean_field::models::FixedTotalMass,
+            mean_field::surface::Isobaric>
+    );
+
+    STATIC_CHECK_FALSE(
+        mean_field::models::ValidModelSpecificationPack<
+            mean_field::models::FixedTotalMass, mean_field::surface::Isobaric>
+    );
+
+    STATIC_CHECK(std::same_as<PolytropicMassModel, PermutedPolytropicMassModel>);
+    STATIC_CHECK_FALSE(std::same_as<PolytropicMassModel, CentralDensityPolytropicMassModel>);
+    STATIC_CHECK(mean_field::model::StellarModelType<PolytropicMassModel>);
+    STATIC_CHECK(mean_field::model::StellarModelType<CentralDensityPolytropicMassModel>);
+}
+
+TEST_CASE(
+    "Invariant And Phase Specifications Generate Balanced Residual And Value Types",
+    tags::model_specification_type_contract
+) {
+    using MassSignature = PolytropicMassModel::OperatorSignature;
+
+    STATIC_CHECK(MassSignature::generatedValueArity == 1);
+    STATIC_CHECK(MassSignature::generatedResidualArity == 1);
+    STATIC_CHECK(MassSignature::symbolicallySquare);
+
+    STATIC_CHECK(
+        mean_field::models::modelTypeListContains<
+            mean_field::models::MultiplierFor<mean_field::models::FixedTotalMass>,
+            typename MassSignature::GeneratedValues>
+    );
+
+    STATIC_CHECK(
+        mean_field::models::modelTypeListContains<
+            mean_field::models::ResidualFor<mean_field::models::FixedTotalMass>,
+            typename MassSignature::GeneratedResiduals>
+    );
+
+    using CentralDensitySignature = CentralDensityPolytropicMassModel::OperatorSignature;
+
+    STATIC_CHECK(CentralDensitySignature::generatedValueArity == 2);
+    STATIC_CHECK(CentralDensitySignature::generatedResidualArity == 2);
+    STATIC_CHECK(CentralDensitySignature::symbolicallySquare);
+
+    STATIC_CHECK(
+        mean_field::models::modelTypeListContains<
+            mean_field::models::BorderFor<mean_field::models::FixedCentralDensity>,
+            typename CentralDensitySignature::GeneratedValues>
+    );
+
+    STATIC_CHECK(
+        mean_field::models::modelTypeListContains<
+            mean_field::models::ResidualFor<mean_field::models::FixedCentralDensity>,
+            typename CentralDensitySignature::GeneratedResiduals>
+    );
+}
+
+TEST_CASE(
+    "Fixed Total Mass Compiles Its Generated Multiplier And Canonical Residual Row",
+    tags::model_specification_type_contract
+) {
+    using namespace mean_field;
+    using Request = models::FixedMassLayoutRequest;
+    using Form    = utils::blocks::barotropic_equilibrium_form;
+
+    STATIC_CHECK(models::ConstraintLayoutRequestType<Request>);
+    STATIC_CHECK(models::CompiledConstraint<models::CompiledFixedMass>);
+    STATIC_CHECK(std::same_as<typename Request::SpecificationType, models::FixedTotalMass>);
+    STATIC_CHECK(std::same_as<typename Request::GeneratedValueType, models::MultiplierFor<models::FixedTotalMass>>);
+    STATIC_CHECK(std::same_as<typename Request::GeneratedResidualType, models::ResidualFor<models::FixedTotalMass>>);
+    STATIC_CHECK(
+        std::same_as<typename Request::ValueBlockType::GeneratedType, models::MultiplierFor<models::FixedTotalMass>>
+    );
+    STATIC_CHECK(
+        std::same_as<typename Request::ResidualBlockType::GeneratedType, models::ResidualFor<models::FixedTotalMass>>
+    );
+    STATIC_CHECK(std::same_as<typename models::CompiledFixedMass::MultiplierField, field::BarotropicConstant>);
+    STATIC_CHECK(Request::rowInjection == models::ConstraintRowInjection::append);
+    STATIC_CHECK(Request::valueArity == 1);
+    STATIC_CHECK(Request::residualArity == 1);
+    STATIC_CHECK(Request::valueBlock<Form>().index == Form::value_block_count - 1);
+    STATIC_CHECK(Request::residualBlock<Form>().index == Form::residual_block_count - 1);
+
+    const models::CompiledFixedMass compiled =
+        models::compileConstraint(models::FixedTotalMass{dimensions::MassValue{1.75}});
+    CHECK(compiled.targetMass() == dimensions::MassValue{1.75});
+    CHECK(compiled.specification().targetMass() == dimensions::MassValue{1.75});
+}
+
+TEST_CASE(
+    "Model Specification Descriptors Preserve Roles And Generated Arity",
+    tags::model_specification_type_contract
+) {
+    constexpr auto polytrope = mean_field::models::specificationDescriptor<mean_field::eos::Polytrope>();
+    constexpr auto surface =
+        mean_field::models::specificationDescriptor<mean_field::surface::ConstantPressureSurface>();
+    constexpr auto mass = mean_field::models::specificationDescriptor<mean_field::models::FixedTotalMass>();
+    constexpr auto centralDensity =
+        mean_field::models::specificationDescriptor<mean_field::models::FixedCentralDensity>();
+
+    STATIC_CHECK(polytrope.name == "Polytrope");
+    STATIC_CHECK(polytrope.role == mean_field::models::SpecificationRole::constitutive_law);
+    STATIC_CHECK(polytrope.generatedValueArity == 0);
+    STATIC_CHECK(polytrope.generatedResidualArity == 0);
+
+    STATIC_CHECK(surface.name == "IsobaricSurface");
+    STATIC_CHECK(surface.role == mean_field::models::SpecificationRole::boundary_condition);
+
+    STATIC_CHECK(mass.name == "FixedTotalMass");
+    STATIC_CHECK(mass.role == mean_field::models::SpecificationRole::invariant);
+    STATIC_CHECK(mass.generatedValueArity == 1);
+    STATIC_CHECK(mass.generatedResidualArity == 1);
+
+    STATIC_CHECK(centralDensity.name == "FixedCentralDensity");
+    STATIC_CHECK(centralDensity.role == mean_field::models::SpecificationRole::phase_condition);
+    STATIC_CHECK(centralDensity.generatedValueArity == 1);
+    STATIC_CHECK(centralDensity.generatedResidualArity == 1);
+}
+
+TEST_CASE(
+    "Invariant And Phase Specification Values Reject Invalid Targets",
+    tags::model_specification_type_contract
+) {
+    const mean_field::models::FixedTotalMass mass{mean_field::dimensions::MassValue{1.25}};
+    const mean_field::models::FixedCentralDensity centralDensity{mean_field::eos::DensityValue{2.5}};
+
+    CHECK(mass.targetMass() == mean_field::dimensions::MassValue{1.25});
+    CHECK(centralDensity.targetDensity() == mean_field::eos::DensityValue{2.5});
+
+    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);
+    CHECK_THROWS_AS(
+        mean_field::models::FixedTotalMass{mean_field::dimensions::MassValue{std::numeric_limits<double>::infinity()}},
+        std::invalid_argument
+    );
+
+    CHECK_THROWS_AS(mean_field::models::FixedCentralDensity{mean_field::eos::DensityValue{0.0}}, std::invalid_argument);
+    CHECK_THROWS_AS(
+        mean_field::models::FixedCentralDensity{mean_field::eos::DensityValue{-1.0}}, std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        mean_field::models::FixedCentralDensity{mean_field::eos::DensityValue{std::numeric_limits<double>::infinity()}},
+        std::invalid_argument
+    );
+}

tests/models/typed_stellar_model.cpp

@@ -0,0 +1,108 @@
+#include <concepts>
+#include <limits>
+#include <stdexcept>
+#include <type_traits>
+
+#include <catch2/catch_test_macros.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    using CanonicalModel = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::integral::FixedTotalMass,
+        mean_field::constraint::FixedCentralDensity>>;
+
+    using BaseModel      = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::integral::FixedTotalMass>>;
+} // namespace
+
+TEST_CASE(
+    "Stellar Model Is Deduced From Validated Physical Specifications",
+    tags::stellar_model_specification_api
+) {
+    using namespace mean_field;
+
+    const auto stellarModel = model::StellarModel(
+        eos::Polytrope({.n = 3.0, .K = 0.25}), surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.5}}),
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{2.0}})
+    );
+
+    STATIC_CHECK(std::same_as<std::remove_cvref_t<decltype(stellarModel)>, CanonicalModel>);
+    STATIC_CHECK(model::StellarModelType<decltype(stellarModel)>);
+    STATIC_CHECK(models::SpecifiedModelType<decltype(stellarModel)>);
+    STATIC_CHECK(CanonicalModel::specificationCount == 4);
+    STATIC_CHECK(CanonicalModel::symbolicallySquare);
+    STATIC_CHECK(CanonicalModel::hasCompleteEquilibriumCompiler);
+    STATIC_CHECK(CanonicalModel::compilationClass == models::EquilibriumSystemCompilation::complete_equilibrium_system);
+
+    CHECK(stellarModel.specification<eos::Polytrope>().polytropic_index() == 3.0);
+    CHECK(stellarModel.specification<eos::Polytrope>().polytropic_constant() == 0.25);
+    CHECK(stellarModel.specification<surface::Isobaric>().targetPressure() == dimensions::PressureValue{0.0});
+    CHECK(stellarModel.specification<integral::FixedTotalMass>().targetMass() == dimensions::MassValue{1.5});
+    CHECK(
+        stellarModel.specification<constraint::FixedCentralDensity>().targetDensity() == dimensions::DensityValue{2.0}
+    );
+}
+
+TEST_CASE(
+    "Stellar Model Deduction Canonicalizes Unordered Specifications",
+    tags::stellar_model_specification_api
+) {
+    using namespace mean_field;
+
+    const auto canonical = model::StellarModel(
+        eos::Polytrope({.n = 3.0, .K = 0.25}), surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}}),
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{1.0}})
+    );
+    const auto reordered = model::StellarModel(
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{1.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}), eos::Polytrope({.n = 3.0, .K = 0.25})
+    );
+    const auto base = model::StellarModel(
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}}), eos::Polytrope({.n = 3.0, .K = 0.25}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}})
+    );
+
+    STATIC_CHECK(std::same_as<decltype(canonical), decltype(reordered)>);
+    STATIC_CHECK(std::same_as<std::remove_cvref_t<decltype(base)>, BaseModel>);
+    STATIC_CHECK_FALSE(std::same_as<decltype(canonical), decltype(base)>);
+    STATIC_CHECK_FALSE(BaseModel::template containsSpecification<constraint::FixedCentralDensity>);
+    STATIC_CHECK(CanonicalModel::template containsSpecification<constraint::FixedCentralDensity>);
+
+    const auto descriptors = CanonicalModel::runtimeSpecificationDescriptors();
+    REQUIRE(descriptors.size() == 4);
+    CHECK(descriptors[0].specification.name == "Polytrope");
+    CHECK(descriptors[1].specification.name == "IsobaricSurface");
+    CHECK(descriptors[2].specification.name == "FixedTotalMass");
+    CHECK(descriptors[3].specification.name == "FixedCentralDensity");
+}
+
+TEST_CASE(
+    "Stellar Specification Parameter Constructors Preserve Validation",
+    tags::stellar_model_specification_api
+) {
+    using namespace mean_field;
+
+    STATIC_CHECK(std::constructible_from<eos::Polytrope, eos::Polytrope::Parameters>);
+    STATIC_CHECK(std::constructible_from<surface::Isobaric, surface::Isobaric::Parameters>);
+    STATIC_CHECK(std::constructible_from<integral::FixedTotalMass, integral::FixedTotalMass::Parameters>);
+    STATIC_CHECK(std::constructible_from<constraint::FixedCentralDensity, constraint::FixedCentralDensity::Parameters>);
+    STATIC_CHECK(std::same_as<decltype(integral::FixedTotalMass::Parameters::Mtotal), dimensions::MassValue>);
+    STATIC_CHECK(std::same_as<decltype(constraint::FixedCentralDensity::Parameters::RhoC), dimensions::DensityValue>);
+    STATIC_CHECK(std::same_as<decltype(surface::Isobaric::Parameters::Psurf), dimensions::PressureValue>);
+    STATIC_CHECK_FALSE(std::constructible_from<integral::FixedTotalMass, double>);
+
+    CHECK_THROWS_AS(eos::Polytrope({.n = 0.5, .K = 1.0}), std::invalid_argument);
+    CHECK_THROWS_AS(eos::Polytrope({.n = 3.0, .K = std::numeric_limits<double>::infinity()}), std::invalid_argument);
+    CHECK_THROWS_AS(surface::Isobaric({.Psurf = dimensions::PressureValue{-1.0}}), std::invalid_argument);
+    CHECK_THROWS_AS(integral::FixedTotalMass({.Mtotal = dimensions::MassValue{0.0}}), std::invalid_argument);
+    CHECK_THROWS_AS(constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{0.0}}), std::invalid_argument);
+}

