MeanField/experiments/PRECONDITIONER_IMPROVEMENT_2026-09-18.md

Preconditioner improvement: frozen systems and nonlinear checks

This experiment changes preconditioning only. The physical equations, complete Newton Jacobian construction, normalization rules, mesh, and linear acceptance gate are unchanged. Within each frozen comparison, the numerical Jacobian, right-hand side, and normalization factors are held fixed; nonlinear runs rebuild them along their respective trajectories. The date in branch and artifact names is an experiment identifier; provenance files contain the actual UTC execution times.

The strongest equal-accuracy timing result is a 1.39548x frozen linear solve speedup for n = 1.5, J = 0. The combined material/gravity inverse also passes a saved rotating system that the default cannot solve within 600 iterations. Neither result establishes convergence of the hard nonlinear models. The easy n = 1, J = 0 physical regression retains all virial errors below 3.74e-7, but takes approximately 4% longer with the stronger inverse.

Reproduction and scope

Integration branch: research/preconditioner-benchmarks-20260918, based on 10adf7e. Hypotheses are retained separately on research/gravity-consistent-schur-20260918, research/preconditioner-cached-cross-actions-20260918, and research/coupled-shape-coarse-20260918.

The experiment workspace is /Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d. Evidence is stored under diagnostics/preconditioner_improvement_2026-09-18 in that workspace. Each campaign retains its clean source commit, build configuration, copied executable and hash, mesh hash, exact commands, input/output hashes, and logs. Runs are serial, without overlapping builds or FEM jobs during timing.

Frozen comparisons start FGMRES from zero on the same saved state and right-hand side. Each role constructs a fresh inverse, including its own specification border responses. The restart length is 160, the initial cap is 200 iterations, and the required relative full-J true defect is 0.03. The independently applied full Jacobian decides acceptance; the Krylov recurrence flag alone does not. All accepted/trial state buffers, normalization factors, preparation generation, and a deterministic full-J probe must remain unchanged.

Costs include inverse construction, normalized binding, Krylov setup, Krylov solve, and full-J verification. Extra diagnostic applications and artifact I/O are recorded separately. A speedup to convergence is reported only if both methods meet the same true gate. Less work spent on a failed solve is not a time-to-convergence improvement. A frozen linear pass is not nonlinear or physical convergence.

Existing two-way coupling: negative controls

screen_v1 uses source 018fc1c, before the cached cross-action optimization.

Saved state	Inverse	Iterations	Full-J relative defect	Complete verified cost (s)
n = 1.5, J = 0 plateau	independent	200	0.0590170471860	87.9302
same	stellar LDU	200	0.0600668598825	599.5473
n = 1, J = 0.1 plateau	independent	200	0.0826534298794	86.9409
same	stellar LDU	200	0.0720550396371	594.2053

All four solves fail the 0.03 gate. Stellar LDU incurs 6.82–6.83 times the total cost; it slightly worsens the n = 1.5 result and reduces the rotating defect by 12.82%. Mean inverse application time rises from approximately 0.19 s to 2.7 s, while full-J actions remain approximately 0.23 s. The uncached gravity-to-material force column explains the application-cost anomaly.

Poisson contributes 95.53% of the squared n = 1.5 baseline defect and 98.28% with stellar LDU. For the rotating state those fractions are 85.00% and 98.10%; LDU reduces the gradient-row norm by 94.16% but the Poisson norm by only 6.35%. These are global normalized block diagnostics, not proof of an exclusively exterior error. No exterior weighting is changed.

The independent artifact audit passes 65 provenance checks and 93 checks per case. Both saved states and the normalization data exactly match their producer artifacts. The two independent baselines reproduce the earlier failed solves.

