Document equal-accuracy gain and preserved virial regression

experiments/PRECONDITIONER_IMPROVEMENT_2026-09-18.md

@@ -141,7 +141,65 @@ 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, and stellar-LDU recipes for ordinary Newton,
 positivity, curvature, and positivity-curvature runs. This adds no public
-physics-facing solver API. The updated driver builds successfully at87569c0.
+physics-facing solver API. The updated driver builds successfully at `87569c0`.
 
-Equal-accuracy hard-state confirmation and Newton/virial regressions are pending.
-The default production recipe remains unchanged during validation.
+## Equal-accuracy confirmation
+
+`confirmation_v1` uses source `87569c0`, restart160 and a maximum of600 iterations
+for both roles. The target remains a freshly verified relative defect of0.03.
+
+| 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 |
+
+The n=1.5 speedup to the same requested accuracy is **1.39548x**, or28.34% less
+complete cost and42.86% fewer iterations. The candidate repeats its earlier132
+iterations and72.3981s timing closely. Both endpoints overshoot the same stopping
+threshold slightly; neither tolerance nor the full-J check was relaxed.
+
+Rotation behaves differently. Refinement reduces the600-step defect by44.74%,
+but costs24.68% more and still fails. The remaining gradient and Poisson defects
+contribute0.02664 and0.03018 to the global relative norm. The older rotating LDU
+control reduced the gradient contribution to0.001813 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.
+
+All73 provenance and186 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 shared1.81187e-12 nonlinear target. Fewer
+iterations do not produce an easy-case speedup: Newton time rises4.35% and full
+run time3.80%. 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 quadrature14 and18) lie between
+2.61811e-7 and3.73092e-7 for both models, comfortably below1e-5. Maximum endpoint
+coefficient difference is2.679e-11; radius difference is3.48e-14. The independent
+paired audit also checks mass, density/enthalpy 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.