…uthor-title convention

Renames the nine PDFs under docs/resources/Drift_Kinetic_Island_References to the
YEAR-Author-Title convention used by the rest of docs/resources, and extends the
.gitignore resources-PDF negation to subdirectories (!docs/resources/**/*.pdf) so
these tracked references match the existing docs/resources/*.pdf policy. Syncs the
filename references in the ISLET reference-library doc (docs/08).

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

…e convention)

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

…n into in-repo layout

Drop the working acronym ISLET in favour of the plain module name Islands, and
record the module-naming convention (simple intuitive names, not standalone-code
acronyms) in the repo-root CLAUDE.md.

Reconcile the design bundle from its standalone-package assumptions into the
in-repo layout:
- module conventions   -> src/Islands/CLAUDE.md
- overview             -> docs/src/islands/index.md
- design docs 00-08    -> docs/src/islands/design/
- Physics Book / state / derivations / papers / notes / LOG / QUESTIONS
                       -> docs/src/islands/{,state,derivations,papers,notes}
- benchmarks + figures -> benchmarks/islands/{,figures}
- verify harness       -> src/Islands/verify/
- tests                -> test/runtests_islands_*.jl (top-level test/)
- regression cases     -> integrated under regression-harness/ (islands_*)

Islands is a submodule of the GeneralizedPerturbedEquilibrium package (no separate
Project.toml), not a standalone subpackage; docs/03/06 updated accordingly.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

…docs

The exported Islands module carries a docstring, which checkdocs=:exports
requires be covered in the manual. Add docs/src/islands.md with an @autodocs
block and wire it into make.jl (API Reference), matching the other submodule
pages. Fixes the missing_docs docs-build failure introduced by the module stub.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

…tings.json

/doctor flagged two deny rules as skipped: the :* prefix wildcard is only valid
at the end of a pattern. Rewrite them with a mid-pattern wildcard so the rules
load and again restrict the protected-branch pushes for any remote or flags.

A loaded OMFIT module leaks the conda env lib dir onto LD_LIBRARY_PATH, shadowing
Julia's bundled artifacts (CHOLMOD, glib/Cairo/GR) and producing false 'broken
build' blocks in the Stop hook. Strip the var for the julia smoke check via
env -u; this is a no-op on a clean shell and in CI.

M1 needs forward-mode AD for the JVP/AD-compatibility checks (design
docs/src/islands/design/04-numerics.md §9). Adds ForwardDiff to Project.toml
(already present transitively in the manifest); removes nothing.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

…ck + verify harness

Land the milestone-M1 numerical skeleton (design 03 §1-2, 04; ladder A1/A2),
structure only — no [VERIFY] physics coefficients assigned in src/.

- phasespace/PhaseSpace.jl: the (x, ξ, y, E, σ) grids with layer-clustered maps.
  Fourier spectral ∂ξ; Fornberg high-order finite differences for ∂x/∂y on
  sinh-stretched grids (per-derivative window widths so D1 and D2 are both
  4th-order including boundaries); composite-Simpson quadrature weights;
  Gauss-Laguerre energy nodes. Pure numerics.
- operators/Operators.jl: AbstractTerm + apply! + residual! and the term structs
  of 03 §2 (ParallelStreaming, MagneticDrift with the :original/:improved toggle,
  ExBDrift as the (x,ξ) Poisson bracket, Collisions, GradientDrive, PerpTransport
  and RadiationSink L4 stubs, Quasineutrality field residual). Every physics
  coefficient is a supplied data field, never a literal; allocation-free and
  generic over eltype so ForwardDiff duals flow through.
- verify/Verify.jl: manufactured-solution (MMS) + AD-vs-finite-difference JVP
  harness, plus allocation probes. Manufactured coefficients are arbitrary
  order-unity test values (not physics), exercising the discretization only.
- Islands.jl: wire the three submodules.

physics-verifier: PASS (no [VERIFY]-policy violation).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

…to runtests

test/runtests_islands_grids.jl and test/runtests_islands_operators.jl, included
in test/runtests.jl:

- A1 per-operator MMS: 4th-order convergence for the ∂x/∂y differential terms,
  machine precision for the Fourier ∂ξ (drift) term; assembled kinetic residual
  converges at 4th order.
- A2: forward-mode-AD JVP of the (nonlinear) assembled residual matches the
  central finite-difference directional derivative to ~1e-9.
- Allocation regression: every apply! and residual! hot path allocates 0 bytes.
- Grid unit tests: Fourier exactness, mapped-FD order, Simpson/Gauss quadrature,
  layer-clustered packing, IslandGrid assembly and input validation.

All 53 Islands tests pass.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

…path)

- docs/src/islands.md: document the M1 skeleton (PhaseSpace/Operators/Verify),
  restating the structure-only [VERIFY] discipline.
- LOG.md: M1 session entry (what moved / blocked / next).
- QUESTIONS.md: Q1 RESOLVED — julia is at the ncl2128 software dir and must be
  invoked with a clean LD_LIBRARY_PATH (OMFIT contamination); used here to run
  the suite locally.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

Set up the next milestone (M2 — L0 solve machinery) for the autonomous loop,
mirroring the M1 launch-prompt bootstrap.