Candidate mechanisms

1. Prepared cross action. Reuse the complete prepared gravity force with zero density and displacement directions to apply its gravity-gradient column. This is the same mathematical coupling with cached geometry; it does not strengthen the inverse. See PRECONDITIONER_CACHED_CROSS_ACTION_2026-09-18.md.
2. Consistent gravity Schur refinement. Keep the existing AMG action, but apply a fixed number of corrections against B R B^T, where R is the same mass inverse used in flux recovery. Fixed work preserves linear stationarity for stationary child inverses. Damping does not guarantee contraction; it must earn its additional cost. See GRAVITY_CONSISTENT_SCHUR_2026-09-18.md.
3. Experimental coupled shape coarse inverse. Use nine fixed geometric modes, complete their material/gravity responses, and construct a checked low-rank correction to the normalized inverse. Modes do not depend on the Newton right-hand side or previous Krylov results. A rank/overlap guard explicitly falls back to the underlying inverse when the correction cannot be trusted. Coarse setup and storage are counted, and actual FEM subspace identities are independently checked before solving.

First candidate screen

candidates_v1 uses source dc8d4b0 and the same n = 1.5, J = 0 plateau state. Every fresh independent control reproduces relative defect 0.059017047186 at 200 iterations. Complete control costs remain between 87.53 and 88.25 s.

Candidate	Iterations	Full-J relative defect	Gate	Complete verified cost (s)
Cached stellar LDU	200	0.0600668598779	fail	117.4845
Consistent gravity Schur, k = 1, omega = 0.5	132	0.0292091549685	pass	72.3981
Cached stellar LDU + consistent gravity Schur	121	0.0285619160766	pass	84.7104
Nine-mode coupled shape coarse correction	200	0.0668657989082	fail	97.0564

At the same iteration cap, the cached LDU reduces total execution time by a factor of 5.1032 relative to the old path, while mean inverse application time drops by a factor of 8.0425. This is an execution improvement, not a convergence improvement. Saved inputs and the full-J probe agree exactly; the final normalized correction changes by only 2.7051e-11 relatively. Both LDU solves fail, and the cached version remains 34.2% more expensive than the independent control on this state.

Gravity refinement is the first candidate to cross the original gate. It lowers the Poisson-row defect by 55.4% and the gradient-row defect by 15.0%. The squared Poisson share drops from 95.53% to 77.59%; surface and hydrostatic defects grow, so the improvement is judged by the verified global norm. Work changes from 400 to 396 mass-inverse applications, 200 to 264 AMG Schur applications, and 200 to 132 outer inverse applications. At this iteration cap, the default does not pass the same gate, so dividing its failed 88 s attempt by 72.4 s is not a convergence speedup. The combined LDU saves another eleven iterations relative to gravity refinement alone but takes 17% more time overall.

The coarse candidate has all nine modes active, minimum overlap lower bound 0.79465, and no fallback. Its extra construction costs 8.6012 s, including mode completion and eighteen setup Jacobian applications; it stores 28,491,680 bytes in persistent vectors. The actual FEM subspace check passes. Nevertheless, its 200-iteration defect worsens by 13.3%. This is a negative numerical result for these modes on this state, not evidence about every possible coarse space or rapidly rotating model. The prototype remains confined to experiments.

Validation status

The integrated library and experiment driver compile with the repository's Release/ThinLTO configuration. Source dc8d4b0 passes 20 focused serial test cases (441,978 assertions), including FEM equivalence, geometry/discretization refresh, fixed-work algebra, and coarse inverse tests. Distributed unit and coarse tests pass on two ranks (13 cases and 213 assertions per rank).

The broad real-FEM MPI run aborts because the existing METIS partition gives one rank no compactified vacuum element, which the mapped mass constructor requires. The failing log is retained; this campaign does not claim distributed FEM Schur validation. See the gravity refinement note for the exact scope and evidence.

The evidence checker passes 22 lightweight tests. Independent reviews of the cached LDU and gravity-only cases validate provenance, state/operator invariants, true gates, and child-work accounting. Reviews are stored outside immutable campaign directories.

