ISLANDS - M2: Level-0 solve machinery (Newton-Krylov, Δ moments, species/frames/fields) — structure, gated physics

[logan-nc]

Milestone M2 of the Islands module per the contract
docs/src/islands/design/M2-launch-prompt.md: the full Level-0 solve
machinery as structure, with every physics coefficient a supplied
[VERIFY]-gated parameter. Stacked on #320 (M1) — this PR retargets to
feature/islands when #320 merges; review only the M2 commits here.

What's here

• solvers/ — matrix-free Newton–Krylov (Decision D2, 04 §5):
  Krylov.jl GMRES on a preallocated ForwardDiff JVP operator, Eisenstat–Walker
  forcing, backtracking line search, convergence on norm and max-norm;
  YBlockJacobi physics-block preconditioner skeleton with TSVD-regularized
  pencil solves (the explicit y_c treatment of 04 §3); dense tiny-grid debug
  Jacobian; pseudo-arclength continuation with fold detection from day one.
• species/ — first-class species plumbing (D3, 02 §1): backgrounds,
  Bulk/Trace roles, trace-criteria checks (warn, never silently degrade).
• frames/ — the ω/normalization conversion forms; every sign a
  NaN-defaulted FrameConvention field: an un-cleared convention poisons
  results instead of guessing (QUESTIONS Q3).
• fields/ — Q(Ω)/h(Ω) flattened-electron closure structure (supplied
  prefactor) + NaN-gated ElectronClosure constants; the coefficient-free
  identity ⟨∂²h/∂x²⟩_Ω = 0.
• moments/ — J̄_∥ assembly, Δ_cos/Δ_sin Ampère projections with
  required gated prefactors (ψ̃ carries an open [VERIFY]; sin
  normalization [DERIVED]-unpinned — QUESTIONS Q4), ⟨·⟩_Ω diagnostics.
• operators/ additions — PitchAngleDiffusion in mimetic divergence
  form (exact discrete particle conservation + entropy sign);
  FarFieldConditions neoclassical-matching BCs (never bare Neumann,
  01 §3); weighted_moment!; flat-index helpers; IslandGrid.y_c.
• Skipped B-ladder benchmarks (benchmarks/islands/): B2/B4/B5a-c each name
  the QUESTIONS IDs gating them. Regression case islands_l0_structural
  (solve-MMS error, Newton/GMRES counts, A7 identity, σ_min(y_c)) — verified
  locally. Paper-I OUTLINE (docs/src/islands/papers/paper-1/): the Level-0
  figure contract, C1–C3 green / C4–C8 gated.

Structural gates (all green; 67 new tests, 120 Islands total)

Gate	Result
A5 zero-drive null	residual exactly machine zero at g ≡ 0; Newton falls back to zero
A1-solve assembled solve-MMS	Newton–Krylov recovers the manufactured state at order 3.98 (nx 17→33)
A4 conservation/entropy	exact to ≲1e-11 / definite sign (mimetic form)
A3 parity	Δ_cos even / Δ_sin odd under ξ-reflection, spectrally exact
A7 closure identity	⟨∂²h/∂x²⟩_Ω ≈ 1e-16 across Ω and prefactors
A8 y_c conditioning	σ_min monitor finite/positive + detects a singularized pencil
Preconditioner	stiff collisional solve: 79.5 s/28 Newton → 0.6 s/7 (same solution to 3e-11)
Continuation	toy fold detected; steps around it, both branches visited

Verification

• Full suite green locally: julia --project=. test/runtests.jl (all 120
  Islands tests incl. the new runtests_islands_solve.jl).
• physics-verifier subagent: PASS — no [VERIFY]-policy violation; the
  literature numbers (−1.173, 1−1.46√ε, 8.73/1.46 ρ_bi, w_ψ/2√2) appear only as
  docstring provenance on NaN-gated fields, never assigned.
• All new kernels pass a forced --check-bounds=yes run (M1 lesson).
• New deps: Krylov.jl (04 §9). Nothing removed.

Explicitly NOT here (gated on the human clearance queue)

The York gates (B5a/b/c), large-w limits (B2), polarization curve (B4), and
Paper-I claims C4–C8 — all require the QUESTIONS.md Q2 (ratify D7/D8),
Q3 (clear the L0 [CHECKED] coefficient set), Q4 (open [VERIFY]s)
clearances. A thin follow-up fills the numbers and un-skips the B-ladder once
cleared.

docs-not-needed: no — docs included: docs/src/islands.md M2 status,
Paper-I OUTLINE, LOG entry.

🤖 Generated with Claude Code

https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr