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+# Independent density order: controlled study
+
+Branch: `research/density-order-study-20260919`.
+Parent: `7ed4d23`, the completed negative-density investigation.
+The separate log-coordinate assessment is on `research/log-density-assessment-20260919` at `90580b2`.
+
+**Finding:** Q3 removes the resolved n=1 density undershoot, and Q4 substantially
+improves higher-index density accuracy. Neither provides positivity at higher n
+or resolves the n=3 coupled-solver stall in the common eight-step budget. All
+n=1 and n=1.5 controls converge and remain well virialized. Density remains an
+independent unknown. Log density was assessed mathematically and in small 1D
+controls, not implemented in the coupled stellar solver.
+
+## Question and controls
+
+Does increasing only the independent density approximation order reduce the
+observed unphysical undershoots, improve internal EOS consistency, and preserve
+virial accuracy at acceptable coupled-solver cost?
+
+The density field remains an independent unknown in every equilibrium solve.
+The frozen EOS projections are observations of approximation spaces; none is
+installed as a replacement field or used to alter a solver state.
+
+The baseline is discontinuous tensor Q2 density. The alternatives are Q3 and
+Q4 in the same Gauss-Legendre nodal family. Enthalpy remains continuous Q3;
+gravity retains its RT2/Q2 pair; displacement remains Q3; the mesh and geometry
+are unchanged. The surface uses the existing degree-four spherical harmonics
+with all azimuthal orders and simultaneous interior/surface solution.
+
+The active density block has 832 times `(p+1)^3` coefficients: 22,464 at Q2,
+53,248 at Q3 and 104,000 at Q4. Other field blocks retain their dimensions.
+The central-density constraint is present only in the n=3 cases.
+
+## Implementation and mathematical scope
+
+`MEAN_FIELD_DENSITY_POLYNOMIAL_ORDER` selects the base density order at build
+time; its default remains two. The existing uniform order increment still
+applies, and is zero throughout this study. The field registry supplies the
+selected order to typed field spaces, quadrature form queries, adapters,
+normalization and compiled block layouts. The physics-facing API is unchanged.
+
+Under the study's default policy the closure selects a rule covering both its
+density mass form and EOS source form. Their nominal degrees differ once density can have higher order
+than enthalpy. Selecting the greater resolved rule preserves policy overrides
+and the Q2 baseline. Adjacent requested orders may use the same MFEM Gauss rule,
+so a missing nominal bound does not by itself prove actual underintegration.
+
+Q3 is a useful n=1 control because the linear EOS image of Q3 enthalpy lies in
+the density space. For n=3, a general Q3 enthalpy cubed is Q9. For n=1.5 it is
+nonpolynomial. Raising density order therefore does not promise positivity for
+the higher-index models.
+
+There is also a discrete consistency benefit at Q3 and above. On each cell,
+Q3 enthalpy belongs to the DG density test space, even through the common
+curved VALUE mapping. If `a` represents that enthalpy in the density basis and
+`R` is the physical closure residual, the implemented polytropic EOS gives
+`sum_q w_q h_h rho_h - (n+1) sum_q w_q P(h_h) = a^T R`.
+A zero closure residual therefore enforces this enthalpy-weighted EOS moment
+cell by cell under the closure quadrature. Q2 does not generally enforce it.
+This algebraic property does not establish positivity or virial balance;
+other quadrature rules and finite residuals introduce their own defects, and
+the central-density border does not change this identity.
+
+## Frozen and coupled observations
+
+Frozen observations retain the saved enthalpy and geometry from the n=1 and
+n=1.5 nonrotating equilibria and the n=3 initial seed. Each target space receives
+an independently assembled physical-volume-weighted L2 projection, with one
+common q26 quadrature, and a nodal interpolant of `rho_EOS(h_h)`. The latter is
+explicitly not the original independently tabulated Lane-Emden seed.
+
+The squared EOS error has a higher degree than the projection equations. A
+separate shared q30 evaluation checks it against q26 without changing or
+re-solving the projected fields. Both rules are also used on coupled final
+states. At fractional n this is a numerical sensitivity check rather than a
+polynomial exactness argument.
+
+The coupled runs retain the same production quadrature policy, whose resolved
+rule can change when density order changes. Actual closure rules are saved for
+each case. The frozen common-rule controls isolate approximation order more
+strictly; the coupled comparison measures the complete consistent
+discretization, including its policy-selected quadrature adjustments.