Source fcb7833 adds owning experiment-only Newton recipes through the existing ADL preparation interface. They use the same concrete prepared inverses as the frozen benchmark and the existing context refresh/border lifecycle. The driver supports default, gravity-Schur, stellar-LDU, and combined stellar-LDU/gravity-Schur recipes for ordinary Newton, positivity, curvature, and positivity-curvature runs. The combined Newton recipe was added at 7b4f674; the driver builds successfully at 5298d88. This adds no public physics-facing solver API.

Equal-accuracy confirmation

confirmation_v1 uses source 87569c0, a restart length of 160, and a maximum of 600 iterations for both roles. The target remains a freshly verified relative defect of 0.03.

State	Inverse	Iterations	Full-J relative defect	Gate	Complete verified cost (s)
n = 1.5, J = 0 plateau	independent	231	0.0290033752814	pass	100.9943
same	gravity Schur k = 1, omega = 0.5	132	0.0292091549685	pass	72.3727
n = 1, J = 0.1 plateau	independent	600	0.0739197073237	fail	257.9399
same	gravity Schur k = 1, omega = 0.5	600	0.0408505998829	fail	321.5918

For n = 1.5, the frozen linear solve speedup to the same requested accuracy is 1.39548x, with 28.34% less complete cost and 42.86% fewer iterations. The candidate repeats its earlier 132 iterations, with timing close to the earlier 72.3981 s. Both endpoints finish slightly below the same stopping threshold; neither tolerance nor the full-J check was relaxed.

Rotation behaves differently. Refinement reduces the 600-step defect by 44.74%, but costs 24.68% more and still fails the gate. The remaining gradient and Poisson defects contribute 0.02664 and 0.03018 to the global relative norm. The older rotating LDU control reduced the gradient contribution to 0.001813 while leaving a large Poisson contribution. This supports one complementary LDU+Schur follow-up; it neither proves a gradient floor nor permits multiplying the separate gains.

All 73 provenance checks and 186 case checks pass. The reproducible figure and plotting provenance are in confirmation_v1_figures; reported Krylov histories are shown separately from independently verified final points. Plotting used an existing compatible Python environment and installed no dependencies.

Easy nonlinear and physical regression

The first two newton_v1 cases use source bed1893, the same n = 1, J = 0 model with no central-density constraint, positivity strategy, and identical tolerances. Both converge in six accepted full steps, with no repairs or rejections. Initial physical diagnostics and layout are byte-identical; all linear true gates pass.

Measure	Independent	Gravity Schur k = 1, omega = 0.5
Total linear iterations	307	258
Summed Newton iteration time (s)	191.068	199.375
Summed linear solve time (s)	125.035	132.028
Complete diagnostic-run wall time (s)	251.720	261.285
Final nonlinear residual	1.65157e-13	1.05839e-13

Both final residuals are below the shared 1.81187e-12 nonlinear target. Fewer iterations do not produce an easy-case speedup: Newton time rises by 4.35% and full run time by 3.80%. The full run includes physical validation and artifact I/O; it is not presented as pure solver cost.

All eight virial errors (four definitions at quadrature orders 14 and 18) lie between 2.61811e-7 and 3.73092e-7 for both models, comfortably below 1e-5. The maximum endpoint coefficient difference is 2.679e-11; the radius difference is 3.48e-14. The independent paired audit also checks mass, sampled density/enthalpy admissibility and local EOS metrics. At quadrature order 18, the density EOS mismatch, scaled by the sampled maximum density, has RMS 8.5654e-5 and maximum 4.2355e-4. The small virial error therefore does not establish pointwise accuracy to 1e-5 or spatial mesh convergence. This preserves the existing physical regression; it does not add independent ESTER validation.

Higher-index nonlinear follow-through

The n = 1.5, J = 0 curvature case uses the same initial physical state, layout, mesh, normalization, and stopping settings as the historical independent preconditioner run. Gravity refinement completes all twelve permitted Newton steps, with all twelve independently checked linear defects below 0.03. The historical run stopped after six accepted steps because its next linear solve failed. The new run therefore removes that immediate linear obstruction.

