MeanField/experiments/PRESSURE_AND_SHAPE_CAMPAIGN_2026-09-17.md

Pressure-preparation obstruction and shape curvature — September 17, 2026

The rotating plateau has a demonstrated local enthalpy-positivity obstruction: the same interior polynomial sample approaches zero, while the surface trace remains exactly zero. The late accepted polynomial is already negative between production quadrature points. Separately, the n=1.5 full step is admissible but nonlinear shape/field curvature raises its residual to 12.24 times the base. The measured small-step behavior supports second-order curvature rather than a first-order Jacobian defect along these directions. No production solver fix or additional virial-qualified model is claimed.

Scope and controlled source

This campaign adds experiment-only diagnostics to the preserved combined solver at 95d9fba. It changes no production operator, admissibility rule, normalization, preconditioner, solver policy, or physics-facing API. The central-density constraint remains optional. The working campaign branch is research/diagnostic-campaign-20260917; pressure and shape helpers are retained on separate hypothesis branches:

Purpose	Branch	Retained revision
Preserved combined solver	research/combined-20260916	95d9fba
Pressure and surface diagnostic helper	research/pressure-obstruction-20260917	c6572d3
Shape/field diagnostic helper	research/shape-curvature-20260917	b24e6e0
Integrated captures and focused probes	research/diagnostic-campaign-20260917	tested fd08f7b, 9e1b65d, 4280c53

The primary checkout remains on the diagnostic campaign branch. main remains at 75cc638. Existing untracked files were preserved. Source additions in this campaign are confined to experiments/.

The capture executable records source fd08f7b. It copies observer spans without calling the prepared operator during Newton. A first-trial direction is reconstructed from candidate minus base divided by alpha; the exact candidate is also saved. A zero candidate-reconstruction error does not prove recovery of every bit of the original internal correction. Roundoff estimates and direct candidate replay distinguish that limitation from the original rejection.

Raw artifacts, copied executables, manifests, input/output hashes and analysis are retained under: /Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/diagnostics/pressure_shape_2026-09-17.

Trajectory fidelity

The full twelve-attempt n=1, J=0.1 replay exactly reproduces every non-timing Newton column, every trial and block record, the initial/final physical CSVs, layout, and accepted-state bytes from the September 16 combined run. Its accepted-state SHA-256 is ba55e7295c05f0c9ac5f32490f74d87556502d1744aa42deb7964729b2be2018. The n=1.5 replay reproduces the first three guarded attempts exactly, including all ten trials and all 42 block records. There is no archived three-step endpoint for an independent final-state comparison with that shortened run.

Seven selected iterations save finite 178073-entry vectors. Independently reconstructing each candidate gives zero componentwise round-trip error. The late n=1 capture nevertheless has a direction-roundoff scale estimate of 1.40534e-8, about 4.88e-7 of its direction norm, because its first trial alpha is 1.90735e-6. The onset capture uses alpha=1 and has an estimate of 2.68e-14. Exact candidate replay remains authoritative for the historical rejection.

Pressure-preparation diagnosis

The exact saved first candidates from iterations 2, 3, 4, 7 and 11 all reproduce thermodynamic_domain / pressure_force / outside_domain (enum values 2 / 6 / 5). The production pressure source reaches this branch for finite negative enthalpy. These are enthalpy-domain failures rather than inverted geometry. No new Krylov solve is involved in the frozen replays.

All iteration labels below are zero-based. Physical quantities use the model units M=R_reference=G=1; h values below are not normalized residual entries. The pressure limiter stays at local stellar element 459, pressure quadrature point 50; the closure limiter is point 36 of its different production rule on the same element.

