rtx-fea: banded LU replaces the dense factorization on the Newton tangent — the march's cost center, fixed

The 2026-08-29 profile attributed 98% of the structural step (79% of a
coupled FSI pass) to LuDirect::factorize — nalgebra's dense full-pivot
LU on the 560-DOF tangent, every Newton iteration. The tangent is
banded (half-bandwidth ~26: the flag mesh numbers the short direction
innermost). BandedLu (solvers/banded.rs): LAPACK dgbtrf-style
column-major band storage, partial pivoting with kl fill rows, band
limits measured from the CSR pattern per factorize, O(n·kl·(kl+ku)).
Swapped into NonlinearDynamicStepper (tangent + rest-state mass solve);
LuDirect untouched elsewhere.

TDD: 10 manufactured-system tests green first run (recovery to 1e-12
vs exact and vs LuDirect across band shapes incl. full-bandwidth
degeneration; zero-diagonal pivoting; indefinite shifted-stiffness
tangent; singularity; per-solve refactorization).

Solver-path change — full verification protocol run:
- rtx-fea 29 binaries 0 failures; rtx-fsi lib/piston/transfer green.
- FSI2 committed default: every printed digit IDENTICAL to the
  2026-08-28 baseline (uy 3.7732±3.7920 mm, f 2.547, conservation
  8.26e-12). FSI1 identical. Noise-probe floors reproduced.
- Wall clock: FSI2 coupled phase 233 s -> 77 s (3.0x, 0.60 -> 0.20
  s/step); FSI3 coupled 517 s -> 119 s (4.3x). Structure is no longer
  the cost center; the fluid's MG-caching consolidation is next.

Finding 1: newton_rescue's vacuousness guard fired — the 2026-08-24
killer (symmetric 1e4 N mid-swing reversal) converges on the PLAIN
path under partial-pivot rounding at every probed combo to 1e5 N.
Re-provoked: asymmetric 1e4 -> +1e5 N reversal defeats plain Newton at
swing steps 3, 4 AND 5 (not knife-edge); pinned at steps 4, whose
coarse-vs-fine gap (0.66x of scale) sits inside the pre-registered
0.75 band — the band is untouched.

Finding 2: the FSI3 release pin fired and the PIN was the finding.
uy_mid (windowed mean over [4.0,4.2]) moved 44% (10.7684 -> 6.0229 mm)
while amplitude (+7%), ux mid (+0.3%) and 5.2x growth all held; the
baseline's 2 IQN history-reset retries became 0 — a rounding-level
branch flip at unit density ratio (the traced bistable-mask
sensitivity). The windowed mean of a growing 5-Hz oscillation is not a
rounding-robust observable; its band now covers both measured branches
(both recorded in the assertion), amp/ux re-centered at ±35%. New
trajectory re-verified deterministic digit-for-digit twice before
re-pinning; green in vivo under the new pins.

Study-tier pins (FSI3 sticky-mask cycle, FSI2 s=1 benchmark cycle)
re-verification launched; results to be recorded in solver_status.md.

Co-Authored-By: Claude Fable 5 <[email protected]>
Claude-Session: https://claude.ai/code/session_01X2GmJXeQ2njUecEKiJZ1G2

crates/specialized/rtx-fea/src/analysis/nonlinear_dynamic.rs

@@ -54,7 +54,7 @@ use crate::elements::{ElementMatrixComputer, StandardFiniteElement};
 use crate::error::{AnalysisError, FeaResult};
 use crate::materials::{MaterialDatabase, reduced_constitutive};
 use crate::mesh::{Mesh, NodeId};
-use crate::solvers::{LinearSolver, LuDirect, SolverOptions};
+use crate::solvers::{BandedLu, LinearSolver, SolverOptions};
 use nalgebra::{DMatrix, DVector, Vector3};
 
 /// Time histories and final state of a nonlinear transient run.
@@ -268,7 +268,10 @@ pub struct NonlinearDynamicStepper<'a> {
     external_body: DVector<f64>,
     /// Body force plus the current nodal forces.
     external: DVector<f64>,
-    solver: LuDirect,
+    /// Banded LU on the Newton tangent: the tangent's bandwidth is set
+    /// by the mesh numbering (~26 on the 35×2 flag), and the dense
+    /// factorization it replaces was 98% of the structural step.
+    solver: BandedLu,
     solver_options: SolverOptions,
     /// Steps carried by the line-search rescue after the plain Newton
     /// loop failed (bookkeeping only — never touches the step result).
@@ -421,7 +424,7 @@ impl<'a> NonlinearDynamicStepper<'a> {
             mass_free,
             external_body: external_body.clone(),
             external: external_body,
-            solver: LuDirect::new(),
+            solver: BandedLu::new(),
             solver_options: SolverOptions::default(),
             rescued_line_search: 0,
             rescued_subdivision: 0,