It does not establish nonlinear convergence. The final residual is 1.3966991e-5 against a target of 2.7365922e-12, only 6.13% below the historical endpoint. After the first full step, accepted step lengths are mostly 1/16 or 1/32; the final two are 1/64 and 1/512. All nineteen rejected trials fail the residual-decrease test. All 31 auxiliary gravity solves pass. This evidence points to the remaining nonlinear step obstruction, rather than another failed outer or auxiliary gravity solve.

The run spends 415.038 s in outer linear solves and 590.003 s in preparation, including 178.937 s in auxiliary gravity solves. Summed Newton time is 1075.368 s and complete diagnostic-run wall time is 1138.002 s. These are costs of an unconverged trajectory, not a nonlinear speedup.

The enthalpy-pressure energy and force virials pass 1e-5, with quadrature-18 errors 4.06909e-6 and 2.57118e-6. Density-pressure virials remain undefined: the sampled density minimum is -8.14767e-4, with 9600 negative samples and negative mass fraction 2.89597e-5. Negative density also occurred in the historical case. Sampled enthalpy stays positive, but this curvature-only run does not provide a global polynomial positivity certificate. Thus the virial numbers alone do not make this an acceptable physical model. The independent audit records these failures separately from successful linear solves and artifact verification.

Rotating nonlinear follow-through

The n = 1, J = 0.1 positivity-curvature run accepts one full trial step after a positivity repair modifies its search direction. Its first raw linear solve passes in thirteen iterations with relative true defect 0.023583; that defect does not certify the subsequently repaired direction. The second reaches 200 iterations with defect 0.0315391 and fails the unchanged 0.03 gate, before a second trial can be tested. There are no line-search rejections. This case therefore retains a linear obstruction as well as an unconverged nonlinear state.

The final nonlinear residual is 0.00896796 against a target of 8.58812e-10. Summed Newton time is 139.198 s, including 113.338 s in outer linear solves and 18.544 s in preparation; full diagnostic-run wall time is 201.245 s. Enthalpy-pressure virials are approximately 0.0108 and fail the physical milestone. Density-pressure virials are undefined, with sampled density minimum -0.00692627 and 2015 negative samples at quadrature order 18. The enthalpy positivity policy passes its checks, but does not establish density admissibility.

The historical rotating run starts from matching physical diagnostics and uses matching physics and linear controls, but permits eight Newton steps instead of twelve. Its endpoint is not used for a controlled timing claim. All 67 full-campaign provenance checks pass; physical and solver failures are retained as experimental outcomes, not treated as successful convergence.

Complementary rotating coupling

rotation_coupling_v1, built from 5298d88, combines cached stellar LDU with one consistent gravity Schur correction at relaxation 0.5. On the saved rotating plateau state it passes the unchanged 0.03 gate in 142 iterations, with relative true defect 0.0299068971 and complete verified cost 96.8236 s. Its fresh independent control reproduces the original 200-iteration defect 0.0826534298794 and costs 85.8991 s. The earlier independent control still failed at 600 iterations, so there is no measured finite baseline time-to-convergence with which to form a speedup ratio.

The independent audit verifies byte-identical saved states, right-hand side, normalization factors, layout, and deterministic Jacobian probe against the earlier rotating campaigns. Its fresh baseline correction and exact residual also match the original 200-iteration control byte for byte. The combination reduces the Poisson defect norm by 63.33% and the gradient defect norm by 94.47%. Surface and hydrostatic defects increase; acceptance still uses the complete norm. Mean inverse application cost rises from 0.18409 to 0.43046 s, so iteration reduction alone must not be presented as an application speedup.