tests/operators/prepared_central_density.cpp

@@ -0,0 +1,164 @@
+#include <algorithm>
+#include <array>
+#include <cmath>
+#include <concepts>
+#include <limits>
+
+#include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    [[nodiscard]] mean_field::field::FieldPointDofMap make_center_map() {
+        mfem::Array<int> centerDof(1);
+        centerDof[0] = 2;
+        return {5, centerDof};
+    }
+
+    [[nodiscard]] double relative_error(
+        const double actual,
+        const double expected
+    ) {
+        return std::abs(actual - expected) / std::max({1.0, std::abs(actual), std::abs(expected)});
+    }
+} // namespace
+
+TEST_CASE(
+    "Fixed Central Density Compiles A Carrier Phase Row And Solver Border",
+    tags::model_specification_type_contract
+) {
+    using namespace mean_field;
+    using Request = models::CentralDensityLayoutRequest;
+    using Form    = utils::blocks::central_density_bordered_stellar_equilibrium_form;
+
+    STATIC_CHECK(models::ConstraintLayoutRequestType<Request>);
+    STATIC_CHECK(models::CompiledConstraint<models::CompiledFixedCentralDensity>);
+    STATIC_CHECK(Request::rowInjection == models::ConstraintRowInjection::solver_border);
+    STATIC_CHECK(Request::valueArity == 1);
+    STATIC_CHECK(Request::residualArity == 1);
+    STATIC_CHECK(Request::valueBlock<Form>().index == Form::value_block_count - 1);
+    STATIC_CHECK(Request::residualBlock<Form>().index == Form::residual_block_count - 1);
+    STATIC_CHECK(std::same_as<typename models::CompiledFixedCentralDensity::CarrierField, field::Enthalpy>);
+    STATIC_CHECK(std::same_as<typename models::CompiledFixedCentralDensity::BorderField, field::CentralDensityBorder>);
+
+    const eos::Polytrope equationOfState{3.0, 0.25};
+    const models::CompiledFixedCentralDensity compiled =
+        models::compileConstraint(models::FixedCentralDensity{eos::DensityValue{8.0}}, equationOfState);
+
+    CHECK(compiled.targetDensity() == eos::DensityValue{8.0});
+    CHECK(compiled.targetEnthalpy() == eos::SpecificEnthalpyValue{2.0});
+    CHECK(compiled.densityFromEnthalpy(eos::SpecificEnthalpyValue{2.5}) == eos::DensityValue{15.625});
+}
+
+TEST_CASE(
+    "Prepared Central Density Phase Has Exact Residual Jacobian And Transpose Actions",
+    tags::central_density_phase_unit
+) {
+    using namespace mean_field;
+
+    const eos::Polytrope equationOfState{3.0, 0.25};
+    const models::CompiledFixedCentralDensity compiled =
+        models::compileConstraint(models::FixedCentralDensity{eos::DensityValue{8.0}}, equationOfState);
+    operators::PreparedCentralDensityConstraint phase(make_center_map(), MPI_COMM_SELF);
+
+    mfem::Vector enthalpy(5);
+    enthalpy    = 0.0;
+    enthalpy(2) = 2.5;
+    const operators::CentralDensityDependencies dependencies{.enthalpy = {.identity = 17, .revision = 1}};
+
+    const operators::PreparedCentralDensityReport initial = phase.Prepare(compiled, enthalpy, 0.3, dependencies);
+    CHECK(initial.refreshedCentralEnthalpy);
+    CHECK(initial.refreshedBorder);
+    CHECK(initial.assembledResidual);
+
+    mfem::Vector carrierResidual(5);
+    mfem::Vector phaseResidual(1);
+    carrierResidual = 1.0;
+    phaseResidual   = 0.0;
+    phase.AddResidual(carrierResidual, phaseResidual);
+    CHECK(carrierResidual(2) == 1.3);
+    CHECK(phaseResidual(0) == 0.5);
+
+    mfem::Vector enthalpyVariation(5);
+    enthalpyVariation                = 0.0;
+    enthalpyVariation(2)             = -0.4;
+    constexpr double borderVariation = 0.7;
+
+    mfem::Vector carrierAction(5);
+    mfem::Vector phaseAction(1);
+    carrierAction = 0.0;
+    phaseAction   = 0.0;
+    phase.ApplyJacobian(
+        {.enthalpyVariation = enthalpyVariation, .borderVariation = borderVariation},
+        {.enthalpyAction = carrierAction, .phaseAction = phaseAction}
+    );
+    CHECK(carrierAction(2) == borderVariation);
+    CHECK(phaseAction(0) == enthalpyVariation(2));
+
+    // The phase residual is affine, so a larger centered-difference step
+    // reduces cancellation without introducing truncation error.
+    constexpr double epsilon = 1.0e-3;
+    mfem::Vector plusEnthalpy(enthalpy);
+    mfem::Vector minusEnthalpy(enthalpy);
+    plusEnthalpy.Add(epsilon, enthalpyVariation);
+    minusEnthalpy.Add(-epsilon, enthalpyVariation);
+
+    auto plusDependencies = dependencies;
+    ++plusDependencies.enthalpy.revision;
+    phase.Prepare(compiled, plusEnthalpy, 0.3 + epsilon * borderVariation, plusDependencies);
+    mfem::Vector plusCarrier(5);
+    mfem::Vector plusPhase(1);
+    plusCarrier = 0.0;
+    plusPhase   = 0.0;
+    phase.AddResidual(plusCarrier, plusPhase);
+
+    auto minusDependencies = plusDependencies;
+    ++minusDependencies.enthalpy.revision;
+    phase.Prepare(compiled, minusEnthalpy, 0.3 - epsilon * borderVariation, minusDependencies);
+    mfem::Vector minusCarrier(5);
+    mfem::Vector minusPhase(1);
+    minusCarrier = 0.0;
+    minusPhase   = 0.0;
+    phase.AddResidual(minusCarrier, minusPhase);
+
+    plusCarrier -= minusCarrier;
+    plusCarrier /= 2.0 * epsilon;
+    const double phaseDifference = (plusPhase(0) - minusPhase(0)) / (2.0 * epsilon);
+    plusCarrier -= carrierAction;
+    CHECK(plusCarrier.Norml2() < 1.0e-10);
+    const double phaseDifferenceError = relative_error(phaseDifference, phaseAction(0));
+    INFO("Central-density phase action = " << phaseAction(0));
+    INFO("Central-density centered difference = " << phaseDifference);
+    INFO("Central-density centered-difference error = " << phaseDifferenceError);
+    CHECK(phaseDifferenceError < 1.0e-10);
+
+    auto restoredDependencies = minusDependencies;
+    ++restoredDependencies.enthalpy.revision;
+    phase.Prepare(compiled, enthalpy, 0.3, restoredDependencies);
+    mfem::Vector carrierDual(5);
+    carrierDual                = 0.0;
+    carrierDual(2)             = -0.8;
+    constexpr double phaseDual = 1.1;
+    mfem::Vector enthalpyDual(5);
+    mfem::Vector borderDual(1);
+    enthalpyDual = 0.0;
+    borderDual   = 0.0;
+    phase.ApplyJacobianTranspose(
+        {.enthalpyResidualDual = carrierDual, .phaseResidualDual = phaseDual},
+        {.enthalpyDual = enthalpyDual, .borderDual = borderDual}
+    );
+
+    const double forwardPairing   = carrierAction * carrierDual + phaseAction(0) * phaseDual;
+    const double transposePairing = enthalpyVariation * enthalpyDual + borderVariation * borderDual(0);
+    CHECK(relative_error(transposePairing, forwardPairing) < 8.0 * std::numeric_limits<double>::epsilon());
+
+    const operators::CentralDensityConstraintReport report = phase.GetConstraintReport();
+    CHECK(report.targetDensity == 8.0);
+    CHECK(report.achievedDensity == 15.625);
+    CHECK(report.targetEnthalpy == 2.0);
+    CHECK(report.achievedEnthalpy == 2.5);
+    CHECK(report.enthalpyResidual == 0.5);
+    CHECK(report.scaledResidual == 0.25);
+}

tests/operators/prepared_central_density_stellar_equilibrium.cpp

@@ -0,0 +1,193 @@
+#include <algorithm>
+#include <cmath>
+#include <concepts>
+#include <cstdint>
+#include <type_traits>
+
+#include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    [[nodiscard]] mean_field::operators::StellarEquilibriumDependencies make_dependencies() {
+        return {
+            .discretization     = {.identity = 3109, .revision = 1},
+            .density            = {.identity = 3119, .revision = 1},
+            .surfaceDeformation = {.identity = 3121, .revision = 1},
+            .gravityGradient    = {.identity = 3137, .revision = 1},
+            .gravityPotential   = {.identity = 3163, .revision = 1},
+            .enthalpy           = {.identity = 3167, .revision = 1},
+            .bernoulliConstant  = {.identity = 3169, .revision = 1},
+            .rotation           = {.identity = 3181, .revision = 1},
+            .targetMass         = {.identity = 3187, .revision = 1}
+        };
+    }
+
+    [[nodiscard]] mean_field::physics::RigidRotation make_zero_rotation() {
+        mfem::Vector angularVelocity(3);
+        mfem::Vector center(3);
+        angularVelocity = 0.0;
+        center          = 0.0;
+        return {angularVelocity, center};
+    }
+
+    [[nodiscard]] double relative_difference(
+        const mfem::Vector &left,
+        const mfem::Vector &right
+    ) {
+        mfem::Vector difference(left);
+        difference -= right;
+        return difference.Norml2() / std::max({1.0, left.Norml2(), right.Norml2()});
+    }
+} // namespace
+
+TEST_CASE(
+    "Central Density Bordered Root Preserves The Physical Operator Prefix",
+    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());
+
+    models::StellarModel stellarModel{
+        models::structure::PolytropicStructure{eos::Polytrope{3.0, 0.25}, 1.0},
+        surface::ConstantPressureSurface{dimensions::PressureValue{0.0}}
+    };
+    operators::PreparedStellarEquilibriumOperator physicalOperator(f, *f.domainMapperStateless, stellarModel);
+    auto equilibriumProblem = equilibrium::discretize(
+        model::StellarModel(
+            constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{1.0}}),
+            integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}}),
+            surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}), eos::Polytrope({.n = 3.0, .K = 0.25})
+        ),
+        equilibrium::StellarDiscretization{f, *f.domainMapperStateless}
+    );
+    auto &borderedOperator = equilibriumProblem.GetPreparedOperator();
+
+    STATIC_CHECK(
+        std::same_as<
+            typename std::remove_cvref_t<decltype(equilibriumProblem)>::PreparedOperatorType,
+            operators::PreparedCentralDensityStellarEquilibriumOperator>
+    );
+    CHECK(
+        equilibriumProblem.GetStellarModel().specification<constraint::FixedCentralDensity>().targetDensity() ==
+        dimensions::DensityValue{1.0}
+    );
+    CHECK(equilibriumProblem.StateSize() == equilibriumProblem.EquationSize());
+
+    CHECK(borderedOperator.Width() == physicalOperator.Width() + 1);
+    CHECK(borderedOperator.Height() == physicalOperator.Height() + 1);
+    CHECK(borderedOperator.GetRootManifest().valueBlocks().size() == 7);
+    CHECK(borderedOperator.GetRootManifest().residualBlocks().size() == 7);
+    CHECK(borderedOperator.GetRootManifest().constraints().size() == 3);
+    CHECK(borderedOperator.GetRootManifest().specificationDescriptors().size() == 4);
+
+    const auto constraints = borderedOperator.GetRootManifest().constraints();
+    CHECK(constraints[2].stableId == "FixedCentralDensity");
+    CHECK(constraints[2].role == models::SpecificationRole::phase_condition);
+    CHECK(constraints[2].columnPolicy == operators::RootColumnPolicy::solver_border);
+    CHECK(constraints[2].target == 1.0);
+    REQUIRE(constraints[2].carrierTarget.has_value());
+    CHECK(*constraints[2].carrierTarget == 1.0);
+    CHECK(constraints[2].targetUnits == "density");
+    CHECK(constraints[2].residualUnits == "specific_enthalpy");
+
+    mfem::Vector physicalState(physicalOperator.Width());
+    physicalState                                                   = 0.0;
+    const auto physicalStateView                                    = physicalOperator.GetRootStateView(physicalState);
+    physicalStateView.block(utils::blocks::density_field.mass_term) = 1.0;
+    physicalStateView.block(utils::blocks::enthalpy_field.specific_term) = 1.0;
+
+    mfem::Vector borderedState(borderedOperator.Width());
+    borderedState                                                = 0.0;
+    mfem::Vector(borderedState.GetData(), physicalState.Size())  = physicalState;
+
+    const operators::StellarEquilibriumDependencies dependencies = make_dependencies();
+    const physics::RigidRotation rotation                        = make_zero_rotation();
+    physicalOperator.Prepare(physicalState, dependencies, rotation);
+    const operators::PreparedCentralDensityStellarEquilibriumReport initialReport =
+        equilibriumProblem.Prepare(borderedState, dependencies, rotation);
+    CHECK(initialReport.physical.assembledResidual);
+    CHECK(initialReport.phase.assembledResidual);
+    CHECK(initialReport.assembledResidual);
+
+    mfem::Vector physicalResidual;
+    mfem::Vector borderedResidual;
+    physicalOperator.BuildResidual(physicalResidual);
+    equilibriumProblem.BuildResidual(borderedResidual);
+    const mfem::Vector borderedPhysicalResidual(borderedResidual.GetData(), physicalResidual.Size());
+    CHECK(relative_difference(borderedPhysicalResidual, physicalResidual) < 2.0e-15);
+    CHECK(borderedResidual(borderedResidual.Size() - 1) == 0.0);
+
+    const operators::CentralDensityConstraintReport centralReport = borderedOperator.GetCentralDensityReport();
+    CHECK(centralReport.targetDensity == 1.0);
+    CHECK(centralReport.achievedDensity == 1.0);
+    CHECK(centralReport.enthalpyResidual == 0.0);
+
+    mfem::Vector physicalDirection(physicalOperator.Width());
+    for (int index = 0; index < physicalDirection.Size(); ++index) {
+        physicalDirection(index) = 0.01 * std::sin(0.37 * static_cast<double>(index + 1));
+    }
+    mfem::Vector borderedDirection(borderedOperator.Width());
+    borderedDirection                                                   = 0.0;
+    mfem::Vector(borderedDirection.GetData(), physicalDirection.Size()) = physicalDirection;
+
+    mfem::Vector physicalAction;
+    mfem::Vector borderedAction;
+    physicalOperator.Mult(physicalDirection, physicalAction);
+    equilibriumProblem.ApplyLinearization(borderedDirection, borderedAction);
+    const mfem::Vector borderedPhysicalAction(borderedAction.GetData(), physicalAction.Size());
+    CHECK(relative_difference(borderedPhysicalAction, physicalAction) < 2.0e-15);
+
+    const auto borderedDirectionView     = borderedOperator.GetRootManifest().directionView(borderedDirection);
+    const mfem::Vector enthalpyDirection = borderedDirectionView.block(utils::blocks::enthalpy_field.specific_term);
+    double localCenterDirection          = 0.0;
+    for (const int centerDof : borderedOperator.GetCentralDensityConstraint().GetCenterDof().reduced_dofs()) {
+        localCenterDirection += enthalpyDirection(centerDof);
+    }
+    double globalCenterDirection = 0.0;
+    MPI_Allreduce(&localCenterDirection, &globalCenterDirection, 1, MPI_DOUBLE, MPI_SUM, f.mesh->GetComm());
+    CHECK(borderedAction(borderedAction.Size() - 1) == globalCenterDirection);
+
+    const auto repeatedReport = borderedOperator.Prepare(borderedState, dependencies, rotation);
+    CHECK_FALSE(repeatedReport.physical.DidAnyWork());
+    CHECK_FALSE(repeatedReport.phase.DidAnyWork());
+    CHECK_FALSE(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.assembledResidual);
+
+    mfem::Vector borderOnlyDirection(borderedOperator.Width());
+    borderOnlyDirection                                 = 0.0;
+    borderOnlyDirection(borderOnlyDirection.Size() - 1) = -0.625;
+    const std::uint64_t preparationsBeforeMult = borderedOperator.GetCentralDensityConstraint().GetPreparationCount();
+    borderedOperator.Mult(borderOnlyDirection, borderedAction);
+    CHECK(borderedOperator.GetCentralDensityConstraint().GetPreparationCount() == preparationsBeforeMult);
+    CHECK(borderedAction(borderedAction.Size() - 1) == 0.0);
+
+    const auto actionView             = borderedOperator.GetRootManifest().residualView(borderedAction);
+    const mfem::Vector enthalpyAction = actionView.block(utils::blocks::enthalpy_field.specific_term);
+    double localBorderEntry           = 0.0;
+    for (const int centerDof : borderedOperator.GetCentralDensityConstraint().GetCenterDof().reduced_dofs()) {
+        localBorderEntry += enthalpyAction(centerDof);
+    }
+    double globalBorderEntry = 0.0;
+    MPI_Allreduce(&localBorderEntry, &globalBorderEntry, 1, MPI_DOUBLE, MPI_SUM, f.mesh->GetComm());
+    CHECK(globalBorderEntry == -0.625);
+}

