rtx-fea: rescue the nonlinear Newmark Newton — line search, then step subdivision

Both FSI3 study deaths were the flag's SVK Newton returning
ConvergenceFailed{60} inside a coupling pass at a violent mid-cycle
load. The plain full-step Newton stays float-op identical (FSI2 and
FSI3 committed defaults re-verified digit-for-digit, Newton rescues
(0,0)); only on failure does the stepper retry: a backtracking line
search on ||R|| (Armijo, alpha down to 2^-29 — 2^-8 was measured too
shallow when the tangent K_T + M/(beta dt^2) is near singular and the
solved direction enormous and inexact), then 2/4/8/16 Newmark substeps
of dt/n, each line-searched. Rescues are counted and surfaced through
MarchResult and both FSI test printouts.

Measured before writing (tests/newton_rescue.rs): a static tip load
from rest NEVER defeats plain Newton (1e6 N converges in 19 its — from
a quiescent state the predictor is the current configuration and
M/(beta dt^2) regularizes the walk) — pinned as a negative result; the
killer is a mid-swing load REVERSAL (1e4 N tip load, 3 steps of swing
at dt 5e-3, then reversed: dead in 60 its), the FSI3 turning-point
shape — now rescued by the line search alone and consistent with a
dt/32 reference march of the same interval (-0.341 vs -0.195 m, same
branch), with determinism (bit-identical re-step) and
march-continuation pinned alongside.

Also: pin the s = 1 FSI2 benchmark cycle at study horizons (iqn /
subcycle 1, t_end >= 16: f in [1.85, 2.0], uy amp in [70e-3, 92e-3] —
the mode-2 s = 2 cycle fails both bands, so losing the benchmark cycle
stays loud).

Co-Authored-By: Claude Fable 5 <[email protected]>
Claude-Session: https://claude.ai/code/session_01Lnyrw33Lu6rUhW42E9KHwq

crates/specialized/rtx-fsi/tests/fsi2_harness/march.rs

@@ -148,6 +148,12 @@ pub struct MarchResult {
     pub worst_conservation: f64,
     pub skipped: usize,
     pub spiked: usize,
+    /// Flag-Newton rescues over the whole march: `(line_search,
+    /// subdivision)` (see `NonlinearDynamicStepper::rescue_counts`).
+    /// Zero on a healthy march — a nonzero count marks passes whose load
+    /// defeated the plain SVK Newton, the failure that used to kill the
+    /// march outright.
+    pub newton_rescues: (usize, usize),
     pub final_state_finite: bool,
     pub elapsed: f64,
 }
@@ -549,6 +555,7 @@ pub fn run_march(case: BenchmarkCase, config: &MarchConfig) -> MarchResult {
         worst_conservation,
         skipped: total_skipped,
         spiked: harness.spiked_total.get(),
+        newton_rescues: flag.borrow().rescue_counts(),
         final_state_finite: flag_state.displacement.iter().all(|v| v.is_finite()),
         elapsed: start.elapsed().as_secs_f64(),
     }