experiment: record pressure positivity and shape curvature evidence

experiments/PRESSURE_AND_SHAPE_CAMPAIGN_2026-09-17.md

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+# 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:
+
+```sh
+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/`:
+
+```sh
+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.