+
+Density order does not enter the pressure-force rule or its enthalpy
+admissibility guard: their field operands are enthalpy and the displacement
+gradient. Pressure-preparation rejection counts therefore measure changed
+reached states and coupled directions, rather than a density-order change in
+pressure sampling. The current preparation order evaluates displacement and
+pressure before the barotropic closure. Closure therefore cannot preempt that
+pressure stage in these runs; all rejection sources are nevertheless retained
+separately rather than interpreting a stage count as total admissibility.
+
+Coupled runs use the original Lane-Emden initialization, the same normalization
+and default preconditioner, an FGMRES restart of 160, at most 200 linear
+iterations per step, relative linear tolerance 0.03, and at most eight Newton
+steps. Nonlinear relative tolerance is 1e-8 and absolute tolerance is zero.
+Cases cover n=1,1.5,3 without rotation and the established n=1,J=0.01 rotating
+control. A failure within this budget is a recorded trajectory outcome, not a
+proof that the discretization has no root.
+
+The initializer tabulates Lane–Emden density and enthalpy before constructing
+either FEM field, then interpolates them independently into their spaces
+(`libmeanfield/impl/seed/lane_emden.cpp`, lines 231–236, and
+`libmeanfield/impl/seed/stellar_equilibrium_projection.cpp`, lines 164–180).
+Gravity uses the represented density; it does not replace enthalpy. Thus the
+production density seed is `I_p(rho_LE_tabulated)`, whereas the frozen nodal
+and L2 controls are `I_p(rho_EOS(h_h))` and `Pi_p(rho_EOS(h_h))`.
+Nonlinear EOS evaluation does not commute with either radial or FEM
+interpolation. Increasing density order does not remove the discrepancy
+between the tabulated density target and the EOS image of represented Q3
+enthalpy. Source structure supports unchanged seeded enthalpy across these
+orders; no cross-run bitwise identity of the initial states is asserted.
+
+For n=3, the existing central-density phase condition constrains central
+enthalpy to its EOS value at the requested central density; it does not directly
+pin the independent DG density's point value. Bordered convergence is assessed
+alongside the residual with artificial central forcing removed and the
+normalized size of that forcing. That unbordered diagnostic retains the scalar
+constraint rows. A small bordered residual or virial alone cannot establish
+that the artificial forcing has vanished.
+
+All expensive builds, numerical cases and artifact audits run serially.
+Reported solver costs must exclude the post-solve observer and distinguish
+Newton steps, Krylov work, preparation and preconditioner refresh.
+
+The normalized residual and its relative stopping threshold belong to each
+order's discrete system. Changing density order changes its test space,
+diagonal Riesz weights and initial interpolant as well as possibly its resolved
+quadrature. A smaller final residual across orders is not a physical-error
+ranking. Krylov totals also reflect different Newton trajectories, warm starts
+and per-action costs; they do not by themselves measure preconditioner quality.
+Physical diagnostics and measured Newton time are therefore reported alongside
+the iteration counts and density block sizes.
+Timings are single observations on this workstation, with serial workloads;
+they are not repeated benchmark means or estimates with error bars.
+
+Density sign regions are integrated by isolating polynomial roots of degree
+two through four along reference-cell lines. The observer also checks surface
+traces, EOS mismatch, signed and negative mass, moments and unchanged source
+fields. It separates resolved negative witnesses from values at reconstruction
+uncertainty. Tangential sampling and mapped integration remain numerical
+observations, not global sign certificates.
+
+Here `M = integral rho_h dV` is signed mass,
+`M_- = -integral min(rho_h,0) dV`, and `V_-` is the volume where density is
+negative. Reported fractions are `M_-/abs(M)` and `V_-/V`; the denominator of
+the mass fraction is not the integral of absolute density. EOS L2 error means
+`sqrt(integral (rho_h-rho_EOS(h_h))^2 dV)` in the model's units, without an
+additional relative normalization.
+
+If the stored-state observer encounters negative enthalpy, complete EOS error
+integrals are marked undefined while density integrals remain reported. No
+density is clipped. Both virial definitions from pressure evaluated at enthalpy
+and, where admissible, at density are retained; the required virial tolerance
+remains 1e-5.
+
+The summary enthalpy minimum and EOS-availability flag refer to integration
+nodes. Surface enthalpy samples and the enthalpy at exported negative-density
+witnesses are reported separately. Sliced mass and volume are cross-checked
+against the physical observer; first moments and inertia are only compared,
+because their additional coordinate factors need not be integrated exactly by
+the fixed eight-point normal rule. Tangential refinement alone cannot establish
+accuracy in that normal direction.