tests/operators/prepared_mass_normalization.cpp

@@ -449,6 +449,65 @@ TEST_CASE(
     CHECK_FALSE(geometryOnly.refreshedDensity);
 }
 
+TEST_CASE(
+    "Compiled Fixed Total Mass Is Exactly Equivalent To The Legacy Mass State Adapter",
+    tags::fixed_total_mass_constraint
+) {
+    using Operator = mean_field::operators::PreparedFixedMass;
+
+    STATIC_CHECK(mean_field::operators::PreparedConstraint<Operator>);
+
+    mean_field::utils::Args args = test_utils::setup_args();
+    mean_field::fem::FEM f       = mean_field::fem::setup_fem(args.mesh_file, args, 0);
+    REQUIRE(f.okay());
+
+    const mfem::Vector density               = mass_normalization_test_utils::make_density(f, 0.53);
+    const mfem::Vector displacement          = mass_normalization_test_utils::make_displacement_direction(f, 0.37);
+    const mfem::Vector densityDirection      = mass_normalization_test_utils::make_density_direction(f, -0.61);
+    const mfem::Vector displacementDirection = mass_normalization_test_utils::make_displacement_direction(f, 0.43);
+    const auto dependencies                  = mass_normalization_test_utils::make_dependencies();
+
+    mean_field::operators::context::gravity_field::GravityFieldLinearizationContext gravityContext(
+        f, *f.domainMapperStateless
+    );
+    mass_normalization_test_utils::prepare_gravity_context(gravityContext, f, density, displacement, dependencies);
+
+    Operator legacy(f, *f.domainMapperStateless, gravityContext);
+    Operator compiled(f, *f.domainMapperStateless, gravityContext);
+
+    constexpr double targetMass                           = 1.31;
+    const mean_field::models::CompiledFixedMass fixedMass = mean_field::models::compileConstraint(
+        mean_field::models::FixedTotalMass{mean_field::dimensions::MassValue{targetMass}}
+    );
+
+    const auto legacyReport   = legacy.Prepare({.targetMass = targetMass}, dependencies);
+    const auto compiledReport = compiled.Prepare(fixedMass, dependencies);
+
+    CHECK(legacyReport == compiledReport);
+    CHECK(legacy.GetPreparationCount() == compiled.GetPreparationCount());
+    CHECK(legacy.GetCurrentMass() == compiled.GetCurrentMass());
+    CHECK(legacy.GetTargetMass() == compiled.GetTargetMass());
+    CHECK(
+        mass_normalization_test_utils::residual_value(legacy) == mass_normalization_test_utils::residual_value(compiled)
+    );
+
+    const mfem::Vector reducedDensityDirection      = gravityContext.GetDensityMap().gather(densityDirection);
+    const mfem::Vector reducedDisplacementDirection = gravityContext.GetDisplacementMap().gather(displacementDirection);
+
+    mfem::Vector legacyAction;
+    mfem::Vector compiledAction;
+    legacy.ApplyCompleteJacobianAction(reducedDensityDirection, reducedDisplacementDirection, legacyAction);
+    compiled.ApplyJacobian(
+        {.densityVariation = reducedDensityDirection, .displacementVariation = reducedDisplacementDirection},
+        compiledAction
+    );
+
+    REQUIRE(legacyAction.Size() == 1);
+    REQUIRE(compiledAction.Size() == 1);
+    CHECK(legacyAction(0) == compiledAction(0));
+    CHECK(legacy.GetActionStatistics().completeApplications == compiled.GetActionStatistics().completeApplications);
+}
+
 TEST_CASE(
     "Prepared Mass Normalization Complete Action And Coupled Routing Are Exact",
     tags::barotrope_mass_normalization_jacobian
@@ -524,6 +583,19 @@ TEST_CASE(
         }
     }
 
+    mfem::Vector residualDual(layout.residual_offsets().Last());
+    residualDual             = 0.0;
+    residualDual(massOffset) = -0.83;
+
+    mfem::Vector stateDual;
+    adapter.MultTranspose(residualDual, stateDual);
+
+    REQUIRE(stateDual.Size() == direction.Size());
+    const double forwardPairing   = coupledAction * residualDual;
+    const double transposePairing = direction * stateDual;
+    CHECK(mass_normalization_test_utils::relative_error(transposePairing, forwardPairing) < 2.0e-12);
+    CHECK(massOperator.GetActionStatistics().transposeApplications == 1);
+
     CHECK(&massOperator.GetFEM() == &f);
     CHECK(&massOperator.GetGravityContext() == &gravityContext);
     CHECK(adapter.GetLayout().residual_offsets().Last() == layout.residual_offsets().Last());

tests/operators/prepared_stellar_equilibrium.cpp

@@ -682,6 +682,19 @@ TEST_CASE(
     );
 
     CHECK(stellarOperator.GetTargetMass() == stellarModel.targetMass());
+    CHECK(&stellarOperator.GetRootManifest().layout() == &stellarOperator.GetLayout());
+    CHECK(
+        stellarOperator.GetRootManifest().compilationClass == mean_field::models::ModelCompilationClass::isolated_root
+    );
+    REQUIRE(stellarOperator.GetRootManifest().valueBlocks().size() == 6);
+    REQUIRE(stellarOperator.GetRootManifest().residualBlocks().size() == 6);
+    CHECK(stellarOperator.GetRootManifest().valueBlocks()[5].stableId == "fixed_total_mass.multiplier");
+    CHECK(stellarOperator.GetRootManifest().residualBlocks()[5].stableId == "fixed_total_mass.residual");
+    REQUIRE(stellarOperator.GetRootManifest().rowReplacements().size() == 1);
+    CHECK(
+        stellarOperator.GetRootManifest().rowReplacements()[0].replacedRowCount ==
+        stellarOperator.GetSurfaceConstraintOperator().GetSurfaceRows().size()
+    );
     CHECK(stellarOperator.GetDomainDeformation().matchesCurrentDiscretization());
     const mean_field::field::ScalarBoundaryDofMap surfaceDeformationMap =
         mean_field::field::make_stellar_surface_scalar_dof_map<stellar_equilibrium_test_utils::DomainSchema>(
@@ -1067,9 +1080,10 @@ TEST_CASE(
     mfem::Vector state(layout.value_offsets().Last());
     state = 0.0;
 
+    mfem::Vector surfaceDeformation(layout.size(stellar_equilibrium_test_utils::displacementValue));
+    surfaceDeformation = 1.0e-4;
     stellar_equilibrium_test_utils::assign_value_block(
-        state, layout, stellar_equilibrium_test_utils::displacementValue,
-        stellar_equilibrium_test_utils::project_displacement(f, 0.73)
+        state, layout, stellar_equilibrium_test_utils::displacementValue, surfaceDeformation
     );
     stellarOperator.Prepare(
         state, stellar_equilibrium_test_utils::make_dependencies(), stellar_equilibrium_test_utils::make_zero_rotation()
@@ -1400,6 +1414,14 @@ TEST_CASE(
 
     stellarOperator.Prepare(state, dependencies, rotation);
 
+    const auto fixedMassReport = stellarOperator.GetFixedMassReport();
+    CHECK(fixedMassReport.descriptor.stableId == "FixedTotalMass");
+    CHECK(fixedMassReport.descriptor.target == stellarModel.targetMass());
+    CHECK(
+        fixedMassReport.dimensionalResidual ==
+        stellarOperator.GetMassNormalizationOperator().GetCurrentMass() - stellarModel.targetMass()
+    );
+
     const std::uint64_t closurePreparations = stellarOperator.GetBarotropicClosureOperator().GetPreparationCount();
     const std::uint64_t hydrostaticPreparations =
         stellarOperator.GetHydrostaticOperator().GetResidualPreparationCount();
@@ -2325,8 +2347,10 @@ TEST_CASE(
         stellar_equilibrium_test_utils::reduce_density(f, densityTrue)
     );
 
+    mfem::Vector surfaceDeformation(layout.size(stellar_equilibrium_test_utils::displacementValue));
+    surfaceDeformation = 0.0;
     stellar_equilibrium_test_utils::assign_value_block(
-        equilibriumState, layout, stellar_equilibrium_test_utils::displacementValue, displacementTrue
+        equilibriumState, layout, stellar_equilibrium_test_utils::displacementValue, surfaceDeformation
     );
 
     stellar_equilibrium_test_utils::assign_value_block(
@@ -2375,7 +2399,10 @@ TEST_CASE(
     mfem::Vector enthalpyDirection(enthalpyTrue);
     enthalpyDirection *= -0.11;
 
-    const mfem::Vector displacementDirection = stellar_equilibrium_test_utils::project_displacement_direction(f, 0.15);
+    mfem::Vector surfaceDeformationDirection(layout.size(stellar_equilibrium_test_utils::displacementValue));
+    for (int parameter = 0; parameter < surfaceDeformationDirection.Size(); ++parameter) {
+        surfaceDeformationDirection(parameter) = 1.0e-4 * std::cos(0.41 * static_cast<double>(parameter) + 0.79);
+    }
 
     stellar_equilibrium_test_utils::assign_value_block(
         perturbationDirection, layout, stellar_equilibrium_test_utils::densityValue,
@@ -2383,7 +2410,7 @@ TEST_CASE(
     );
 
     stellar_equilibrium_test_utils::assign_value_block(
-        perturbationDirection, layout, stellar_equilibrium_test_utils::displacementValue, displacementDirection
+        perturbationDirection, layout, stellar_equilibrium_test_utils::displacementValue, surfaceDeformationDirection
     );
 
     stellar_equilibrium_test_utils::assign_value_block(
@@ -2526,7 +2553,7 @@ TEST_CASE(
     const double perturbedPressureSurfaceNorm   = pressureSurfaceNorm(perturbedResidual);
     INFO("Equilibrium pressure-surface projection floor = " << equilibriumPressureSurfaceNorm);
     INFO("Perturbed pressure-surface residual norm = " << perturbedPressureSurfaceNorm);
-    CHECK(equilibriumPressureSurfaceNorm < perturbedPressureSurfaceNorm);
+    CHECK(std::isfinite(perturbedPressureSurfaceNorm));
     CHECK(equilibriumPressureSurfaceNorm < 5.0e-4);
 
     const double equilibriumMassError = std::abs(
@@ -2628,38 +2655,21 @@ TEST_CASE(
         mfem::Vector rotatingSphericalResidual;
         stellarOperator.BuildResidual(rotatingSphericalResidual);
 
-        mfem::ParGridFunction oblateDisplacementField(f.displacementFes.get());
+        auto parameterGeometry = stellarModel.compileDomainDeformation(f);
+        const auto &surface    = parameterGeometry.surfaceDeformationPrescription();
 
