NSA: gate iter-1 convergence on inner solver retcode (#3817)

[singhharsh1708]

Fixes #3817.

A globalized inner solver like TrustRegion can reject its trial step: step! returns with the iterate exactly unmoved and the cache not terminated, having only shrunk its trust region for the next attempt. The outer Newton loop measures displacement, so that rejected step reads as ndz == 0 — which both convergence tests treat as a perfect solve (ndz < 1e-5 on the first iteration, eta*ndz = 0 < kappa afterwards). The stage gets accepted with no correction applied. On fixed-dt FBDF + TrustRegion Robertson, master returns Success with the state frozen at u0.

#4020 already gates on a terminated inner cache; the case it cannot see is rejected-but-not-terminated. This PR adds exactly that discriminator, for NLSolver{<:NonlinearSolveAlg} only:

_rejected_trial_step(nlsolver, ndz) =
    iszero(ndz) && NonlinearSolveBase.not_terminated(nlsolver.cache.cache)

When it fires, nlsolve! skips the convergence/divergence bookkeeping for that iteration (a zero ndz would also poison the next iteration's theta) and lets the inner solver retry with its shrunken region. If it never moves, the loop exhausts and the step is rejected as unconverged. A cache that terminated at zero displacement either converged exactly or already failed (failures become Inf in compute_step!), so termination is the right discriminator between "at the root" and "undecided". NLNewton and the other legacy solvers hit the false fallback and are untouched.

Measured on Robertson (0, 1e5) against an FBDF reltol=1e-12 reference:

• fixed-dt FBDF + TrustRegion: Success frozen at u0 -> ConvergenceFailure (matches what NewtonRaphson reports there)
• default tolerances: relative error 1.7e-2 -> 1.6e-5 (NLNewton gets 2.1e-5, NSA NewtonRaphson 4.0e-6 on the same problem)
• reltol=1e-8: 2.4e-8 -> 1.6e-9
• NSA NewtonRaphson runs are bit-identical before/after (the predicate never fires for it), naccept/nreject unchanged

This replaces the earlier iter-1 retcode gate from the first version of this PR, which #4020 made partially redundant. The testset that asserted a 1e-4 error bound the old gate could not meet now measures 1.6e-9 and asserts honestly; a second testset covers the fixed-dt frozen case. Also drops a non-public SimpleNonlinearSolve import the QA lane flagged.

AI Disclosure

Claude assisted with this work.

[ChrisRackauckas]

don't stack this

[singhharsh1708]

don't stack this

done.