This combination succeeds where either isolated change failed on this frozen rotating problem. The follow-through in rotation_newton_v1, source 83afd82, demonstrates the limit of that result. With the historical eight-step budget and otherwise matching controls, its first raw linear solve passes in twelve iterations at relative true defect 0.0171651. A positivity repair changes the search direction; the accepted full trial reduces the nonlinear residual from 0.0858812 to 0.00685562. At this new state, the next linear solve reaches 200 iterations with defect 0.0339685 and fails. There is no second trial and no nonlinear convergence.

Summed Newton time is 172.396 s, including 141.631 s in outer linear solves; full diagnostic-run wall time is 234.738 s. The quadrature-18 enthalpy-pressure energy and force virial errors are 0.00712711 and 0.00709672. Both fail 1e-5. Density-pressure virials are undefined, with 1975 negative density samples and negative mass fraction 4.04801e-5. The successful frozen linear solve therefore does not certify a physically acceptable rotating solution, or even passage through every linear system along a new Newton trajectory.

The historical independent endpoint has a larger nonlinear residual, 0.00840387, but a smaller enthalpy energy virial error, 0.000544539. Both physical endpoints fail. This is direct evidence that improvement in the normalized nonlinear residual need not improve virial balance along distinct trajectories. The combined run passes all 38 current-case artifact checks and 32 historical-control checks, including matching initial diagnostics, layout, physics and solver settings.

Production defaults remain unchanged: the stronger inverse is an optional typed policy, and the coarse candidate remains an experiment.

Here J = 0.1 denotes the angular-momentum input, not a measured fraction of critical rotation. This campaign establishes neither a 50% nor a 99% critical rotating equilibrium.

Integration decision and reproducibility

Keep the prepared cross-action optimization: it preserves the mathematical action and removes the demonstrated cost of rebuilding geometry inside the two-way stellar inverse. Keep consistent gravity Schur refinement as an optional typed policy. The measured setting is one sweep at relaxation 0.5; use it alone for the tested n = 1.5 state and combine it with stellar LDU for the tested rotating state. These are measured choices for these states, not universal parameter recommendations. The nine-mode coarse prototype has not earned promotion, and no production default is changed.

The production policy and cross-action files, with their tests, match their isolated hypothesis branches. Changes are committed on the integration branch; the original working checkout and its unrelated quadrature formatting edit are preserved. No central-density constraint or physics-facing API requirement was added.

The retained campaign source revisions and tracked recipes are:

Evidence directory	Source	Recipe in experiments/
screen_v1	018fc1c	preconditioner_screen_cases_2026-09-18.json
candidates_v1	dc8d4b0	preconditioner_candidates_2026-09-18.json
confirmation_v1	87569c0	preconditioner_confirmation_2026-09-18.json
newton_v1	bed1893	preconditioner_newton_cases_2026-09-18.json
rotation_coupling_v1	5298d88	preconditioner_rotation_coupling_2026-09-18.json
rotation_newton_v1	83afd82	preconditioner_rotation_newton_2026-09-18.json

Use run_plateau_campaign.py with the selected recipe, a build configured from the recorded provenance.json, and a fresh output directory. It requires a clean committed source tree, checks the build's source directory, copies the executable before running cases, records hashes, and refuses to overwrite an earlier campaign. The exact build and execution commands are retained in each campaign. The mesh is versioned as sandbox.smesh; saved-state input paths and hashes identify the earlier producer artifacts.

For frozen campaigns, check_preconditioner_benchmark.py independently checks the artifacts. Place its --output outside the campaign directory to retain the original evidence unchanged. The Newton audits and their source scripts are retained alongside the campaigns, with solver and physical failures reported separately from artifact-integrity checks. Six lightweight audit tests cover failed solves, raw versus repaired directions, incorrect policy labels, hash failures, incomplete producers, and unmatched Newton budgets.

The final comparison figure, script, and input hashes are in final_comparison_figures/ under the evidence directory. It shows reported Krylov histories and independently verified endpoints separately, includes setup and verification cost, and labels the unequal rotating iteration caps and failed controls explicitly.