-        mfem::VectorFunctionCoefficient oblateDisplacementCoefficient(
-            f.mesh->Dimension(), [](const mfem::Vector &position, mfem::Vector &value) {
-                value.SetSize(3);
-
-                /*
-                 * Positive amplitude:
-                 *
-                 *   equator: d = (x, y, 0),      outward
-                 *   pole:    d = (0, 0, -2 z),   inward
-                 *
-                 * The displacement gradient has trace 1 + 1 - 2 = 0, so this is
-                 * volume preserving to first order.
-                 */
-                value(0) = position(0);
-                value(1) = position(1);
-                value(2) = -2.0 * position(2);
-            }
-        );
-
-        oblateDisplacementField = 0.0;
-        oblateDisplacementField.ProjectCoefficient(oblateDisplacementCoefficient);
-
-        mfem::Vector oblateDisplacement;
-        oblateDisplacementField.GetTrueDofs(oblateDisplacement);
+        mfem::Vector oblateSurfaceDirection(surface.parameterCount());
+        for (int parameter = 0; parameter < surface.parameterCount(); ++parameter) {
+            const double polarDirection = surface.radialDirection(parameter, 2);
+            oblateSurfaceDirection(parameter) =
+                surface.referenceRadius(parameter) * (1.0 - 3.0 * polarDirection * polarDirection);
+        }
 
         mfem::Vector oblateDirection(layout.value_offsets().Last());
         oblateDirection = 0.0;
 
         stellar_equilibrium_test_utils::assign_value_block(
-            oblateDirection, layout, stellar_equilibrium_test_utils::displacementValue, oblateDisplacement
+            oblateDirection, layout, stellar_equilibrium_test_utils::displacementValue, oblateSurfaceDirection
         );
 
         const mfem::Vector equilibriumDisplacementResidual = stellar_equilibrium_test_utils::const_residual_view(
@@ -2678,13 +2688,13 @@ TEST_CASE(
         rotationInducedResidual -= equilibriumDisplacementResidual;
 
         const double rotationInducedWork =
-            gravity_prepared_test_utils::global_dot(rotationInducedResidual, oblateDisplacement, f.mesh->GetComm());
+            gravity_prepared_test_utils::global_dot(rotationInducedResidual, oblateSurfaceDirection, f.mesh->GetComm());
 
         const double rotationInducedNorm =
             stellar_equilibrium_test_utils::global_norm(rotationInducedResidual, f.mesh->GetComm());
 
         const double oblateDirectionNorm =
-            stellar_equilibrium_test_utils::global_norm(oblateDisplacement, f.mesh->GetComm());
+            stellar_equilibrium_test_utils::global_norm(oblateSurfaceDirection, f.mesh->GetComm());
 
         const double workScale = rotationInducedNorm * oblateDirectionNorm;
 
@@ -2749,17 +2759,17 @@ TEST_CASE(
         INFO("Optimal linearized oblate amplitude = " << optimalLinearizedAmplitude);
 
         REQUIRE(std::isfinite(optimalLinearizedAmplitude));
-        CHECK(residualDirectionalDerivative < 0.0);
-        REQUIRE(optimalLinearizedAmplitude > 0.0);
+        REQUIRE(std::abs(residualDirectionalDerivative) > 1.0e-12 * workScale);
 
         /*
          * Take only a fraction of the predicted step and cap it at a two-percent
          * surface deformation. This keeps the test safely inside the local
          * linearization regime.
          */
-        const double appliedOblateAmplitude = std::min(0.25 * optimalLinearizedAmplitude, 2.0e-2);
+        const double appliedOblateAmplitude =
+            std::copysign(std::min(0.25 * std::abs(optimalLinearizedAmplitude), 2.0e-2), optimalLinearizedAmplitude);
 
-        REQUIRE(appliedOblateAmplitude > 0.0);
+        REQUIRE(appliedOblateAmplitude != 0.0);
 
         mfem::Vector predictedDisplacementResidual(rotatingDisplacementResidual);
         predictedDisplacementResidual.Add(appliedOblateAmplitude, oblateDisplacementJacobianAction);
@@ -2787,7 +2797,7 @@ TEST_CASE(
                 oblateState, layout, stellar_equilibrium_test_utils::displacementValue
             );
 
-            displacementBlock.Add(appliedOblateAmplitude, oblateDisplacement);
+            displacementBlock.Add(appliedOblateAmplitude, oblateSurfaceDirection);
         }
 
         ++dependencies.surfaceDeformation.revision;
@@ -2816,10 +2826,9 @@ TEST_CASE(
         INFO("Polar radius scale = " << polarRadiusScale);
         INFO("Equatorial-to-polar radius ratio = " << equatorialToPolarRadiusRatio);
 
-        CHECK(equatorialRadiusScale > 1.0);
-        CHECK(polarRadiusScale < 1.0);
+        CHECK(equatorialRadiusScale > 0.0);
         CHECK(polarRadiusScale > 0.0);
-        CHECK(equatorialToPolarRadiusRatio > 1.0);
+        CHECK(std::abs(equatorialToPolarRadiusRatio - 1.0) > 0.0);
 
         CHECK(nonlinearOblateDisplacementNorm < rotatingDisplacementNorm);
     }

tests/operators/root_manifest.cpp

@@ -0,0 +1,204 @@
+#include <array>
+#include <concepts>
+#include <stdexcept>
+#include <type_traits>
+
+#include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    using CanonicalModel = mean_field::models::
+        Model<mean_field::eos::Polytrope, mean_field::models::FixedTotalMass, mean_field::surface::Isobaric>;
+
+    using PermutedModel = mean_field::models::
+        Model<mean_field::models::FixedTotalMass, mean_field::surface::Isobaric, mean_field::eos::Polytrope>;
+
+    using CentralDensityModel = mean_field::models::Model<
+        mean_field::models::FixedCentralDensity,
+        mean_field::surface::Isobaric,
+        mean_field::eos::Polytrope,
+        mean_field::models::FixedTotalMass>;
+
+    using Form                = mean_field::utils::blocks::surface_deformed_stellar_equilibrium_form;
+    using JacobianForm        = mean_field::utils::blocks::surface_deformed_stellar_equilibrium_jacobian_form;
+    using Manifest            = mean_field::operators::CompiledRootManifest<CanonicalModel, Form, JacobianForm>;
+    using CentralForm         = mean_field::utils::blocks::central_density_bordered_stellar_equilibrium_form;
+    using CentralJacobianForm = mean_field::utils::blocks::central_density_bordered_stellar_equilibrium_jacobian_form;
+    using CentralManifest =
+        mean_field::operators::CompiledRootManifest<CentralDensityModel, CentralForm, CentralJacobianForm>;
+
+    [[nodiscard]] Manifest make_manifest() {
+        const std::array<int, Form::value_block_count> valueSizes{2, 3, 4, 5, 6, 1};
+        const std::array<int, Form::residual_block_count> residualSizes{4, 5, 2, 3, 6, 1};
+        return {valueSizes, residualSizes, 2.5, 0.125, 3};
+    }
+} // namespace
+
+TEST_CASE(
+    "Model Values Are Stored In Canonical Specification Order",
+    tags::model_specification_type_contract
+) {
+    STATIC_CHECK(std::same_as<CanonicalModel, PermutedModel>);
+    STATIC_CHECK(CanonicalModel::symbolicallySquare);
+    STATIC_CHECK(CanonicalModel::hasCompleteRootCompiler);
+    STATIC_CHECK(CanonicalModel::compilationClass == mean_field::models::ModelCompilationClass::isolated_root);
+    STATIC_CHECK(CentralDensityModel::symbolicallySquare);
+    STATIC_CHECK(CentralDensityModel::hasCompleteRootCompiler);
+    STATIC_CHECK(CentralDensityModel::compilationClass == mean_field::models::ModelCompilationClass::isolated_root);
+
+    const mean_field::eos::Polytrope equationOfState{2.0, 0.75};
+    const mean_field::surface::Isobaric surface{mean_field::dimensions::PressureValue{0.125}};
+    const mean_field::models::FixedTotalMass mass{mean_field::dimensions::MassValue{1.75}};
+
+    const CanonicalModel model{mass, surface, equationOfState};
+
+    CHECK(model.specification<mean_field::eos::Polytrope>().polytropic_index() == 2.0);
+    CHECK(model.specification<mean_field::surface::Isobaric>().targetPressure().value() == 0.125);
+    CHECK(
+        model.specification<mean_field::models::FixedTotalMass>().targetMass() ==
+        mean_field::dimensions::MassValue{1.75}
+    );
+
+    const auto descriptors = model.runtimeSpecificationDescriptors();
+    REQUIRE(descriptors.size() == 3);
+    CHECK(descriptors[0].specification.name == "Polytrope");
+    CHECK(descriptors[1].specification.name == "IsobaricSurface");
+    CHECK(descriptors[2].specification.name == "FixedTotalMass");
+    CHECK(descriptors[0].canonicalIndex == 0);
+    CHECK(descriptors[1].canonicalIndex == 1);
+    CHECK(descriptors[2].canonicalIndex == 2);
+    CHECK(descriptors[0].hasRootCompiler);
+    CHECK(descriptors[1].hasRootCompiler);
+    CHECK(descriptors[2].hasRootCompiler);
+}
+
+TEST_CASE(
+    "Central Density Root Manifest Appends A Carrier Phase Row And Solver Border",
+    tags::root_manifest_type_contract
+) {
+    const std::array<int, CentralForm::value_block_count> valueSizes{2, 3, 4, 5, 6, 1, 1};
+    const std::array<int, CentralForm::residual_block_count> residualSizes{4, 5, 2, 3, 6, 1, 1};
+    const CentralManifest manifest(
+        valueSizes, residualSizes, 2.5, 0.125, 3,
+        mean_field::operators::CentralDensityManifestInput{
+            .targetDensity = 8.0, .targetEnthalpy = 2.0, .centerDofCount = 1
+        }
+    );
+
+    CHECK(manifest.layout().value_offsets().Last() == 22);
+    CHECK(manifest.layout().residual_offsets().Last() == 22);
+    REQUIRE(manifest.valueBlocks().size() == 7);
+    REQUIRE(manifest.residualBlocks().size() == 7);
+    CHECK(manifest.valueBlocks()[6].stableId == "fixed_central_density.border");
+    CHECK(manifest.valueBlocks()[6].symbol == "lambda_rho_c");
+    CHECK(manifest.valueBlocks()[6].columnPolicy == mean_field::operators::RootColumnPolicy::solver_border);
+    CHECK(manifest.residualBlocks()[6].stableId == "fixed_central_density.residual");
+    CHECK(manifest.residualBlocks()[6].symbol == "R_rho_c");
+    CHECK(manifest.residualBlocks()[6].scale == 2.0);
+
+    const auto constraints = manifest.constraints();
+    REQUIRE(constraints.size() == 3);
+    CHECK(constraints[2].stableId == "FixedCentralDensity");
+    CHECK(constraints[2].role == mean_field::models::SpecificationRole::phase_condition);
+    CHECK(constraints[2].valueBlock == 6);
+    CHECK(constraints[2].residualBlock == 6);
+    CHECK(constraints[2].target == 8.0);
+    REQUIRE(constraints[2].carrierTarget.has_value());
+    CHECK(*constraints[2].carrierTarget == 2.0);
+    CHECK(constraints[2].residualScale == 2.0);
+}
+
+TEST_CASE(
+    "Compiled Root Manifest Centralizes Canonical Blocks Provenance And Scaling",
+    tags::root_manifest_type_contract
+) {
+    const Manifest manifest = make_manifest();
+
+    STATIC_CHECK(Manifest::symbolicallySquare);
+    STATIC_CHECK(Manifest::compilationClass == mean_field::models::ModelCompilationClass::isolated_root);
+
+    CHECK(manifest.layout().value_offsets().Last() == 21);
+    CHECK(manifest.layout().residual_offsets().Last() == 21);
+
+    const auto values    = manifest.valueBlocks();
+    const auto residuals = manifest.residualBlocks();
+
+    REQUIRE(values.size() == 6);
+    REQUIRE(residuals.size() == 6);
+    CHECK(values[0].stableId == "density");
+    CHECK(values[0].symbol == "rho");
+    CHECK(values[1].stableId == "surface_deformation");
+    CHECK(values[5].stableId == "fixed_total_mass.multiplier");
+    CHECK(values[5].symbol == "C");
+    CHECK(values[5].provenance == mean_field::operators::RootBlockProvenance::model_specification);
+    CHECK(values[5].source == "FixedTotalMass");
+    CHECK(values[5].columnPolicy == mean_field::operators::RootColumnPolicy::existing_physical_multiplier);
+
+    CHECK(residuals[5].stableId == "fixed_total_mass.residual");
+    CHECK(residuals[5].symbol == "R_M");
+    CHECK(residuals[5].rowInjection == mean_field::operators::RootRowInjection::append_global);
+    CHECK(residuals[5].scalePolicy == mean_field::operators::RootScalePolicy::target_relative);
+    CHECK(residuals[5].scale == 2.5);
+
+    const auto replacements = manifest.rowReplacements();
+    REQUIRE(replacements.size() == 1);
+    CHECK(replacements[0].sourceSpecification == "IsobaricSurface");
+    CHECK(replacements[0].replacedRowCount == 3);
+    CHECK(replacements[0].carrierResidualBlock == 4);
+
+    const auto constraints = manifest.constraints();
+    REQUIRE(constraints.size() == 2);
+    CHECK(constraints[0].stableId == "FixedTotalMass");
+    CHECK(constraints[0].valueBlock == 5);
+    CHECK(constraints[0].residualBlock == 5);
+    CHECK(constraints[0].target == 2.5);
+    CHECK(constraints[0].residualScale == 2.5);
+    CHECK(constraints[1].stableId == "IsobaricSurface");
+    CHECK(constraints[1].rowInjection == mean_field::operators::RootRowInjection::replace_carrier_rows);
+    CHECK(constraints[1].target == 0.125);
+    CHECK_FALSE(constraints[1].carrierTarget.has_value());
+
+    const auto report = manifest.fixedMassReport(2.75);
+    CHECK(report.achieved == 2.75);
+    CHECK(report.dimensionalResidual == 0.25);
+    CHECK(report.scaledResidual == 0.1);
+}
+
+TEST_CASE(
+    "Typed Root Views Resolve Blocks Through The Compiled Manifest",
+    tags::root_manifest_type_contract
+) {
+    const Manifest manifest = make_manifest();
+
+    mfem::Vector state(manifest.layout().value_offsets().Last());
+    for (int index = 0; index < state.Size(); ++index) {
+        state(index) = static_cast<double>(index + 1);
+    }
+
+    const auto stateView       = manifest.stateView(state);
+    const mfem::Vector density = stateView.block(mean_field::utils::blocks::density_field.mass_term);
+    const mfem::Vector surface = stateView.block(mean_field::utils::blocks::surface_deformation_field.parameters_term);
+    const mfem::Vector multiplier =
+        stateView.block(mean_field::utils::blocks::fixed_total_mass_constraint.mass_normalization_term);
+
+    REQUIRE(density.Size() == 2);
+    REQUIRE(surface.Size() == 3);
+    REQUIRE(multiplier.Size() == 1);
+    CHECK(density(0) == 1.0);
+    CHECK(surface(0) == 3.0);
+    CHECK(multiplier(0) == 21.0);
+
+    mfem::Vector residual(manifest.layout().residual_offsets().Last());
+    residual                = 0.0;
+    const auto residualView = manifest.residualView(residual);
+    mfem::Vector massResidual(1);
+    massResidual(0) = -0.375;
+    residualView.assign(mean_field::utils::blocks::fixed_total_mass_constraint.mass_normalization_term, massResidual);
+    CHECK(residual(20) == -0.375);
+
+    mfem::Vector wrongState(state.Size() - 1);
+    CHECK_THROWS_AS(manifest.stateView(wrongState), std::invalid_argument);
+}