Iteration	Pressure-point h	Direction dh there	Pressure alpha bound	Closure alpha bound	Live accepted alpha
2	9.63674e-4	-1.31441e-2	7.33159e-2	7.27489e-2	6.25e-2
3	1.42165e-4	-2.11119e-3	6.73390e-2	7.15585e-2	6.25e-2
4	1.02159e-5	-1.89704e-3	5.38520e-3	1.01581e-2	3.90625e-3
7	7.04574e-8	-1.86569e-3	3.77648e-5	4.90600e-3	3.05176e-5
11	1.07227e-9	-1.86493e-3	5.74963e-7	4.87083e-3	4.76837e-7

The pressure step budget falls by 127,514 times. These are physical enthalpy values and corrections evaluated with unchanged normalization at the same production points. For consecutive captured states, the next h agrees with h + accepted_alpha * dh to 2.01e-18. At the late state, the closure bound is about 8,472 times looser than the pressure bound. The shrinking pressure h explains the vanishing admissible steps along these measured directions; it does not yet explain why the computed directions keep decreasing h there.

The element touches the stellar surface. Its limiting pressure point has reference Cartesian radius 0.997553 and lies near the negative rotation pole (about 8.66 degrees from the negative z axis at onset). This is an interior quadrature point in a surface-adjacent element, not a surface boundary row. Topology establishes adjacency, not distance to the deformed surface.

Both onset and late probes bracket the production enthalpy bound with 0.99 and 1.01 times that bound. Enthalpy-only steps reproduce the crossing rejection; shape-only steps prepare successfully. Across the two 19-trial grids, no geometry rejection occurs. Affine limiting-point h predictions agree with independent point evaluations within 1.05e-17. Preparation success alone does not imply merit acceptance.

At onset the closure bound is slightly tighter. The original large candidate crosses both bounds, and pressure is prepared first, so its rejection stage cannot identify the tighter family. The supplemental candidate at alpha=0.07303236911639208 lies strictly between the two bounds. Its predicted closure h is -5.01979e-6 while the limiting pressure h remains +3.72619e-6. The production replay confirms that sequence: pressure preparation passes and closure preparation rejects with thermodynamic_domain / barotropic_closure / outside_domain (2 / 3 / 5). This distinguishes the first reported failure stage from the tightest sampled bound.

The supplemental trace inspection uses the actual compiled surface carrier map. All 866 surface enthalpy coefficients and directions are exactly zero at onset and late plateau. The exported indices also establish exact zero surface enthalpy in all five captured physical bases and their five exact first-trial candidates. Their contributions to h and dh at both limiting interior points are exactly zero. The surface condition exists and is satisfied here; boundary leakage does not explain this wall.

The limiting element is cubic, with 64 enthalpy FE nodes. At onset, a step of 1.01 times the sampled union bound leaves all 64 nodal h values nonnegative, while closure and pressure interior samples have h=-1.28819e-5 and -2.11e-6, respectively. At the late state, even the full unit step leaves all nodes nonnegative, but h at the pressure limiting point is -1.86493e-3. This establishes interior polynomial undershoot along these corrections. Nodal positivity or exact zero surface trace is insufficient to prevent it. It does not establish an interpolation implementation bug. The onset full unit step also makes five nodes negative, so that larger step must not be described as an exclusively between-node failure.

A final offline check reconstructs the two local tensor-Q3 polynomials from those 64 physical FE nodal values. It reproduces all nodal values exactly and both production limiting-point h/dh values within 6.94e-18. The same interior sampling grid has 495,264 points per inspected state, including uniform spacing 1/64 and logarithmic distances from 1e-12 to 1e-1 from the zero-trace z=0 face.

Local polynomial state	Minimum sampled h	Reference-element location	Interpretation
Onset base	positive on this grid	—	No negativity detected; not a positivity certificate
Onset accepted alpha=0.0625	positive on this grid	—	Same limited conclusion
Onset alpha=0.99 times production bound	-2.36406e-5	(0.0130467, 0.515625, 0.01)	Negative between production samples
Late captured base	-2.87114e-5	(0.0130467, 0.515625, 0.01)	Existing pointwise violation before the next step

The late base has 2,401 negative samples on that grid. This conclusion uses only the captured base polynomial and is independent of direction-reconstruction roundoff. A separate line evaluation at the pressure point's x,y coordinates and reference z=0.005 gives h=-1.54933e-5. An independent 60-digit Decimal reconstruction gives -1.5493345386780819e-5 at that point; its input hashes, formula and production-point checks are retained in tensor_q3_independent_check.json. The negative magnitudes are many orders above interpolation-check discrepancies. They prove the existence of negative values in this element; the sampled minimum is not a certified global minimum.