+
+## Results
+
+### Frozen approximation controls
+
+The frozen campaign completed all three cases and its final artifact audit
+passed. The following uses the common-q26 L2 projection; EOS L2 errors use the
+separate unsliced q30 measurement. Negative measures use the finest root-sliced
+tangential rule. These are approximation controls, not new equilibrium solves.
+
+| Frozen model | Density order | M_- / abs(M) | Negative volume fraction | EOS L2 error |
+|---|---:|---:|---:|---:|
+| n=1 equilibrium | Q2 | 6.09585e-8 | 3.27346e-4 | 1.25266e-4 |
+| n=1 equilibrium | Q3 | 1.05e-31 (unresolved roundoff) | 2.49e-16 | 1.56e-15 |
+| n=1 equilibrium | Q4 | 1.25e-31 (unresolved roundoff) | 2.73e-16 | 1.60e-15 |
+| n=1.5 equilibrium | Q2 | 2.24299e-5 | 1.49181e-2 | 3.67965e-4 |
+| n=1.5 equilibrium | Q3 | 9.33602e-6 | 1.04476e-2 | 1.12325e-4 |
+| n=1.5 equilibrium | Q4 | 3.18362e-6 | 6.83440e-3 | 4.80663e-5 |
+| n=3 seed | Q2 | 2.92063e-5 | 1.21064e-1 | 1.06409e-2 |
+| n=3 seed | Q3 | 2.63486e-6 | 3.73200e-2 | 6.86223e-4 |
+| n=3 seed | Q4 | 1.02066e-8 | 1.69628e-2 | 3.84980e-5 |
+
+The n=1 control matches the representability argument: Q3 and Q4 contain the
+linear EOS image of Q3 enthalpy, and no resolved negative witnesses remain.
+Higher-order density does not guarantee positivity for either higher-index
+case. The n=3 Q4 negative mass is about 2,862 times smaller than Q2, yet negative
+density still occupies about 1.70% of the sampled stellar volume.
+
+The difference between n=1.5 and n=3 is consistent with surface regularity.
+Where enthalpy vanishes linearly, the fractional-index density behaves as
+distance-to-surface raised to 1.5 and has a singular second derivative. For
+n=3, cubing the represented enthalpy gives a polynomial Q9 target. Increasing
+density order can therefore improve the integer-index approximation very
+rapidly without resolving the conditioning of the coupled nonlinear equations.
+
+At n=1.5, changing the error rule from q26 to q30 changes the Q4 EOS L2 norm by
+about 0.18%; the corresponding fixed eight-point root-sliced norm differs by
+about 2.2%. This is why the unsliced q30 values are used above. Tangential
+refinement gives stable negative masses but cannot establish convergence of
+the fractional EOS error in the normal direction. The integral-index norms
+agree across q26/q30 to arithmetic accuracy, except for relative comparisons of
+n=1 Q3/Q4 errors that are already at roundoff.
+
+### Coupled results
+
+All nine n=1 and n=1.5 controls converged with full Newton steps and no rejected
+trials. Their energy and force virial errors from `P(h)` are all below 6.6e-7.
+The following times sum Newton iterations and exclude initialization and the
+post-solve density observer.
+
+| Model | Density | Newton updates | Krylov iterations | Newton seconds | M_- / abs(M) | EOS L2 error, q30 |
+|---|---:|---:|---:|---:|---:|---:|
+| n=1, J=0 | Q2 | 5 | 212 | 147.5 | 6.09585e-8 | 1.25266e-4 |
+| n=1, J=0 | Q3 | 6 | 286 | 204.3 | unresolved roundoff | 3.13581e-16 |
+| n=1, J=0 | Q4 | 6 | 286 | 215.1 | unresolved roundoff | 3.51021e-16 |
+| n=1, J=0.01 | Q2 | 6 | 296 | 196.1 | 6.11256e-8 | 1.25327e-4 |
+| n=1, J=0.01 | Q3 | 6 | 305 | 217.8 | unresolved roundoff | 7.26203e-16 |
+| n=1, J=0.01 | Q4 | 6 | 304 | 229.1 | unresolved roundoff | 3.90198e-16 |
+| n=1.5, J=0 | Q2 | 6 | 335 | 222.0 | 2.24535e-5 | 3.67965e-4 |
+| n=1.5, J=0 | Q3 | 6 | 329 | 241.7 | 9.35454e-6 | 1.12195e-4 |
+| n=1.5, J=0 | Q4 | 7 | 397 | 303.7 | 3.20115e-6 | 4.80640e-5 |
+
+For n=1, Q3/Q4 remove the resolved undershoot and reduce the EOS mismatch to
+roundoff. Q3 and Q4 also agree in their virial diagnostics to about 1e-14. The
+nonrotating Q3/Q4 total times are 39%/46% above Q2, but both took an extra
+Newton step: their fifth residuals were 1.91e-12 and 1.80e-12 against thresholds
+of about 1.55e-12 and 1.54e-12. Q2's fifth residual narrowly passed its own
+threshold. Those total-time differences should not be interpreted as pure
+per-iteration order overhead. The rotating control takes six updates at every
+order; Q3/Q4 cost about 11%/17% more there.
+
+For n=1.5, Q3 reduces negative mass by 2.40 times and EOS L2 error by 3.28 times
+for 8.9% more Newton time. Q4 reduces them by 7.01 and 7.66 times, respectively,
+for 36.8% more time. Its sixth residual, 2.26e-12, narrowly missed its 1.93e-12
+threshold; the seventh update cost 58.8 seconds. Negative volume fractions
+remain 1.493%, 1.046% and 0.685% at Q2, Q3 and Q4. These are converged discrete
+roots with a resolved density defect, not merely incomplete Newton iterates.
+
+Density/EOS consistency is not the only internal accuracy measure. At n=1.5,
+`hydrostatic_gradient_rms` is 0.00727475, 0.00731970 and 0.00729901 at Q2/Q3/Q4.
+It is the unscaled volume RMS of `grad(h) + g_h - grad(Psi)` in model
+acceleration units, using the independent gravity-gradient field `g_h`.
+It is neither a relative error nor the normalized algebraic residual. Thus
+density enrichment has not appreciably improved this local balance diagnostic.
+
+None of the three n=3 cases converged within eight updates. Their endpoints
+must not be treated as equally accurate equilibrium models or their times as
+time to solution.
+
+| Density | F_final / F_initial | Smallest accepted alpha | Krylov iterations | Newton seconds | Energy virial, P(h) | Force virial, P(h) |
+|---|---:|---:|---:|---:|---:|---:|
+| Q2 | 0.430062 | 1/64 | 221 | 301.6 | 1.77163e-4 | 1.34597e-4 |
+| Q3 | 0.985301 | 1/256 | 452 | 478.8 | 8.07504e-6 | 2.28169e-5 |
+| Q4 | 0.987949 | 1/512 | 458 | 508.2 | 1.37586e-5 | 2.84372e-5 |
+
+Q3 reaches the energy-virial milestone alone; its force virial and nonlinear
+residual still fail their respective criteria. Q4's better approximation
+space does not produce a better virial endpoint than Q3. Q2 recovers larger
+steps during this window, while Q3/Q4 make only about 1–1.5% residual progress.
+This is evidence of a stalled short trajectory, not a proof against eventual
+convergence with a longer run or different initialization.
+
+All recorded n=3 linear solves meet their true-residual tolerance. Q2 has four
+pressure-force rejections and four insufficient-decrease rejections; Q3 and
+Q4 each have five and nine. No other rejection stages occur in these runs.
+Both enriched cases first accept only 1/256 of their Newton direction. The
+unchanged pressure quadrature makes this a change in reached states and
+directions, not a change in pressure sampling. Increasing density order alone
+has not removed pressure-admissibility and insufficient-decrease failures.
+This campaign records those failures but does not independently localize the
+nonlinear mechanism responsible for insufficient decrease.
+
+The n=3 endpoint negative-mass fractions are 2.22117e-6, 9.13593e-7 and
+5.21562e-7, with negative volume fractions 3.440%, 2.931% and 2.619%.
+EOS L2 errors are 9.78088e-3, 8.99701e-4 and 5.08742e-4. These still-unconverged
+fields show much less improvement than the frozen projections; their
+different trajectories prevent interpreting those ratios as equilibrium
+convergence rates.
+
+The unbordered n=3 residuals remain 1.33327e-3, 7.18640e-4 and 7.26009e-4.
+Normalized central-border forcing is 3.17039e-5, 1.27669e-5 and 1.20966e-5.
+Thus neither a physical root nor vanishing artificial forcing has been
+demonstrated. Signed multipliers and both virial definitions are retained in
+the full tables.
+
+The initializer distinction also matters. At n=3, the production Q4 seed has
+an observed EOS L2 mismatch of about 5.26e-4 under q18, while the frozen Q4
+projection of represented enthalpy has 3.85e-5 under q30. These are different
+fields and quadratures, not a controlled improvement ratio. Together with the
+initializer's source structure, they show why refining the density space need
+not make the independently interpolated seed close to the weak closure
+manifold. A closure-consistent initial guess is a separate hypothesis; it
+could populate an independent density unknown without deriving it from
+enthalpy during the subsequent coupled solve.
+
+The actual closure rules also differ only at Q4 in this study:
+
+| Model index | Q2 rule: achieved order / points per cell | Q3 rule | Q4 rule |
+|---|---:|---:|---:|
+| n=1 | 17 / 729 | 17 / 729 | 19 / 1,000 |
+| n=1.5 | 19 / 1,000 | 19 / 1,000 | 21 / 1,331 |
+| n=3 | 23 / 1,728 | 23 / 1,728 | 25 / 2,197 |
+
+All 832 stellar elements use the listed rule in each case. Q2/Q3 therefore
+share closure sampling, while Q4's coupled accuracy/cost includes the policy-selected
+quadrature enrichment as well as the larger space.
+
+### Implications and recommendation
+
+Q3 is the economical consistency control for n=1: it contains the EOS image
+and Q4 adds no resolved accuracy benefit there. Q4 is a useful higher-index
+accuracy reference, but neither higher order is a positivity mechanism or a
+demonstrated n=3 convergence remedy. The production default remains Q2; these
+are isolated, reproducible research variants.
+
+The next density-representation experiment should test a pointwise exponential
+with an independently chosen vacuum reference, retaining density as an
+independent unknown and solving surface and interior together. Judge it by
+EOS consistency, accepted Newton steps, both virial diagnostics and cost—not
+positivity alone. A separate conservative, closure-consistent seed control
+would distinguish initialization effects from representation effects. Neither
+of these further stellar experiments was performed here.
+
+The complete tables and checked PNG/PDF figure are in
+`../../diagnostics/density_order_2026-09-19/final_analysis/` relative to the
+worktree. `trial_stages.csv` retains every source/disposition count, including
+accepted trials; `coupled_outcomes.csv` retains all endpoint diagnostics.
+
+[Comparison figure](../../../diagnostics/density_order_2026-09-19/final_analysis/density_order_comparison.png)
+and [coupled-result table](../../../diagnostics/density_order_2026-09-19/final_analysis/coupled_outcomes.csv).
+
+## Relation to the independent log-density hypothesis
+
+The separate `LOG_DENSITY_ASSESSMENT_2026-09-19.md` in the log-density worktree
+contains the derivation, primary-source MESA comparison and reproducible
+one-dimensional controls. It does not contain a coupled stellar log solve.
+
+Exponentiating polynomial nodal coefficients cannot remove the observed
+between-node undershoots: the current negative fields can already have positive
+nodal values. A pointwise representation `rho = rho_scale * exp(u_h)` changes
+the trial family and gives positive interior density in exact arithmetic while
+keeping density independent. A finite polynomial log field cannot also have an
+exact zero-density trace.
+
+For the present vacuum polytrope, the more suitable hypothesis is
+`rho = rho_ref * exp(u_h)`, where a reference chosen independently of live
+enthalpy is positive inside and vanishes at the surface with the correct
+leading exponent. That exponent is a polytropic assumption to test, not a
+universal prescription for the future general EOS. Geometry dependencies of
+the reference and all mapped operators must be differentiated consistently.
+
+The log-coordinate closure derivative is a density-weighted mass matrix.
+Consequently normalization, local solves and globalization must account for
+its small response near vacuum. Exponentiation alone does not fix those
+conditioning issues, nor does it make the separately represented enthalpy
+admissible in every Newton trial. Floating-point underflow also needs an
+explicit rejection/range policy; silently clipping density would change the
+equations. A typed physical-density evaluator and its directional action would
+allow polynomial and log representations to be exchanged internally without
+exposing representation details in the physics-facing API.
+
+## Validation
+
+All three builds passed the same 48 focused contract tests with the same RNG
+seed. They cover field registration and spaces, quadrature policy, prepared
+closure values and derivatives, gravity coupling, physical mass diagonals, and
+the normalized complete operator.
+
+| Density order | Test cases passed | Assertions passed |
+|---|---:|---:|
+| Q2 | 48 | 9,766 |
+| Q3 | 48 | 9,822 |
+| Q4 | 48 | 9,894 |
+
+The final frozen artifact audit and all three coupled artifact audits pass.
+All twelve coupled cases completed with their declared outcome. The audit helper
+has nine passing small tests; the comparison helper has seven, including a check
+that separately audited orders retain the same mesh and non-density blocks.
+The independent log-coordinate control has four passing small tests.
+
+No ESTER comparison or moderately rotating n=3 validation is part of this
+study. The n=1,J=0.01 case is a rotation control; its numerical J value is not a
+fraction of critical rotation.
+
+## Reproduction and retained evidence
+
+Raw campaigns, immutable diagnostic executables, input/output hashes, build
+logs and test logs are retained under
+`../../diagnostics/density_order_2026-09-19/` relative to this worktree.
+Every campaign records its source commit, mesh hash, complete CMake cache,
+arguments and runtime environment. The mesh SHA256 is
+`ce11ae99e21e6c3bbfbee402acd2e191c1da0d8261d2227b4203f73f2f337a74`.
+
+Producer commits are `241621c` for frozen controls, `c0e8551` for coupled Q2,
+`cd71f37` for coupled Q3 and `e054c74` for coupled Q4. Later changes to reporting
+scripts and this document do not replace those immutable executables or their
+recorded provenance. The final comparison generator is from `c0a496e`.
+The log control's source and CSV hashes were separately verified in
+`log_reference_validation.json`.
+
+The committed presets are `density_order_frozen_cases_2026-09-19.json` and
+`density_order_coupled_cases_2026-09-19.json` in `experiments/`. Frozen replay
+requires the earlier harmonic-surface campaign retained alongside these data;
+the runner verifies the saved state, layout, metadata and original provenance.
+
+For example, the Q3 configuration uses:
+
+```sh
+/Applications/CLion.app/Contents/bin/cmake/mac/aarch64/bin/cmake -S . -B cmake-build-release-density-q3 -G Ninja \
+  -DCMAKE_BUILD_TYPE=Release \
+  -DCMAKE_CXX_COMPILER=/opt/homebrew/opt/llvm/bin/clang++ \
+  -DCMAKE_C_COMPILER=/opt/homebrew/opt/llvm/bin/clang \
+  -DCMAKE_MAKE_PROGRAM=/opt/homebrew/bin/ninja \
+  -DBoost_DIR=/opt/homebrew/anaconda3/lib/cmake/Boost-1.82.0 \
+  -DMEAN_FIELD_DENSITY_POLYNOMIAL_ORDER=3 \
+  -DMEAN_FIELD_UNIFORM_POLYNOMIAL_ORDER_INCREMENT=0 -DBUILD_TESTING=ON
+```
+
+Build `plateau_diagnostic` and `density_order_contract_tests` with two compile
+workers. The focused test filter includes all non-normalization cases in that
+target plus the reference-mass-diagonal comparison and complete normalized
+operator test:
+
+```sh
+env OMPI_MCA_btl=self,sm cmake-build-release-density-q3/density_order_contract_tests \
+  --catch2 --reporter compact --rng-seed 1218739852 \
+  '~[normalization],Direct Reference Mass Diagonals Match Full True DOF Assembly,Normalized Stellar Equilibrium Preserves Physical Residuals Jacobians And Frozen Solve Coordinates'
+```
+
+The same fixed seed is used for the other density orders. Campaign runtime settings are
+`OMPI_MCA_btl=self,sm`, `OPENBLAS_NUM_THREADS=1`, `OMP_NUM_THREADS=1` and one MPI
+rank. The campaign runner uses a fresh output directory for each invocation;
+its `--timeout 1800` applies per case. Use the order-specific build directory
+and the coupled preset with `experiments/run_plateau_campaign.py`; all other
+arguments are preserved in each `provenance.json` and `*_run.json`.
+
+`experiments/audit_density_order.py` checks the retained evidence independently
+of the producer's summary tables. It validates saved polynomial algebra,
+bindings, sums, trial histories and declared gates, but cannot reconstruct the
+full FEM integrals because local matrices, RHS vectors and all mapped weights
+are not exported. `experiments/summarize_density_order.py` consumes only passing
+audit JSON files and emits comparison tables and scientific figures. It keeps
+undefined values and unresolved sign witnesses explicit.