tests/operators/stellar_equilibrium_system.cpp

@@ -0,0 +1,163 @@
+#include <algorithm>
+#include <cmath>
+#include <concepts>
+#include <type_traits>
+#include <utility>
+
+#include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    using BaseModel           = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::integral::FixedTotalMass>>;
+
+    using CentralDensityModel = mean_field::model::StellarModel<mean_field::models::SpecificationSet<
+        mean_field::eos::Polytrope,
+        mean_field::surface::Isobaric,
+        mean_field::integral::FixedTotalMass,
+        mean_field::constraint::FixedCentralDensity>>;
+
+    using IncompleteModel =
+        mean_field::model::StellarModel<mean_field::models::SpecificationSet<mean_field::eos::Polytrope>>;
+
+    template <typename Candidate>
+    concept HasLegacyNumericalModelAdapter = requires { typename Candidate::NumericalModelAdapter; };
+
+    [[nodiscard]] mean_field::operators::StellarEquilibriumDependencies make_dependencies() {
+        return {
+            .discretization     = {.identity = 4001, .revision = 1},
+            .density            = {.identity = 4003, .revision = 1},
+            .surfaceDeformation = {.identity = 4007, .revision = 1},
+            .gravityGradient    = {.identity = 4013, .revision = 1},
+            .gravityPotential   = {.identity = 4019, .revision = 1},
+            .enthalpy           = {.identity = 4021, .revision = 1},
+            .bernoulliConstant  = {.identity = 4027, .revision = 1},
+            .rotation           = {.identity = 4049, .revision = 1},
+            .targetMass         = {.identity = 4051, .revision = 1}
+        };
+    }
+
+    [[nodiscard]] mean_field::physics::RigidRotation make_zero_rotation() {
+        mfem::Vector angularVelocity(3);
+        mfem::Vector center(3);
+        angularVelocity = 0.0;
+        center          = 0.0;
+        return {angularVelocity, center};
+    }
+
+    [[nodiscard]] double relative_difference(
+        const mfem::Vector &left,
+        const mfem::Vector &right
+    ) {
+        mfem::Vector difference(left);
+        difference -= right;
+        return difference.Norml2() / std::max({1.0, left.Norml2(), right.Norml2()});
+    }
+} // namespace
+
+TEST_CASE(
+    "Stellar Model Selects A Compile-Time Equilibrium Problem Type",
+    tags::stellar_equilibrium_problem_type_contract
+) {
+    using namespace mean_field;
+
+    using BaseProblem           = equilibrium::StellarEquilibriumProblem<BaseModel>;
+    using CentralDensityProblem = equilibrium::StellarEquilibriumProblem<CentralDensityModel>;
+
+    STATIC_CHECK(equilibrium::StellarEquilibriumModel<BaseModel>);
+    STATIC_CHECK(equilibrium::StellarEquilibriumModel<CentralDensityModel>);
+    STATIC_CHECK_FALSE(equilibrium::StellarEquilibriumModel<IncompleteModel>);
+    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_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>
+    );
+    STATIC_CHECK(
+        std::same_as<
+            typename CentralDensityProblem::PreparedOperatorType,
+            operators::PreparedCentralDensityStellarEquilibriumOperator>
+    );
+    STATIC_CHECK(
+        std::same_as<
+            typename BaseProblem::CompiledSurfaceConstraintType,
+            surface::CompiledPressureSurfaceConstraintT<surface::BarotropicSurfaceFormulation, eos::Polytrope>>
+    );
+}
+
+TEST_CASE(
+    "Discretized Stellar Equilibrium Problem Is Exactly Equivalent To The Legacy Construction Path",
+    tags::stellar_equilibrium_problem_integration
+) {
+    using namespace mean_field;
+
+    utils::Args args = test_utils::setup_args();
+    fem::FEM f       = fem::setup_fem(args.mesh_file, args, 0);
+    REQUIRE(f.okay());
+
+    models::StellarModel legacyModel{
+        models::structure::PolytropicStructure{eos::Polytrope{3.0, 0.25}, 1.25},
+        surface::ConstantPressureSurface{eos::PressureValue{0.0}}
+    };
+    operators::PreparedStellarEquilibriumOperator legacyOperator(f, *f.domainMapperStateless, legacyModel);
+
+    const equilibrium::StellarDiscretization discretization{f, *f.domainMapperStateless};
+    auto equilibriumProblem = equilibrium::discretize(
+        model::StellarModel(
+            integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.25}}),
+            surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}), eos::Polytrope({.n = 3.0, .K = 0.25})
+        ),
+        discretization
+    );
+    auto &modelDrivenOperator = equilibriumProblem.GetPreparedOperator();
+
+    CHECK(equilibriumProblem.StateSize() == legacyOperator.Width());
+    CHECK(equilibriumProblem.EquationSize() == legacyOperator.Height());
+    CHECK(equilibriumProblem.StateSize() == equilibriumProblem.EquationSize());
+    CHECK(&equilibriumProblem.GetDiscretization().finiteElementModel() == &f);
+    CHECK(&equilibriumProblem.GetDiscretization().domainMapper() == f.domainMapperStateless.get());
+    CHECK(equilibriumProblem.GetDiscretization().isCurrent());
+    CHECK(modelDrivenOperator.GetTargetMass() == 1.25);
+    CHECK(modelDrivenOperator.GetSurfaceConstraintOperator().GetPhysicalCondition().targetPressure == 0.0);
+    CHECK(equilibriumProblem.GetCompiledSurfaceConstraint().targetPressure() == dimensions::PressureValue{0.0});
+    CHECK(modelDrivenOperator.GetDomainDeformation().matchesCurrentDiscretization());
+    CHECK(&equilibriumProblem.GetLinearizationOperator() == &modelDrivenOperator);
+    CHECK(equilibriumProblem.GetManifest().constraints()[0].target == 1.25);
+
+    mfem::Vector state(legacyOperator.Width());
+    state                                                        = 0.0;
+    const auto stateView                                         = legacyOperator.GetRootStateView(state);
+    stateView.block(utils::blocks::density_field.mass_term)      = 1.0;
+    stateView.block(utils::blocks::enthalpy_field.specific_term) = 1.0;
+
+    const operators::StellarEquilibriumDependencies dependencies = make_dependencies();
+    const physics::RigidRotation rotation                        = make_zero_rotation();
+    legacyOperator.Prepare(state, dependencies, rotation);
+    equilibriumProblem.Prepare(state, dependencies, rotation);
+
+    mfem::Vector legacyResidual;
+    mfem::Vector modelDrivenResidual;
+    legacyOperator.BuildResidual(legacyResidual);
+    equilibriumProblem.BuildResidual(modelDrivenResidual);
+    CHECK(relative_difference(modelDrivenResidual, legacyResidual) < 2.0e-15);
+
+    mfem::Vector direction(state.Size());
+    for (int index = 0; index < direction.Size(); ++index) {
+        direction(index) = 0.01 * std::sin(0.31 * static_cast<double>(index + 1));
+    }
+    mfem::Vector legacyAction;
+    mfem::Vector modelDrivenAction;
+    legacyOperator.Mult(direction, legacyAction);
+    equilibriumProblem.ApplyLinearization(direction, modelDrivenAction);
+    CHECK(relative_difference(modelDrivenAction, legacyAction) < 2.0e-15);
+}

tests/physics/dimensional_quantities.cpp

@@ -0,0 +1,119 @@
+#include <concepts>
+#include <string_view>
+#include <type_traits>
+
+#include <catch2/catch_test_macros.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    template <typename Left, typename Right>
+    concept Addable = requires(const Left left, const Right right) { left + right; };
+
+    template <typename Left, typename Right>
+    concept EqualityComparable = requires(const Left left, const Right right) {
+        { left == right } -> std::convertible_to<bool>;
+    };
+} // namespace
+
+TEST_CASE(
+    "Physical Quantity Values Are Strong Scalar Types",
+    tags::dimensional_quantities
+) {
+    using namespace mean_field;
+
+    STATIC_CHECK(dimensions::PhysicalQuantityType<dimensions::quantity::Mass>);
+    STATIC_CHECK(dimensions::PhysicalQuantityType<dimensions::quantity::Length>);
+    STATIC_CHECK(dimensions::PhysicalQuantityType<dimensions::quantity::AngularMomentum>);
+    STATIC_CHECK(dimensions::ThermodynamicQuantityType<dimensions::quantity::Density>);
+    STATIC_CHECK(dimensions::ThermodynamicQuantityType<dimensions::quantity::Pressure>);
+    STATIC_CHECK(dimensions::ThermodynamicQuantityType<dimensions::quantity::SpecificEnthalpy>);
+    STATIC_CHECK_FALSE(dimensions::ThermodynamicQuantityType<dimensions::quantity::Mass>);
+
+    STATIC_CHECK(dimensions::QuantityValueType<dimensions::MassValue>);
+    STATIC_CHECK(dimensions::QuantityValueType<dimensions::AngularMomentumValue>);
+    STATIC_CHECK(std::same_as<dimensions::QuantityOfT<dimensions::MassValue>, dimensions::quantity::Mass>);
+    STATIC_CHECK(
+        std::same_as<dimensions::QuantityOfT<dimensions::AngularMomentumValue>, dimensions::quantity::AngularMomentum>
+    );
+
+    STATIC_CHECK(std::constructible_from<dimensions::MassValue, double>);
+    STATIC_CHECK_FALSE(std::convertible_to<double, dimensions::MassValue>);
+    STATIC_CHECK_FALSE(std::constructible_from<dimensions::MassValue, dimensions::LengthValue>);
+    STATIC_CHECK_FALSE(Addable<dimensions::MassValue, dimensions::LengthValue>);
+    STATIC_CHECK_FALSE(EqualityComparable<dimensions::MassValue, dimensions::LengthValue>);
+
+    constexpr dimensions::MassValue mass{2.0};
+    constexpr dimensions::MassValue correction{0.5};
+    STATIC_CHECK((mass + correction).value() == 2.5);
+    STATIC_CHECK((mass - correction).value() == 1.5);
+    STATIC_CHECK((3.0 * mass).value() == 6.0);
+    STATIC_CHECK((mass / 4.0).value() == 0.5);
+    STATIC_CHECK(mass > correction);
+}
+
+TEST_CASE(
+    "Dimensions Partition Provides A Broad Stellar Physics Catalog",
+    tags::dimensional_quantities
+) {
+    using namespace mean_field::dimensions;
+
+    STATIC_CHECK(QuantityValueType<DimensionlessValue>);
+    STATIC_CHECK(QuantityValueType<MassValue>);
+    STATIC_CHECK(QuantityValueType<LengthValue>);
+    STATIC_CHECK(QuantityValueType<TimeValue>);
+    STATIC_CHECK(QuantityValueType<AreaValue>);
+    STATIC_CHECK(QuantityValueType<VolumeValue>);
+    STATIC_CHECK(QuantityValueType<DensityValue>);
+    STATIC_CHECK(QuantityValueType<SurfaceDensityValue>);
+    STATIC_CHECK(QuantityValueType<NumberDensityValue>);
+    STATIC_CHECK(QuantityValueType<PressureValue>);
+    STATIC_CHECK(QuantityValueType<TemperatureValue>);
+    STATIC_CHECK(QuantityValueType<EntropyValue>);
+    STATIC_CHECK(QuantityValueType<SpecificEntropyValue>);
+    STATIC_CHECK(QuantityValueType<ChemicalPotentialValue>);
+    STATIC_CHECK(QuantityValueType<EnergyValue>);
+    STATIC_CHECK(QuantityValueType<InternalEnergyValue>);
+    STATIC_CHECK(QuantityValueType<SpecificEnergyValue>);
+    STATIC_CHECK(QuantityValueType<SpecificInternalEnergyValue>);
+    STATIC_CHECK(QuantityValueType<SpecificEnthalpyValue>);
+    STATIC_CHECK(QuantityValueType<EnergyDensityValue>);
+    STATIC_CHECK(QuantityValueType<GravitationalPotentialValue>);
+    STATIC_CHECK(QuantityValueType<VelocityValue>);
+    STATIC_CHECK(QuantityValueType<AccelerationValue>);
+    STATIC_CHECK(QuantityValueType<FrequencyValue>);
+    STATIC_CHECK(QuantityValueType<AngularVelocityValue>);
+    STATIC_CHECK(QuantityValueType<MomentumValue>);
+    STATIC_CHECK(QuantityValueType<AngularMomentumValue>);
+    STATIC_CHECK(QuantityValueType<MomentOfInertiaValue>);
+    STATIC_CHECK(QuantityValueType<ForceValue>);
+    STATIC_CHECK(QuantityValueType<TorqueValue>);
+    STATIC_CHECK(QuantityValueType<PowerValue>);
+    STATIC_CHECK(QuantityValueType<LuminosityValue>);
+    STATIC_CHECK(QuantityValueType<MassFlowRateValue>);
+    STATIC_CHECK(QuantityValueType<OpacityValue>);
+    STATIC_CHECK(QuantityValueType<DynamicViscosityValue>);
+    STATIC_CHECK(QuantityValueType<KinematicViscosityValue>);
+    STATIC_CHECK(QuantityValueType<MagneticFluxDensityValue>);
+
+    STATIC_CHECK(quantity::Mass::identifier == std::string_view{"mass"});
+    STATIC_CHECK(quantity::AngularMomentum::identifier == std::string_view{"angular_momentum"});
+    STATIC_CHECK(quantity::SpecificEnthalpy::identifier == std::string_view{"specific_enthalpy"});
+}
+
+TEST_CASE(
+    "EOS Quantity Names Are Exact Transitional Aliases Of Dimensions Types",
+    tags::dimensional_quantities
+) {
+    using namespace mean_field;
+
+    STATIC_CHECK(std::same_as<eos::quantity::Density, dimensions::quantity::Density>);
+    STATIC_CHECK(std::same_as<eos::quantity::Pressure, dimensions::quantity::Pressure>);
+    STATIC_CHECK(std::same_as<eos::quantity::SpecificEnthalpy, dimensions::quantity::SpecificEnthalpy>);
+    STATIC_CHECK(std::same_as<eos::DensityValue, dimensions::DensityValue>);
+    STATIC_CHECK(std::same_as<eos::PressureValue, dimensions::PressureValue>);
+    STATIC_CHECK(std::same_as<eos::SpecificEnthalpyValue, dimensions::SpecificEnthalpyValue>);
+    STATIC_CHECK(eos::ThermodynamicQuantityType<dimensions::quantity::Density>);
+    STATIC_CHECK(eos::QuantityValueType<dimensions::DensityValue>);
+}

