Saloon: Hermitian (%cplx) eigendecomposition (eig Phase A2)

[sigilante]

Summary

Phase A2 of Saloon eigendecomposition: complex Hermitian matrices via complex cyclic Jacobi. +eig is now polymorphic — it dispatches on kind (%cplx → Hermitian path, else the symmetric A1 path) — so eig/eigvals/eigvecs work for both.

The unitary rotation J that zeros a_pq has real diagonal c and complex off-diagonal b = s·(a_pq/|a_pq|) with J[q,p] = −conj(b); updates are A ← Jᴴ·A·J, V ← V·J. Returns real %i754 eigenvalues and %cplx (unitary) eigenvectors — matching NumPy eigh. Works on @cs (bloq 6, single) and @cd (bloq 7, double); asserts exact Hermitian.

Adds complex scalar helpers (cadd/csub/cmul/cdiv/cconj/cre/cpak/cabs-re over /lib/complex) and the Hermitian arms (hermitian, off-norm-h, frob-h, diag-real, rot-cols-h, rot-rows-h, sweep-herm, eig-herm).

Durability fix (also hardens A1)

Replaced the unbounded /lib/math Newton sqrt with an iteration-capped fsqt (50-step backstop, width-fixed feps). A sub-ULP rtol otherwise makes Newton oscillate between two adjacent floats forever — which OOM-crashed the ship (recover: dig: meme + segfault) while developing this. All 18 symmetric (A1) tests remain green.

Tests

tests/lib/saloon-eigh (generated by tools/eigh_check.py vs numpy.linalg.eigvalsh) covers @cs 2×2/3×3/4×4 and @cd 3×3/4×4 Hermitian matrices; checks sort(computed) = sort(eigh). All 5 pass. The canonical [[2,i],[−i,2]] → {1,3} is included.

Caveats (documented in EIG-DESIGN.md)

• rtol's width must match the component width: @rs for @cs, @rd for @cd.
• The exact Hermitian assert is bit-exact, so ±0.0 mismatches in conjugate pairs are rejected (NumPy's -1j packs real as −0.0); callers should canonicalize/symmetrize. The oracle does this via x + 0.0.

⚠️ Stacking

This branch stacks on %cplx (PR #46) and eig A1 (PR #47) — it merges sigilante/lagoon-cplx to get %cplx + /lib/complex. The diff vs main therefore includes those; review the Saloon: Hermitian ... commit for A2 itself. Best merged after #46 → #47; then rebase.

Follow-on

• Phase B: general real → complex (Hessenberg + Francis double-shift QR), its own design + PR.

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