For the onset direction, the finer sampled step bound is 0.0665364 versus the production union bound 0.0727489. For the late base, a positivity-preserving step bound from an everywhere-admissible base cannot be asserted because the base already has negative h. Passing production quadrature preparation currently does not ensure internal pointwise enthalpy positivity. The inspected late state is nonconverged and already fails the virial milestone; this finding does not assert that the previously converged nonrotating/low-J controls share the same defect. No new quadrature or EOS rule was installed.

Shape and field curvature

Let p be the captured correction, p_s its actual surface-deformation block, and p_f all remaining fields and scalars. The diagnostic prepares the same frozen base before every Jacobian action and measures

R_d(alpha) = F(x + alpha*p_d) - F(x) - alpha*J*p_d.

The mixed component is the vector R_full - R_shape - R_fields, formed before any norm. Each case uses six common alphas separated by decades; the largest is capped by geometry and both enthalpy rules so all three directions can be compared at the same step. Directions are not renormalized separately.

Frozen case	Common alpha	‖Ftrial‖/‖F0‖	‖L‖/‖F0‖	‖R‖ / alpha²	Gravity fraction of ‖R‖²
n=1, J=0.1, iteration 2	0.058199106	0.94307844	0.94180757	0.062255721	99.8390%
n=1.5, J=0, iteration 1	1	12.243161	0.02576812	0.0021890295	99.7747%
n=3 constrained seed	0.12194828	1.0237652	0.87826014	0.11177311	90.7812%

For n=1.5 at iteration 1, the exact first candidate is preparation-admissible, with positive minimum Jacobian 0.896116. Its residual norm reproduces the original merit-rejected trial within 1.43e-17. The full-step linear prediction would leave only 2.5768% of the starting residual, whereas the actual trial has 12.243 times the starting residual. At alpha=0.1, the actual ratio is still 1.02052 versus the linear prediction 0.900070; at 0.01 it falls to 0.991209. This directly measures curvature-driven merit rejection separately from the rotating enthalpy wall.

At this n=1.5 base, gravity-gradient shape and field action norms are 0.115959 and 0.116055, with cosine -0.999999831; their combined action norm is only 1.16990e-4. Poisson actions likewise nearly cancel (cosine -0.999996055). Truncating just the shape block would destroy this measured first-order cancellation. A full-step second-order remainder of 2.18903e-3 survives, with 99.7747% of its squared norm in the two gravity residual blocks.

The remainder divided by alpha squared stabilizes as alpha decreases, while resolved gravity remainder slopes approach two. With shape held fixed, gravity is affine in its other fields in the source, and the measured fields-only gravity remainders remain at arithmetic scale. The n=1.5 maximum is 2.96e-14, 1.042 times the diagnostic's heuristic floor; the strict below-floor flag is therefore false, but that floor is not a certified assembly-error bound. This slight excess is not evidence of nonlinear fixed-shape gravity. The smallest steps that reach the arithmetic floor are excluded from slope claims.

Frozen case	2⟨L,R_shape⟩	2⟨L,R_fields⟩	2⟨L,R_mixed⟩	‖R_full‖²	Total curvature cost
n=1, J=0.1, iteration 2	-1.952454e-08	-3.042692e-10	9.736527e-08	4.446556e-08	1.220020e-07
n=1.5, J=0, iteration 1	-7.601828e-10	1.587389e-12	1.185527e-09	4.791850e-06	4.792277e-06
n=3 constrained seed	1.458147e-09	-9.452314e-08	-9.285962e-09	2.762978e-06	2.660627e-06

Here L = F0 + alpha*Jp and the exact scalar identity is ||Ftrial||² = ||L||² + 2<L,Rfull> + ||Rfull||². The three signed projection terms add to 2<L,Rfull>; the last squared term contains cross terms between remainder components. Positive values worsen the linear squared-merit prediction. At the n=1.5 full step, the dominant cost is the squared remainder itself, not the small signed projections onto the nearly zero linear prediction. At alpha=0.1, both shape and mixed projections are adverse. At the admissible rotating onset step, curvature modestly worsens the prediction but still allows residual decrease; its larger historical steps instead fail enthalpy preparation.