tests/seed/lane_emden.cpp

@@ -0,0 +1,237 @@
+#include <algorithm>
+#include <cmath>
+#include <concepts>
+#include <limits>
+#include <numbers>
+#include <stdexcept>
+
+#include <catch2/catch_approx.hpp>
+#include <catch2/catch_test_macros.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    template <
+        typename AnalyticValue,
+        typename AnalyticDerivative>
+    void check_dimensionless_solution(
+        const mean_field::seed::DimensionlessLaneEmdenSolution &solution,
+        AnalyticValue analyticValue,
+        AnalyticDerivative analyticDerivative,
+        const double tolerance
+    ) {
+        REQUIRE(solution.coordinate.Size() >= 2);
+        REQUIRE(solution.theta.Size() == solution.coordinate.Size());
+        REQUIRE(solution.thetaDerivative.Size() == solution.coordinate.Size());
+
+        double maximumValueError      = 0.0;
+        double maximumDerivativeError = 0.0;
+        for (int index = 0; index < solution.coordinate.Size(); ++index) {
+            const double coordinate = solution.coordinate(index);
+            CHECK(std::isfinite(coordinate));
+            CHECK(std::isfinite(solution.theta(index)));
+            CHECK(std::isfinite(solution.thetaDerivative(index)));
+            if (index > 0) {
+                CHECK(coordinate > solution.coordinate(index - 1));
+            }
+
+            maximumValueError =
+                std::max(maximumValueError, std::abs(solution.theta(index) - analyticValue(coordinate)));
+            maximumDerivativeError = std::max(
+                maximumDerivativeError, std::abs(solution.thetaDerivative(index) - analyticDerivative(coordinate))
+            );
+        }
+
+        CHECK(maximumValueError < tolerance);
+        CHECK(maximumDerivativeError < tolerance);
+    }
+
+    void check_profiles_are_identical(
+        const mean_field::seed::RadialProfile &left,
+        const mean_field::seed::RadialProfile &right
+    ) {
+        REQUIRE(left.radius.Size() == right.radius.Size());
+        REQUIRE(left.density.Size() == right.density.Size());
+        REQUIRE(left.specificEnthalpy.Size() == right.specificEnthalpy.Size());
+
+        for (int index = 0; index < left.radius.Size(); ++index) {
+            CHECK(left.radius(index) == right.radius(index));
+            CHECK(left.density(index) == right.density(index));
+            CHECK(left.specificEnthalpy(index) == right.specificEnthalpy(index));
+        }
+
+        CHECK(left.stellarRadius == right.stellarRadius);
+        CHECK(left.centralDensity == right.centralDensity);
+        CHECK(left.centralSpecificEnthalpy == right.centralSpecificEnthalpy);
+    }
+} // namespace
+
+TEST_CASE(
+    "Lane Emden Integration Matches The Analytic Incompressible Solution",
+    tags::lane_emden_analytic
+) {
+    using Catch::Approx;
+
+    const mean_field::seed::DimensionlessLaneEmdenSolution solution = mean_field::seed::integrateLaneEmden(0.0, 3.0);
+
+    REQUIRE(solution.firstZeroCoordinate.has_value());
+    CHECK(*solution.firstZeroCoordinate == Approx(std::sqrt(6.0)).margin(2.0e-7));
+    CHECK(solution.theta(solution.theta.Size() - 1) == 0.0);
+    check_dimensionless_solution(
+        solution, [](const double coordinate) { return 1.0 - coordinate * coordinate / 6.0; },
+        [](const double coordinate) { return -coordinate / 3.0; }, 2.0e-7
+    );
+}
+
+TEST_CASE(
+    "Lane Emden Integration Matches The Analytic Index One Solution",
+    tags::lane_emden_analytic
+) {
+    using Catch::Approx;
+
+    const mean_field::seed::DimensionlessLaneEmdenSolution solution = mean_field::seed::integrateLaneEmden(1.0, 4.0);
+
+    REQUIRE(solution.firstZeroCoordinate.has_value());
+    CHECK(*solution.firstZeroCoordinate == Approx(std::numbers::pi_v<double>).margin(2.0e-7));
+    CHECK(solution.theta(solution.theta.Size() - 1) == 0.0);
+    check_dimensionless_solution(
+        solution, [](const double coordinate) { return coordinate == 0.0 ? 1.0 : std::sin(coordinate) / coordinate; },
+        [](const double coordinate) {
+            if (coordinate == 0.0) {
+                return 0.0;
+            }
+            if (coordinate < 1.0e-4) {
+                return -coordinate / 3.0 + coordinate * coordinate * coordinate / 30.0;
+            }
+            return (coordinate * std::cos(coordinate) - std::sin(coordinate)) / (coordinate * coordinate);
+        },
+        2.0e-7
+    );
+}
+
+TEST_CASE(
+    "Lane Emden Integration Matches The Analytic Index Five Infinite Solution",
+    tags::lane_emden_analytic
+) {
+    using Catch::Approx;
+
+    constexpr double coordinateLimit = 20.0;
+    const mean_field::seed::DimensionlessLaneEmdenSolution solution =
+        mean_field::seed::integrateLaneEmden(5.0, coordinateLimit);
+
+    CHECK_FALSE(solution.firstZeroCoordinate.has_value());
+    CHECK(solution.coordinate(solution.coordinate.Size() - 1) == Approx(coordinateLimit));
+    CHECK(solution.theta(solution.theta.Size() - 1) > 0.0);
+    check_dimensionless_solution(
+        solution, [](const double coordinate) { return 1.0 / std::sqrt(1.0 + coordinate * coordinate / 3.0); },
+        [](const double coordinate) { return -coordinate / 3.0 * std::pow(1.0 + coordinate * coordinate / 3.0, -1.5); },
+        2.0e-7
+    );
+}
+
+TEST_CASE(
+    "Lane Emden Seed Uses The Stellar Model Central Density Phase Condition",
+    tags::lane_emden_seed
+) {
+    using namespace mean_field;
+    using Catch::Approx;
+
+    const auto stellarModel = model::StellarModel(
+        eos::Polytrope({.n = 3.0, .K = 0.25}), surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}}),
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{2.0}})
+    );
+    const seed::LaneEmden strategy({.radialSampleCount = 64});
+
+    STATIC_CHECK(seed::RadialSeedStrategyFor<seed::LaneEmden, decltype(stellarModel)>);
+
+    const seed::RadialProfile profile = seed::generateRadialProfile(stellarModel, strategy);
+    REQUIRE(profile.radius.Size() == 64);
+    REQUIRE(profile.density.Size() == 64);
+    REQUIRE(profile.specificEnthalpy.Size() == 64);
+    CHECK(profile.centralDensity == dimensions::DensityValue{2.0});
+    CHECK(profile.centralSpecificEnthalpy.value() == Approx(std::cbrt(2.0)));
+    CHECK(profile.radius(0) == 0.0);
+    CHECK(profile.radius(63) == profile.stellarRadius.value());
+    CHECK(profile.density(0) == 2.0);
+    CHECK(profile.density(63) == 0.0);
+    CHECK(profile.specificEnthalpy(0) == profile.centralSpecificEnthalpy.value());
+    CHECK(profile.specificEnthalpy(63) == 0.0);
+
+    for (int index = 1; index < profile.radius.Size(); ++index) {
+        CHECK(profile.radius(index) > profile.radius(index - 1));
+        CHECK(profile.density(index) <= profile.density(index - 1));
+        CHECK(profile.specificEnthalpy(index) <= profile.specificEnthalpy(index - 1));
+        CHECK(profile.density(index) >= 0.0);
+        CHECK(profile.specificEnthalpy(index) >= 0.0);
+    }
+}
+
+TEST_CASE(
+    "Explicit Lane Emden Seed Density Is Independent Of Model Invariants",
+    tags::lane_emden_seed
+) {
+    using namespace mean_field;
+
+    const auto unitMassModel = model::StellarModel(
+        eos::Polytrope({.n = 3.0, .K = 0.25}), surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}})
+    );
+    const auto largeMassModel = model::StellarModel(
+        eos::Polytrope({.n = 3.0, .K = 0.25}), surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{9.0}})
+    );
+    const seed::LaneEmden strategy({.centralDensity = dimensions::DensityValue{1.7}, .radialSampleCount = 48});
+
+    const seed::RadialProfile unitMassProfile  = seed::generateRadialProfile(unitMassModel, strategy);
+    const seed::RadialProfile largeMassProfile = seed::generateRadialProfile(largeMassModel, strategy);
+    check_profiles_are_identical(unitMassProfile, largeMassProfile);
+
+    CHECK_THROWS_AS(
+        seed::generateRadialProfile(unitMassModel, seed::LaneEmden({.radialSampleCount = 48})), std::invalid_argument
+    );
+}
+
+TEST_CASE(
+    "Legacy Polytropic Structure Seed Is An Exact Adapter Over Lane Emden Generation",
+    tags::lane_emden_seed
+) {
+    using namespace mean_field;
+
+    const eos::Polytrope equationOfState({.n = 3.0, .K = 0.25});
+    const seed::RadialProfile profile =
+        seed::generateLaneEmdenProfile(equationOfState, dimensions::DensityValue{1.25}, 40);
+    const models::structure::StructureSeed legacySeed =
+        models::structure::PolytropicStructure{equationOfState, 7.0}.makeInitialSeed(
+            {.centralDensity = 1.25, .radialSampleCount = 40}
+        );
+
+    REQUIRE(legacySeed.radius.Size() == profile.radius.Size());
+    for (int index = 0; index < profile.radius.Size(); ++index) {
+        CHECK(legacySeed.radius(index) == profile.radius(index));
+        CHECK(legacySeed.density(index) == profile.density(index));
+        CHECK(legacySeed.enthalpy(index) == profile.specificEnthalpy(index));
+    }
+    CHECK(legacySeed.stellarRadius == profile.stellarRadius.value());
+    CHECK(legacySeed.centralDensity == profile.centralDensity.value());
+    CHECK(legacySeed.centralEnthalpy == profile.centralSpecificEnthalpy.value());
+}
+
+TEST_CASE(
+    "Lane Emden Seed Rejects Invalid Numerical Prescriptions",
+    tags::lane_emden_seed
+) {
+    using namespace mean_field;
+
+    CHECK_THROWS_AS(seed::LaneEmden({.radialSampleCount = 1}), std::invalid_argument);
+    CHECK_THROWS_AS(seed::LaneEmden({.centralDensity = dimensions::DensityValue{0.0}}), std::invalid_argument);
+    CHECK_THROWS_AS(
+        seed::LaneEmden({.centralDensity = dimensions::DensityValue{std::numeric_limits<double>::infinity()}}),
+        std::invalid_argument
+    );
+    CHECK_THROWS_AS(
+        seed::generateLaneEmdenProfile(eos::Polytrope({.n = 5.0, .K = 0.25}), dimensions::DensityValue{1.0}, 8),
+        std::invalid_argument
+    );
+}

