MeanField/experiments/PRECONDITIONER_IMPROVEMENT_2026-09-18.md

Preconditioner improvement: matched frozen Newton systems

This experiment changes preconditioning only. The complete Newton Jacobian, normalization, physical residual, mesh, and linear acceptance gate are held fixed. The date in branch and artifact names is an experiment identifier; provenance files contain the actual UTC execution times.

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 under diagnostics/preconditioner_improvement_2026-09-18 there. 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 miss the 0.03 gate. Stellar LDU spends 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 costs rise 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.

Candidate numerical results and execution validation are pending. Defaults remain unchanged until measurements support a different choice.