These observations support genuine nonlinear shape/field curvature along the measured directions, with no detected first-order Jacobian inconsistency there. They do not certify every Jacobian column. The gravity-block aggregates include both stellar and exterior domains and do not establish where the nonlinear remainder originates. No physical importance is assigned to exterior accuracy beyond its influence on the interior solution.

The n=3 seed retains its explicit central-density constraint. Its saved restart-160 correction has true relative linear defect 0.0402590212946322, exactly matching the earlier failed linear solve and exceeding the 0.03 gate. No new solve was substituted. At the common cap the trial residual is 2.3765% above the base despite a predicted 12.174% decrease. The mixed component is large, but its signed projection is favorable; the positive squared-remainder term outweighs all favorable projections. At one tenth the cap, the residual decreases and net curvature slightly improves the linear prediction. This prevents a blanket attribution of every failed step to an adverse mixed projection.

Consequences and next experiments

The two mechanisms should be tested and treated separately.

1. Pressure/closure feasibility. A production-rule enthalpy step cap would avoid known inadmissible trials. It would not, by itself, remove a wall where the same h tends to zero while dh stays negative: the existing accepted steps are already 72.5–92.8% of the sampled union bound. The next discriminating numerical experiment is a tighter frozen linear solve, with local h-direction and component defects measured explicitly, to test how strongly the local direction depends on linear accuracy. A persistent negative direction would still permit nonlinear-path or basin effects; it would not prove that no admissible discrete root exists. Preserve EOS admissibility and use quadrature/interior polynomial checks when testing a positivity representation or constrained step; clipping nodal values cannot prevent the demonstrated between-node undershoot. Test interior polynomial positivity at accepted states as a separate validation check. Increasing quadrature coverage can detect the violation earlier, but detection alone is not a convergence remedy. A trial starting from the late base also needs feasibility restoration if positivity is to be maintained everywhere. Clipping h would change the equations and would not establish internal correctness.
2. Shape/field nonlinear balance. Test a trial-only gravity completion at fixed candidate shape and density, solving the linear mixed-gravity subsystem there before reevaluating every coupled residual. Gravity completion changes the trial path and reduces the gravity defect by construction. The test is whether that produces a better coupled trial, interior solution and virial after every residual is reevaluated. It is a hypothesis, not an implemented production change; other residuals, pointwise interior checks and virial must all be measured. Shape-only damping is poorly motivated by the strong first-order cancellation in these cases.
3. Where the merit is spent. Before changing normalization or exterior weighting, partition the dominant gravity remainder by stellar/exterior contribution and measure its effect on interior hydrostatic balance. These whole-block probes do not justify demanding exterior accuracy for its own sake, nor do they justify dropping exterior residuals that influence the star.

The present data do not single out preconditioner failure for the rotating or n=1.5 captured steps: their current true linear defects still satisfy the 0.03 forcing gate. That gate is a global norm criterion, not proof that the local enthalpy direction is adequate near a vanishing h. The n=3 frozen correction fails that gate and is reported separately. No production preconditioner, normalization, EOS admissibility or line-search rule has been altered here.

Verification and reproducibility limits

The driver was built in Release with Homebrew Clang 22.1.8, IPO enabled, profiling disabled and uniform polynomial order increment zero. Runs were serial, on one MPI rank, using the same 1216-hex mesh (832 stellar, 384 vacuum) and the unchanged production quadrature rules. Mesh SHA-256: ce11ae99e21e6c3bbfbee402acd2e191c1da0d8261d2227b4203f73f2f337a74. Case manifests and copied immutable executables retain the tested sources even while the experiment branch acquires later diagnostic additions. Performance was not compared in this campaign.

