Safeguard reused FGMRES guesses against the current true residual

experiments/KRYLOV_HYPOTHESIS_2026-09-16.md

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+# Krylov information retention and correction reuse
+
+Baseline: `80cc3d8`. This branch evaluates two distinct effects without adding
+public options or changing physical specifications, normalization, preconditioner,
+linear tolerances, or nonlinear acceptance.
+
+The first control changes the existing FGMRES restart setting from 40 to 160.
+Frozen solves in the audit established linear benefits; live Newton runs must
+establish whether the resulting corrections remain useful and physically accurate.
+
+The production candidate safeguards a supplied linear initial guess using its
+true residual under the current operator. It retains the guess when
+`||b-Jp_guess|| <= ||b||`, otherwise starts at zero. A nonzero guess costs one
+extra operator action, included in reported counts and solve time. Zero guesses
+do not incur an extra action. This is neither Krylov basis recycling nor a
+guarantee of fewer iterations: a smaller initial residual can still excite a
+harder set of modes. The candidate must earn its cost in matched live runs.
+
+The retained `(1-alpha)p` heuristic from damped Newton remains intact, as does
+the existing backend API. The safeguard only declines a guess that increases
+the current true defect. It also handles a nonfinite operator action on an
+otherwise finite guess by choosing the finite zero guess, collectively across
+the communicator. Existing input checks still reject nonfinite supplied vectors.
+
+Required comparisons:
+
+- Same no-phase n=1.5 seed at restart 160 with baseline and candidate; record
+  each initial linear residual relative to its RHS, accepted step lengths,
+  true defects, total time and physical endpoint diagnostics.
+- n=1 controls with full steps, where the Newton heuristic already produces
+  zero guesses; their physical trajectory should remain unchanged.
+- Existing beneficial-warm-start test plus the new adverse-guess regression.
+
+Do not promote this change merely because the small regression passes. A live
+improvement, unchanged validation, and sensible overhead are required. Virial
+success means both measured definitions satisfy 1e-5 where pressure is defined;
+interior EOS/Bernoulli consistency and algebraic convergence remain separate.