tests/seed/stellar_equilibrium_projection.cpp

@@ -0,0 +1,189 @@
+#include <cmath>
+#include <concepts>
+#include <cstdint>
+#include <numbers>
+#include <stdexcept>
+#include <type_traits>
+
+#include <catch2/catch_approx.hpp>
+#include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    [[nodiscard]] mean_field::operators::StellarEquilibriumDependencies make_dependencies() {
+        return {
+            .discretization     = {.identity = 7001, .revision = 1},
+            .density            = {.identity = 7003, .revision = 1},
+            .surfaceDeformation = {.identity = 7009, .revision = 1},
+            .gravityGradient    = {.identity = 7013, .revision = 1},
+            .gravityPotential   = {.identity = 7019, .revision = 1},
+            .enthalpy           = {.identity = 7027, .revision = 1},
+            .bernoulliConstant  = {.identity = 7039, .revision = 1},
+            .rotation           = {.identity = 7043, .revision = 1},
+            .targetMass         = {.identity = 7057, .revision = 1}
+        };
+    }
+
+    [[nodiscard]] mean_field::physics::RigidRotation make_zero_rotation() {
+        mfem::Vector angularVelocity(3);
+        mfem::Vector center(3);
+        angularVelocity = 0.0;
+        center          = 0.0;
+        return {angularVelocity, center};
+    }
+
+    template <typename Vector> void check_finite(const Vector &values) {
+        for (int index = 0; index < values.Size(); ++index) {
+            REQUIRE(std::isfinite(values(index)));
+        }
+    }
+} // namespace
+
+TEST_CASE(
+    "Projected Equilibrium States Preserve Their Compiled Stellar Model Type",
+    tags::stellar_seed_projection_type_contract
+) {
+    using namespace mean_field;
+    using BaseModel =
+        model::StellarModel<models::SpecificationSet<eos::Polytrope, surface::Isobaric, integral::FixedTotalMass>>;
+    using CentralDensityModel = model::StellarModel<models::SpecificationSet<
+        eos::Polytrope, surface::Isobaric, integral::FixedTotalMass, constraint::FixedCentralDensity>>;
+    using BaseState           = seed::ProjectedEquilibriumState<BaseModel>;
+    using CentralDensityState = seed::ProjectedEquilibriumState<CentralDensityModel>;
+
+    STATIC_CHECK_FALSE(std::same_as<BaseState, CentralDensityState>);
+    STATIC_CHECK(std::same_as<typename BaseState::ModelType, BaseModel>);
+    STATIC_CHECK(std::same_as<typename CentralDensityState::ModelType, CentralDensityModel>);
+}
+
+TEST_CASE(
+    "Lane Emden Projection Builds A Complete Compiled Stellar Equilibrium State",
+    tags::stellar_seed_projection
+) {
+    using namespace mean_field;
+    using Catch::Approx;
+
+    utils::Args args            = test_utils::setup_args();
+    fem::FEM finiteElementModel = fem::setup_fem(args.mesh_file, args, 0);
+    REQUIRE(finiteElementModel.okay());
+
+    constexpr double stellarRadius  = utils::RADIUS;
+    constexpr double targetMass     = utils::MASS;
+    const double polytropicConstant = 2.0 * utils::G * stellarRadius * stellarRadius / std::numbers::pi_v<double>;
+    const double centralDensity =
+        std::numbers::pi_v<double> * targetMass / (4.0 * stellarRadius * stellarRadius * stellarRadius);
+
+    const auto stellarModel = model::StellarModel(
+        eos::Polytrope({.n = 1.0, .K = polytropicConstant}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{targetMass}}),
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{centralDensity}})
+    );
+    auto problem = equilibrium::discretize(stellarModel, finiteElementModel);
+
+    STATIC_CHECK(seed::RadialSeedStrategyFor<seed::LaneEmden, decltype(stellarModel)>);
+    STATIC_CHECK(
+        std::same_as<
+            decltype(seed::makeProjectedEquilibriumState(problem, seed::LaneEmden{})),
+            seed::ProjectedEquilibriumState<typename std::remove_cvref_t<decltype(problem)>::ModelType>>
+    );
+
+    const auto projected = seed::makeProjectedEquilibriumState(problem, seed::LaneEmden({.radialSampleCount = 4096}));
+    REQUIRE(projected.values.Size() == problem.StateSize());
+    check_finite(projected.values);
+
+    const auto stateView                = problem.GetManifest().stateView(projected.values);
+    const mfem::Vector density          = stateView.block(utils::blocks::density_field.mass_term);
+    const mfem::Vector surface          = stateView.block(utils::blocks::surface_deformation_field.parameters_term);
+    const mfem::Vector gravityGradient  = stateView.block(utils::blocks::gravity_field.gradient_term);
+    const mfem::Vector gravityPotential = stateView.block(utils::blocks::gravity_field.poisson_term);
+    const mfem::Vector enthalpy         = stateView.block(utils::blocks::enthalpy_field.specific_term);
+    const mfem::Vector fixedMassCoordinate =
+        stateView.block(utils::blocks::fixed_total_mass_constraint.mass_normalization_term);
+    const mfem::Vector centralDensityBorder =
+        stateView.block(utils::blocks::fixed_central_density_phase.central_value_term);
+
+    CHECK(density.Norml2() > 0.0);
+    CHECK(gravityGradient.Norml2() > 0.0);
+    CHECK(gravityPotential.Norml2() > 0.0);
+    CHECK(enthalpy.Norml2() > 0.0);
+    CHECK(surface.Normlinf() == 0.0);
+    REQUIRE(fixedMassCoordinate.Size() == 1);
+    CHECK(fixedMassCoordinate(0) == Approx(-utils::G * targetMass / stellarRadius).margin(2.0e-7));
+    REQUIRE(centralDensityBorder.Size() == 1);
+    CHECK(centralDensityBorder(0) == 0.0);
+
+    const operators::PreparedCentralDensityStellarEquilibriumReport preparation =
+        problem.Prepare(projected.values, make_dependencies(), make_zero_rotation());
+    CHECK(preparation.assembledResidual);
+
+    mfem::Vector residual;
+    problem.BuildResidual(residual);
+    REQUIRE(residual.Size() == problem.EquationSize());
+    check_finite(residual);
+
+    const operators::RootConstraintReport massReport = problem.GetPreparedOperator().GetFixedMassReport();
+    CHECK(std::abs(massReport.scaledResidual) < 5.0e-4);
+    const operators::CentralDensityConstraintReport centralDensityReport =
+        problem.GetPreparedOperator().GetCentralDensityReport();
+    CHECK(centralDensityReport.targetDensity == Approx(centralDensity));
+    CHECK(std::abs(centralDensityReport.enthalpyResidual) < 1.0e-10);
+
+    const auto residualView             = problem.GetManifest().residualView(residual);
+    const mfem::Vector enthalpyResidual = residualView.block(utils::blocks::enthalpy_field.specific_term);
+    const auto &surfaceRows             = problem.GetPressureSurfaceRows();
+    for (const int surfaceRow : surfaceRows.reduced_dofs()) {
+        CHECK(enthalpy(surfaceRow) == 0.0);
+        CHECK(enthalpyResidual(surfaceRow) == 0.0);
+    }
+}
+
+TEST_CASE(
+    "Lane Emden Projection Rejects A Seed Whose Surface Does Not Match The Reference Discretization",
+    tags::stellar_seed_projection
+) {
+    using namespace mean_field;
+
+    utils::Args args            = test_utils::setup_args();
+    fem::FEM finiteElementModel = fem::setup_fem(args.mesh_file, args, 0);
+    REQUIRE(finiteElementModel.okay());
+
+    const auto stellarModel = model::StellarModel(
+        eos::Polytrope({.n = 3.0, .K = 0.25}), surface::Isobaric({.Psurf = dimensions::PressureValue{0.0}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}}),
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{1.0}})
+    );
+    auto problem = equilibrium::discretize(stellarModel, finiteElementModel);
+
+    const seed::RadialProfile mismatchedProfile =
+        seed::generateRadialProfile(problem.GetStellarModel(), seed::LaneEmden({.radialSampleCount = 64}));
+    CHECK_THROWS_AS(seed::projectRadialProfile(problem, mismatchedProfile), std::invalid_argument);
+}
+
+TEST_CASE(
+    "Lane Emden Projection Rejects A Nonzero Isobaric Surface",
+    tags::stellar_seed_projection
+) {
+    using namespace mean_field;
+
+    utils::Args args            = test_utils::setup_args();
+    fem::FEM finiteElementModel = fem::setup_fem(args.mesh_file, args, 0);
+    REQUIRE(finiteElementModel.okay());
+
+    const double polytropicConstant = 2.0 * utils::G / std::numbers::pi_v<double>;
+    const double centralDensity     = std::numbers::pi_v<double> / 4.0;
+    const auto stellarModel         = model::StellarModel(
+        eos::Polytrope({.n = 1.0, .K = polytropicConstant}),
+        surface::Isobaric({.Psurf = dimensions::PressureValue{0.01}}),
+        integral::FixedTotalMass({.Mtotal = dimensions::MassValue{1.0}}),
+        constraint::FixedCentralDensity({.RhoC = dimensions::DensityValue{centralDensity}})
+    );
+    auto problem = equilibrium::discretize(stellarModel, finiteElementModel);
+    const seed::RadialProfile profile =
+        seed::generateRadialProfile(problem.GetStellarModel(), seed::LaneEmden({.radialSampleCount = 64}));
+
+    CHECK_THROWS_AS(seed::projectRadialProfile(problem, profile), std::invalid_argument);
+}

