Document higher-index nonlinear limits after Schur refinement

experiments/PRECONDITIONER_IMPROVEMENT_2026-09-18.md

@@ -215,7 +215,39 @@ maximum endpoint coefficient difference is 2.679e-11; the radius difference is
 admissibility and local EOS metrics. This preserves the existing physical
 regression, rather than adding independent ESTER validation.
 
-The two hard nonlinear cases and the targeted rotating coupling follow-up are
-still running or pending. Production defaults remain unchanged: the stronger
-inverse is an optional typed policy, and the coarse candidate remains an
-experiment.
+## 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.
+
+The hard rotating nonlinear case and the targeted rotating coupling
+follow-up are still running or pending. Production defaults remain unchanged:
+the stronger inverse is an optional typed policy, and the coarse candidate
+remains an experiment.