Two instrumented trajectory runs reached their requested iteration caps (expected model exit 3). All eight focused cases and both supplemental cases completed successfully (exit 0), with matching binary and input/output hashes. The n=1.5 rejected-candidate residual norms agree with their historical values to 1.43e-17 and 1.84e-18; the two norm implementations have different reduction orders, so those numbers are roundoff comparisons rather than byte comparisons. All captured base residual vectors reproduce exactly.

All ten frozen cases restore their final base residual exactly. Each shape case has 22 admissible preparation events, for 66 total; all accepted normalized states, physical states and residuals are unchanged. Direction partitions are exact. The largest Jacobian partition error is 8.33e-17. Aggregate signed-merit and projection identities agree to better than 6.8e-15 relative. All 38 pressure crossing trials have complete, unique records and no geometry rejection. Independent audits check vector copies, quadrature-bound identity, CSV coverage, all surface coefficient statistics and nodal identities.

Implicit neighboring capture state/residual inputs are covered by the capture output hashes as well as independent checks; the runner's explicit-input hashes alone do not cover that auxiliary-read path. Late reconstructed directions retain the subtraction-roundoff limitation documented above. The late recomputed true linear defect differs from its historical value by 2.25e-9, while both remain below 0.03; exact-candidate replay is independent of that reconstruction.

No production/library or public-header code changed relative to 95d9fba; the existing production C++ suite was not rerun for these experiment-only changes. Validation instead includes compilation of the diagnostic executables, exact trajectory comparisons, production candidate replays, independently recomputed CSV identities, input/output hashes, and frozen-state restoration. A zero diagnostic process exit means the observations completed. It does not mean a stellar equilibrium converged.

This campaign establishes no additional virial-qualified model. The captured rotating endpoint is exactly the previous nonconverged endpoint: its P(h)-based virial errors remain approximately 4.75e-4 and 8.18e-4, above the required 1e-5. The n=1.5 capture is intentionally only a three-step prefix; its eventual convergence and pointwise accuracy are not inferred from these probes. J=0.1 is an angular-momentum input, not 10% of critical rotation. Critical-rotation fractions were not computed.

Reproduction and retained analyses

The three case manifests are plateau_capture_cases_2026-09-17.json, plateau_focused_cases_2026-09-17.json, and plateau_supplemental_cases_2026-09-17.json in experiments/. Their tested source commits are respectively fd08f7b, 9e1b65d, and 4280c53. Each artifact folder contains provenance.json, exact per-case commands, case input/output hashes, logs and the copied executable. The midpoint candidate also retains a derivation record and SHA-256. No raw state files or binary build products are committed.

Run a new numerical campaign with a compatible configured build and a fresh output directory:

python3 experiments/run_plateau_campaign.py \
  --source . --build cmake-build-release-homebrew --mesh sandbox.smesh \
  --cases experiments/plateau_focused_cases_2026-09-17.json \
  --cmake /Applications/CLion.app/Contents/bin/cmake/mac/aarch64/bin/cmake \
  --jobs 8 --output /absolute/path/to/new-results

Use the corresponding tested source checkout for exact source reproduction. The supplied case manifests reference the retained absolute capture paths; relocating artifacts requires changing those paths consistently. The runner checks committed source, rebuilds, records provenance and refuses reused output directories. Replaying the copied executables with the recorded arguments and a new output path is another way to retain the exact tested binary.

The portable analysis scripts under experiments/plateau_analysis/ take the artifact root containing captures/, focused/, and supplemental/:

python3 experiments/plateau_analysis/analyze_pressure.py /absolute/path/to/artifact-root
python3 experiments/plateau_analysis/analyze_shape.py --campaign /absolute/path/to/artifact-root
python3 experiments/plateau_analysis/audit_focused_outputs.py /absolute/path/to/artifact-root
python3 experiments/plateau_analysis/audit_surface_outputs.py /absolute/path/to/artifact-root
python3 experiments/plateau_analysis/sample_pressure_polynomial.py /absolute/path/to/artifact-root

They do not run the solver or alter raw case data. They regenerate compact Markdown/JSON analyses alongside those data. See pressure_analysis.md, shape_analysis.md, pressure_polynomial_analysis.md, capture_validation.md, and focused_method_validation.md in the retained artifact root for the detailed per-case checks.