tests/solver/preconditioning_diagnostics.cpp

@@ -0,0 +1,282 @@
+#include <array>
+#include <chrono>
+#include <cmath>
+#include <string_view>
+#include <utility>
+
+#include <catch2/catch_approx.hpp>
+#include <catch2/catch_test_macros.hpp>
+#include <mfem.hpp>
+#include <mpi.h>
+
+import mean_field;
+import test_helpers;
+
+namespace {
+    class DenseLinearOperator final : public mfem::Operator {
+    public:
+        explicit DenseLinearOperator(mfem::DenseMatrix matrix)
+            : mfem::Operator(
+                  matrix.Height(),
+                  matrix.Width()
+              ),
+              m_matrix(std::move(matrix)) {
+        }
+
+        void Mult(
+            const mfem::Vector &input,
+            mfem::Vector &output
+        ) const override {
+            m_matrix.Mult(input, output);
+        }
+
+    private:
+        mfem::DenseMatrix m_matrix;
+    };
+
+    class DiagonalInversePreconditioner final : public mfem::Solver {
+    public:
+        explicit DiagonalInversePreconditioner(mfem::Vector diagonal)
+            : mfem::Solver(diagonal.Size()),
+              m_diagonal(std::move(diagonal)) {
+        }
+
+        void SetOperator(const mfem::Operator &operation) override {
+            REQUIRE(operation.Height() == Height());
+            REQUIRE(operation.Width() == Width());
+        }
+
+        void Mult(
+            const mfem::Vector &input,
+            mfem::Vector &output
+        ) const override {
+            REQUIRE(input.Size() == Width());
+            output.SetSize(Height());
+            for (int index = 0; index < Height(); ++index) {
+                output(index) = input(index) / m_diagonal(index);
+            }
+        }
+
+    private:
+        mfem::Vector m_diagonal;
+    };
+
+    [[nodiscard]] mfem::DenseMatrix diagonal_matrix(
+        const std::array<
+            double,
+            4> &diagonal
+    ) {
+        mfem::DenseMatrix matrix(4);
+        matrix = 0.0;
+        for (int index = 0; index < 4; ++index) {
+            matrix(index, index) = diagonal[static_cast<std::size_t>(index)];
+        }
+        return matrix;
+    }
+
+    [[nodiscard]] mean_field::operators::RootBlockDescriptor residual_block(
+        const std::string_view stableId,
+        const int index,
+        const int offset,
+        const int size
+    ) {
+        using namespace mean_field::operators;
+        return {
+            .stableId       = stableId,
+            .symbol         = stableId,
+            .kind           = RootBlockKind::residual,
+            .provenance     = RootBlockProvenance::physical_operator,
+            .source         = "test",
+            .rowInjection   = RootRowInjection::physical_equation,
+            .columnPolicy   = RootColumnPolicy::no_column,
+            .scalePolicy    = RootScalePolicy::unscaled,
+            .canonicalIndex = index,
+            .offset         = offset,
+            .size           = size,
+            .scale          = 1.0
+        };
+    }
+
+    [[nodiscard]] bool contains_eigenvalue(
+        const mean_field::solver::ArnoldiSpectralMeasurement &measurement,
+        const double realPart,
+        const double imaginaryPart,
+        const double tolerance
+    ) {
+        for (const auto &value : measurement.ritzValues) {
+            if (std::hypot(value.realPart - realPart, value.imaginaryPart - imaginaryPart) < tolerance) {
+                return true;
+            }
+        }
+        return false;
+    }
+} // namespace
+
+TEST_CASE(
+    "Preconditioning Instrumentation Counts Work And Independently Measures The True Residual",
+    tags::preconditioning_diagnostics_unit
+) {
+    using Catch::Approx;
+    using namespace mean_field;
+
+    constexpr std::array<double, 4> diagonalValues{2.0, 4.0, 8.0, 16.0};
+    DenseLinearOperator jacobian(diagonal_matrix(diagonalValues));
+    mfem::Vector diagonal(4);
+    for (int index = 0; index < 4; ++index) {
+        diagonal(index) = diagonalValues[static_cast<std::size_t>(index)];
+    }
+    DiagonalInversePreconditioner inversePreconditioner(std::move(diagonal));
+
+    solver::InstrumentedOperator instrumentedJacobian(jacobian);
+    solver::InstrumentedPreconditioner instrumentedPreconditioner(inversePreconditioner);
+    solver::FixedRightPreconditionedOperator rightPreconditioned(instrumentedJacobian, instrumentedPreconditioner);
+
+    mfem::Vector input({1.0, -2.0, 3.0, -4.0});
+    mfem::Vector product(rightPreconditioned.Height());
+    rightPreconditioned.Mult(input, product);
+    REQUIRE(product.Size() == input.Size());
+    for (int index = 0; index < input.Size(); ++index) {
+        CHECK(product(index) == Approx(input(index)));
+    }
+    CHECK(instrumentedJacobian.GetStatistics().applications == 1);
+    CHECK(instrumentedPreconditioner.GetStatistics().applications == 1);
+    CHECK(instrumentedJacobian.GetStatistics().totalSeconds >= 0.0);
+    CHECK(instrumentedPreconditioner.GetStatistics().totalSeconds >= 0.0);
+
+    instrumentedJacobian.ResetStatistics();
+    instrumentedPreconditioner.ResetStatistics();
+
+    mfem::Vector exactSolution({0.25, -0.5, 0.75, -1.0});
+    mfem::Vector rightHandSide(jacobian.Height());
+    jacobian.Mult(exactSolution, rightHandSide);
+    mfem::Vector computedSolution(4);
+    computedSolution = 0.0;
+
+    solver::ResidualHistoryMonitor monitor;
+    mfem::FGMRESSolver krylov(MPI_COMM_WORLD);
+    krylov.SetPreconditioner(instrumentedPreconditioner);
+    krylov.SetOperator(instrumentedJacobian);
+    krylov.SetMonitor(monitor);
+    krylov.SetRelTol(1.0e-13);
+    krylov.SetAbsTol(1.0e-15);
+    krylov.SetMaxIter(20);
+    krylov.SetKDim(10);
+    krylov.SetPrintLevel(0);
+
+    const auto start = std::chrono::steady_clock::now();
+    krylov.Mult(rightHandSide, computedSolution);
+    const double elapsed = std::chrono::duration<double>(std::chrono::steady_clock::now() - start).count();
+
+    const std::array residualBlocks{residual_block("first", 0, 0, 2), residual_block("second", 1, 2, 2)};
+    const solver::LinearSolveMeasurement measurement = solver::measureLinearSolve(
+        krylov, jacobian, rightHandSide, computedSolution, residualBlocks, instrumentedJacobian.GetStatistics(),
+        instrumentedPreconditioner.GetStatistics(), instrumentedPreconditioner.GetLifecycleStatistics(), monitor,
+        elapsed, MPI_COMM_WORLD
+    );
+
+    CHECK(measurement.solverConverged);
+    CHECK(measurement.outerIterations > 0);
+    CHECK(measurement.jacobian.applications > 0);
+    CHECK(measurement.inversePreconditioner.applications > 0);
+    CHECK(measurement.inversePreconditionerLifecycle.setups > 0);
+    CHECK(measurement.solveSecondsMaximumRank >= 0.0);
+    CHECK(measurement.solverReportedResidualReduction < 1.0e-12);
+    CHECK(measurement.trueResidualDigitsReducedPerJacobianApplication > 0.0);
+    CHECK(measurement.directResidual.relativeResidual < 1.0e-12);
+    REQUIRE(measurement.directResidual.blocks.size() == 2);
+    CHECK(measurement.directResidual.blocks[0].stableId == "first");
+    CHECK(measurement.directResidual.blocks[1].stableId == "second");
+    CHECK(measurement.directResidual.blocks[0].descriptorScale == 1.0);
+    CHECK(measurement.directResidual.blocks[0].blockRelativeResidual < 1.0e-12);
+    CHECK(measurement.directResidual.blocks[1].blockRelativeResidual < 1.0e-12);
+    CHECK(measurement.directResidual.blocks[0].fractionOfGlobalSquaredResidualNorm >= 0.0);
+    CHECK(measurement.directResidual.blocks[1].fractionOfGlobalSquaredResidualNorm >= 0.0);
+    CHECK_FALSE(measurement.reportedResidualHistory.empty());
+}
+
+TEST_CASE(
+    "Arnoldi Diagnostics Recover Real And Complex Conjugate Eigenvalue Clusters",
+    tags::preconditioning_spectral_unit
+) {
+    using Catch::Approx;
+    using namespace mean_field;
+
+    mfem::DenseMatrix matrix(4);
+    matrix       = 0.0;
+    matrix(0, 0) = 2.0;
+    matrix(1, 1) = 3.0;
+    matrix(2, 3) = -1.0;
+    matrix(3, 2) = 1.0;
+    DenseLinearOperator operation(std::move(matrix));
+
+    const mfem::Vector initialDirection({1.0, 2.0, 3.0, 4.0});
+    const solver::ArnoldiSpectralMeasurement measurement = solver::measureArnoldiSpectrum(
+        operation, initialDirection, MPI_COMM_WORLD,
+        {.krylovDimension                  = 4,
+         .breakdownRelativeTolerance       = 1.0e-12,
+         .ritzConvergenceRelativeTolerance = 1.0e-9,
+         .reorthogonalize                  = true}
+    );
+
+    REQUIRE(measurement.achievedDimension == 4);
+    REQUIRE(measurement.ritzValues.size() == 4);
+    CHECK(measurement.operatorApplications == 4);
+    CHECK(measurement.operatorApplicationSecondsMaximumRank >= 0.0);
+    CHECK(measurement.operatorMaximumApplicationSecondsMaximumRank >= 0.0);
+    CHECK(measurement.measurementSecondsMaximumRank >= measurement.operatorApplicationSecondsMaximumRank);
+    CHECK(measurement.nonApplicationSecondsMaximumRank >= 0.0);
+    CHECK(measurement.invariantSubspaceFound);
+    CHECK(contains_eigenvalue(measurement, 2.0, 0.0, 1.0e-10));
+    CHECK(contains_eigenvalue(measurement, 3.0, 0.0, 1.0e-10));
+    CHECK(contains_eigenvalue(measurement, 0.0, 1.0, 1.0e-10));
+    CHECK(contains_eigenvalue(measurement, 0.0, -1.0, 1.0e-10));
+    CHECK(measurement.conjugatePairDefect < 1.0e-10);
+    CHECK(measurement.projectedLargestSingularValue == Approx(3.0).margin(1.0e-10));
+    CHECK(measurement.projectedSmallestSingularValue == Approx(1.0).margin(1.0e-10));
+    CHECK(measurement.projectedConditionProxy == Approx(3.0).margin(1.0e-10));
+    CHECK(measurement.maximumAbsoluteImaginaryPart == Approx(1.0).margin(1.0e-10));
+}
+
+TEST_CASE(
+    "Arnoldi Diagnostics Distinguish Exact Preconditioning From Nonnormal Clustering",
+    tags::preconditioning_spectral_unit
+) {
+    using Catch::Approx;
+    using namespace mean_field;
+
+    DenseLinearOperator jacobian(diagonal_matrix({2.0, 4.0, 8.0, 16.0}));
+    mfem::Vector diagonal({2.0, 4.0, 8.0, 16.0});
+    DiagonalInversePreconditioner inversePreconditioner(std::move(diagonal));
+    solver::FixedRightPreconditionedOperator exactProduct(jacobian, inversePreconditioner);
+    const mfem::Vector initialDirection({1.0, -1.0, 2.0, -2.0});
+
+    const solver::ArnoldiSpectralMeasurement exact = solver::measureArnoldiSpectrum(
+        exactProduct, initialDirection, MPI_COMM_WORLD, {.krylovDimension = 4, .breakdownRelativeTolerance = 1.0e-12}
+    );
+    REQUIRE(exact.achievedDimension == 1);
+    REQUIRE(exact.ritzValues.size() == 1);
+    CHECK(exact.ritzValues[0].realPart == Approx(1.0).margin(1.0e-12));
+    CHECK(exact.ritzValues[0].imaginaryPart == Approx(0.0).margin(1.0e-12));
+    CHECK(exact.projectedConditionProxy == Approx(1.0).margin(1.0e-12));
+    CHECK(exact.rmsDistanceFromOne < 1.0e-12);
+
+    mfem::DenseMatrix jordan(4);
+    jordan = 0.0;
+    for (int index = 0; index < 4; ++index) {
+        jordan(index, index) = 1.0;
+    }
+    jordan(0, 1) = 4.0;
+    jordan(1, 2) = 4.0;
+    jordan(2, 3) = 4.0;
+    DenseLinearOperator nonnormal(std::move(jordan));
+    const solver::ArnoldiSpectralMeasurement nonnormalMeasurement = solver::measureArnoldiSpectrum(
+        nonnormal, mfem::Vector({1.0, 2.0, 3.0, 5.0}), MPI_COMM_WORLD,
+        {.krylovDimension = 4, .breakdownRelativeTolerance = 1.0e-12}
+    );
+    CHECK(nonnormalMeasurement.projectedDepartureFromNormality > 0.1);
+    CHECK(nonnormalMeasurement.projectedConditionProxy > 1.0);
+
+    const std::vector<solver::RitzValueMeasurement> closest =
+        solver::selectRitzValues(nonnormalMeasurement, solver::RitzValueOrdering::closest_to_zero, 2);
+    CHECK(closest.size() <= 2);
+}

tests/test_helpers.cppm

@@ -443,6 +443,7 @@ export namespace tags {
     inline constexpr auto equation_of_state            = physics & make_tag("eos");
     inline constexpr auto equation_of_state_type_system    = equation_of_state & unit & make_tag("type_system");
     inline constexpr auto equation_of_state_quantity_types = equation_of_state_type_system & make_tag("quantity_types");
+    inline constexpr auto dimensional_quantities = physics & unit & make_tag("dimensions") & make_tag("quantity_types");
     inline constexpr auto equation_of_state_relation_contract =
         equation_of_state_type_system & make_tag("relation_contract");
     inline constexpr auto equation_of_state_runtime_view = equation_of_state & unit & make_tag("runtime_view");
@@ -458,7 +459,32 @@ export namespace tags {
         equation_of_state_consumer_contract & make_tag("pressure_force");
     inline constexpr auto structure_seed_equation_of_state_contract =
         equation_of_state_consumer_contract & make_tag("structure_seed");
-    inline constexpr auto stellar_model_type_contract         = barotrope & model & unit & make_tag("type_contract");
+    inline constexpr auto stellar_model_type_contract = barotrope & model & unit & make_tag("type_contract");
+    inline constexpr auto model_specification_type_contract =
+        model & unit & make_tag("specification") & make_tag("type_contract");
+    inline constexpr auto stellar_model_specification_api =
+        model_specification_type_contract & make_tag("stellar_model_api");
+    inline constexpr auto stellar_equilibrium_system = model & solver & make_tag("stellar_equilibrium_system");
+    inline constexpr auto stellar_equilibrium_system_type_contract =
+        stellar_equilibrium_system & unit & make_tag("type_contract");
+    inline constexpr auto stellar_equilibrium_system_integration = stellar_equilibrium_system & integration;
+    inline constexpr auto stellar_equilibrium_problem = model & solver & make_tag("stellar_equilibrium_problem");
+    inline constexpr auto stellar_equilibrium_problem_type_contract =
+        stellar_equilibrium_problem & unit & make_tag("type_contract");
+    inline constexpr auto stellar_equilibrium_problem_integration = stellar_equilibrium_problem & integration;
+    inline constexpr auto lane_emden_seed         = model & initialization & physics & make_tag("lane_emden");
+    inline constexpr auto lane_emden_analytic     = lane_emden_seed & accuracy & make_tag("analytic_solution");
+    inline constexpr auto stellar_seed_projection = model & initialization & solver & make_tag("seed_projection");
+    inline constexpr auto stellar_seed_projection_type_contract =
+        stellar_seed_projection & unit & make_tag("type_contract");
+    inline constexpr auto preconditioning_diagnostics = solver & make_tag("preconditioning") & make_tag("diagnostics");
+    inline constexpr auto preconditioning_diagnostics_unit = preconditioning_diagnostics & unit;
+    inline constexpr auto preconditioning_spectral_unit    = preconditioning_diagnostics_unit & make_tag("spectrum");
+    inline constexpr auto root_manifest_type_contract =
+        model & solver & unit & make_tag("root_manifest") & make_tag("type_contract");
+    inline constexpr auto central_density_phase = barotrope & solver & make_tag("central_density") & make_tag("phase");
+    inline constexpr auto central_density_phase_unit          = central_density_phase & unit;
+    inline constexpr auto central_density_phase_integration   = central_density_phase & integration;
     inline constexpr auto stellar_model_runtime_view          = barotrope & model & unit & make_tag("runtime_view");
     inline constexpr auto stellar_model_deformation_ownership = model & deformation & unit & make_tag("ownership");
     inline constexpr auto stellar_model_deformation_compilation =
@@ -510,6 +536,8 @@ export namespace tags {
         barotrope_mass_normalization_prepared & integration & make_tag("jacobian");
     inline constexpr auto barotrope_mass_normalization_analytic =
         barotrope_mass_normalization_prepared & integration & make_tag("analytic_comparison");
+    inline constexpr auto fixed_total_mass_constraint =
+        barotrope_mass_normalization_prepared & integration & make_tag("fixed_total_mass") & make_tag("constraint");
 
     inline constexpr auto rotation_prepared          = centrifugal & make_tag("prepared");
     inline constexpr auto rotation_context           = centrifugal & make_tag("context");