ForceFreeStates - FEATURE! - Cross ideal singular surfaces with a Frobenius-basis jump #669

Summary

The ideal crossing in both integrators is first order in singfac_min. Mapping the solution across the surface
through the Frobenius bases that are already computed removes that dependence: in a scratch prototype the edge
Riccati matrix and the free-boundary energy are flat in singfac_min to ~1e-9 and ~1e-7, and the existing
crossings converge linearly to the same values.

Current behavior

riccati_cross_ideal_singular_surf! (Riccati/Crossings.jl) and cross_ideal_singular_surf!
(EulerLagrange.jl) zero the resonant column, take a two-point trapezoidal step across the surface, and zero
the resonant row before inserting the small solution. The step and the row zeroing each leave an
O(singfac_min) error.

DIII-D-like n = 1 example, eulerlagrange_tolerance = 1e-10, free-boundary et[1]:

Edge S (Riccati, relative to the jump at 1e-6): current 1.3e-2 / 1.6e-3 / 1.9e-4 at 1e-3 / 1e-4 / 1e-5 with an
anti-Hermitian part of the same order; jump ≤ 6e-8 with anti-Hermitian part ~1e-10. The Solovev example behaves
the same way (6.1e-3 → 6.7e-5 against ≤ 1e-8).

Proposed change

At a crossing, with ua_L, ua_R from sing_get_ua:

• ca = ua_L \ u, the coefficients in the left Frobenius basis (what sing_get_ca returns).
• For each resonance, eliminate the big-solution coefficient from every column using the column where it is
  largest, and move the surviving column out of the slot the small solution will take.
• Zero the big and small coefficient rows.
• u_R = ua_R · ca, then place the right small solution in the freed column. No resonant-row zeroing.

Both the elimination and the basis transfer are needed: elimination alone halves the error, and the transfer
without elimination is worse than the trapezoid.

Scope

• Riccati path (_predict_across_singular_surface!, _capture_right_crossing_data!).
• Forward path, recording the elimination and permutation for transform_u!.
• Multi-n shared surfaces, reversed shear (double crossing), Mercier-unstable (complex α).
• Regression cases: a singfac_min scan of et[1] on diiid_n1 and solovev.
• Revisit the singfac_min defaults (1e-4 for Riccati Δ′, 1e-6 for Forward decks feeding
  PerturbedEquilibrium) once the crossing no longer sets them.

Not covered

• The Δ′ matrix is unchanged by the crossing (it comes from the propagator BVP) and still drifts with
  singfac_min: first diagonal entry 9.089 at 1e-3, 9.241 at 1e-4. Separate issue.
• Kinetic crossings (cross_kinetic_singular_surf!) are untouched.

Impact

Results change at every current singfac_min by the size of the present defect (about 3e-5 in et[1] for
Forward at 1e-6, about 5e-3 for Riccati at 1e-4), so regression baselines move. Run time per solve is unchanged;
a larger singfac_min becomes usable (2.6 s at 1e-3 against 3.9 s at 1e-6 on the case above).

Evidence and caveats

Prototype: method overrides in a scratch script on v2.0.1, two equilibria, n = 1, Riccati path only for the
jump. The Forward numbers above are the unmodified code. Not yet tested: the Forward implementation, multi-n,
complex α, and PerturbedEquilibrium outputs.

Background: Schiff & Shnider, SIAM J. Numer. Anal. 36, 1392 (1999). Their Möbius-scheme result needs continuous
coefficients, so it does not remove the ideal crossing, but it motivates treating the crossing as a map on the
solution subspace.