MeanField/experiments/NONLINEAR_STEP_INVESTIGATION_2026-09-18.md

# Nonlinear step obstruction after gravity preconditioning

## Recorded decision from the completed preconditioner investigation

The completed report is `PRECONDITIONER_IMPROVEMENT_2026-09-18.md`, committed
at `dc1da62` on `research/preconditioner-benchmarks-20260918`. Its immutable
campaigns live under the task workspace's
`diagnostics/preconditioner_improvement_2026-09-18/` directory.

- Consistent gravity Schur refinement solves the saved n = 1.5, J = 0 linear
  system in 132 iterations and 72.3727 s, versus 231 iterations and 100.9943 s
  for the independent control. Both satisfy the same independently verified
  relative full-J defect of 0.03: a measured 1.39548x speedup.
- Combining this refinement with stellar LDU solves the saved rotating
  n = 1, J = 0.1 system in 142 iterations and 96.8236 s. The independent and
  gravity-only controls still fail after 600 iterations. Their finite
  time-to-convergence is unknown, so no rotating speedup ratio is reported.
- Prepared geometry reuse makes the existing stellar LDU application 8.0425x
  cheaper without improving its convergence by itself. Retain this
  implementation optimization independently of policy selection.
- The tested nine-mode shape correction does not improve the n = 1.5 case.
  It remains an experimental negative result, not a rejection of all coarse
  spaces.
- The easy n = 1, J = 0 model preserves all eight virials below 3.74e-7,
  while the stronger inverse costs about 4% more. Defaults remain unchanged.
- Neither hard nonlinear model is physically acceptable. Negative sampled
  density makes density-pressure virials undefined. In rotation, a smaller
  nonlinear residual accompanied a larger enthalpy-pressure virial error.
  Residual decrease is not a substitute for virial and field-admissibility
  validation. No angular momentum has been calibrated to critical rotation.

## New investigation and controls

Branch: `research/nonlinear-step-obstruction-20260918`, based on `dc1da62`.
The configured worktree is reused to avoid an unrelated full rebuild; the
preceding branch and copied campaign executables retain their exact source
and binary identities. This branch adds experimental diagnostics only.

Start with the n = 1.5, J = 0 endpoint of
`newton_v1/n1p5_J0_curvature_gravity_schur`. All twelve outer linear solves and
all 31 auxiliary gravity solves passed, but the final nonlinear residual was
1.3966991e-5 against a target of 2.7365922e-12. The final accepted alpha was
1/512. Every trial prepared successfully; all nineteen rejections were
residual-decrease failures, and geometry never limited the step.

That campaign did not retain every raw Newton direction. The new late-state
probe computes a fresh direction at its saved final endpoint; it does not
claim to replay the preceding iteration's rejected trials. A plateau in a
bounded twelve-step run is also not proof that all subsequent steps must
fail. The fresh trial measurements can test that possibility.

The final squared-residual shares are approximately 42.42% hydrostatic,
26.23% closure, 15.87% surface, 10.43% gravity, and 5.05% mass. This motivates
examining retained-equation curvature after the gravity correction, rather
than assuming the earlier gravity or pressure-preparation failure persists.
The last auxiliary prepared defect, approximately 3.33e-14, measures agreement
with the affine gravity target, not satisfaction of the gravity equations.
The final gravity-block residual norm is approximately 4.51e-6.

Keep the complete physical equations, fixed normalization, optional
central-density specification (absent here), mesh, and acceptance thresholds
unchanged. Use one gravity Schur refinement at relaxation 0.5 and the same
restart length 160. Fresh cold solves and independently applied full-J true
defects determine direction quality. Trial probes never commit a model.

## Questions and discriminating evidence

1. **Does the prepared trial path have the expected tangent?** For frozen
   residual F and computed direction p, measure Jp and actual prepared
   differences. The curvature-corrected path should preserve that tangent
   because its gravity correction is second order in alpha. A discrepancy
   that remains first order above arithmetic noise would implicate the
   linearization or preparation path.
2. **Does the inexact linear direction cause the obstruction?** Compare
   directions at the same saved state with forcing tolerances 0.03 and 0.003,
   subject to a recorded finite iteration budget. A tighter requested
   tolerance counts only if its independent true gate passes. Failed solves
   are recorded and do not substantiate this comparison.
3. **Which nonlinear terms oppose the linear prediction?** Split p into its
   actual surface coordinates and all remaining coordinates. Compare raw
   and gravity-corrected trials for full, shape-only and fields-only
   directions. Form mixed remainders by vector subtraction. Signed
   projections against the full linear prediction distinguish adverse
   curvature from large but cancelling terms.
4. **Is the behavior specific to the late state?** Use the same-polytrope
   Lane–Emden seed as an early-state control, where the previous production
   run accepted its first full Newton step. Record preparation and enthalpy
   feasibility separately from the residual metric.

For phi = ||F||^2/2 and ||Jp + F|| <= eta ||F||,

    phi'(0) = <F,Jp> <= -(1-eta) ||F||^2.

At eta = 0.03 the slope must be strongly negative if J differentiates the
prepared path. The production sufficient-decrease test therefore ought to
accept sufficiently small resolved steps. The investigation measures whether
large adverse higher-order terms explain how small those steps become.

Each block's vector remainder R = F_trial - F - alpha Jp must satisfy

    ||F_trial||^2 = ||F + alpha Jp||^2
                   + 2 <F + alpha Jp, R> + ||R||^2.

For the corrected trial map, its mixed remainder includes interactions
introduced by gravity correction; it is not identified solely with the raw
physical shape Hessian. Below-floor differences are unresolved, not evidence
of exact linearity.

The EOS provides an additional interpretation limit: rho(h) is proportional
to max(h,0)^1.5 here. At an exact zero with a positive enthalpy direction,
the closure remainder can scale as alpha^1.5 while its first derivative is
correct. A superlinear slope different from two does not alone establish a
Jacobian defect; the direct tangent check is whether R/alpha decreases in a
resolved regime. The strong zero-pressure boundary row is already expressed
as h-h(P_surface), with derivative delta h, so it does not inherit a vanishing
dP/dh boundary row.

The paired directions also permit a conditioning diagnostic without another
operator solve: w = p_tight - p_loose and Jw = Jp_tight - Jp_loose. A large
change in direction with a small action demonstrates a weak response in that
direction, not a proved nullspace or a particular symmetry. Measure the
shape/fields action cancellation as well. Fixed-K, fixed-M n = 1.5 does not
have the familiar exact n = 3 mass-preserving homology; no additional gauge
or central-density constraint is justified by that analogy.

## Execution and validation

Run a late-state probe first, then use its evidence to choose the matched
tighter-direction and early-state controls. All model jobs and builds run
serially. The campaign runner records clean source commits, exact commands,
input/output and executable hashes, and refuses existing output directories.
The diagnostic saves the frozen vectors, layout and normalization, checks
restoration and accepted-buffer invariance, and reports failed preparations
and inner solves without inventing values for missing trials.

Results are pending. No numerical remedy or new convergence claim follows
from the investigation design alone.