- docs/src/islands/design/M2-launch-prompt.md: the M2 milestone contract. Scope
  = full L0 solve machinery (solvers/, moments/, frames/, fields/, species/,
  neoclassical-matching BCs) as AD-compatible allocation-free structure with every
  physics coefficient a [VERIFY]-gated parameter. DoD = the physics-free structural
  gates (A5 null, assembled solve-MMS, A8 y_c conditioning, A4 conservation, A3
  parity, A7 ⟨∂²h/∂x²⟩=0) + suite + PR + Paper-I OUTLINE + the clearance queue.
  The York gates (B5a/b/c, B2, B4) are explicitly OUT — they need human-cleared
  physics; reaching one by hardcoding is a policy violation, not completion.
- QUESTIONS.md: seed the parallel-human clearance queue — Q2 (ratify D7/D8),
  Q3 (clear the L0 [CHECKED] coefficient set with exact cites), Q4 (open [VERIFY]s:
  the psi-tilde q_s'/q_s typo, B5a collisionality, acquire WCHH96 + Park 2022).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

Drop the overnight-loop framing (dontAsk/--continue header, unattended language,
Stop-hook exit-criteria) — M2 is driven interactively via /goal in auto mode.
Blocked coefficients are surfaced to the user to clear live rather than only
parked in QUESTIONS.md. The contract (scope, DoD, [VERIFY] hard rule, gated York
gates) is unchanged.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

…ylov solver

Matrix-free GMRES backend for the L0 solve (design docs/src/islands/design/04-numerics.md $5, $9).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

…es, fields, moments

Land the Level-0 solve structure (design 03 §1-3, 04 §3-5; M2 contract), every
physics coefficient a supplied [VERIFY]-gated parameter — no literal in src/:

- solvers/Solvers.jl: matrix-free Newton-Krylov (Krylov.jl GMRES on a
  preallocated ForwardDiff JVP operator; Eisenstat-Walker forcing; backtracking
  line search; convergence on both norm and max-norm per 04 §5), YBlockJacobi
  physics-block preconditioner skeleton with TSVD-regularized pencil solves
  (the explicit y_c treatment of 04 §3), dense tiny-grid debug Jacobian, and a
  pseudo-arclength continuation scaffold with fold detection from day one.
- species/Species.jl: Species{AbstractBackground}, Maxwellian/SlowingDown,
  Bulk/Trace roles, validation + trace-criteria check (warn, never degrade) —
  the D3 first-class species plumbing (02 §1).
- frames/Frames.jl: Level0Parameters input vector (01 §5) and the frame
  conversion FORMS with every sign/normalization a NaN-defaulted gated
  FrameConvention field — an un-cleared convention poisons results instead of
  guessing (QUESTIONS Q3).
- fields/Fields.jl: Q(Ω)/h(Ω) flattened-electron closure structure (prefactor
  supplied), the coefficient-free A7 identity ⟨∂²h/∂x²⟩_Ω = 0, and the
  NaN-gated ElectronClosure constant set.
- moments/Moments.jl: J̄_∥ assembly from charge-scaled weighted moments,
  Δ_cos/Δ_sin projections with REQUIRED gated prefactors (ψ̃ open [VERIFY],
  sin normalization unpinned — QUESTIONS Q4), Ω label + ⟨·⟩_Ω average +
  channel-split diagnostics (pinned half-width convention).
- operators/: PitchAngleDiffusion in mimetic divergence form (exact discrete
  particle conservation + entropy sign, ladder A4), FarFieldConditions
  matching BCs (never bare Neumann, 01 §3), weighted_moment!, flat-index
  helpers; IslandGrid now records y_c for the A8 monitor.
- verify/: solve-level MMS + zero-drive configurations (A5/A1-solve) and the
  yc_block_sigma_min conditioning monitor (A8, the L23 §4.2 silent-noise
  regression tripwire).

All new kernels pass a --check-bounds=yes run (M1 lesson).

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…/A8 + solve-MMS)

test/runtests_islands_solve.jl (67 tests, wired into runtests.jl):
- A5 zero-drive null: residual exactly machine zero at g ≡ 0; Newton falls back
  to the zero state from a perturbation.
- A1-solve: assembled Newton-Krylov solve recovers the manufactured state at
  design order (observed 3.98 on the nx = 17→33 pair).
- A4: exact discrete particle conservation (≲1e-11) + entropy sign of the
  mimetic pitch-angle operator; allocation-free apply!.
- A3: Δ_cos even / Δ_sin odd under ξ-reflection, spectrally exact projections.
- A7: ⟨∂²h/∂x²⟩_Ω = 0 to 1e-10 across Ω and prefactors (coefficient-free).
- A8: y_c-block σ_min monitor is finite/positive and detects an artificially
  singularized pencil.
- Preconditioner gate: YBlockJacobi cuts GMRES iterations >2× at identical
  solutions; far-field BC rows; pseudo-arclength fold detection; species +
  gated-frames plumbing.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…ctural regression case

- benchmarks/islands/ (+ figures/): the docs/05 B-ladder scripts B2 (large-w
  limits), B4 (polarization ω_E structure), B5a/b/c (York threshold triangle) —
  all SKIPPED, each naming the QUESTIONS.md entries (Q2-Q4) whose human
  clearance un-gates it. Un-skipping by filling in a coefficient is the
  forbidden action; the scripts say so.
- regression-harness: new computed case islands_l0_structural tracking the
  structural numbers that silently move if the discretization/solver/quadratures
  drift — solve-MMS error + Newton/GMRES iteration counts (nx=17), the A7
  closure identity, and the A8 y_c-block sigma_min. Verified locally:
  err=5.254e-2, 6 Newton / 1210 GMRES, A7=8.0e-17, sigma_min=0.1139.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

- docs/src/islands/papers/paper-1/OUTLINE.md: the Level-0 figure contract
  (docs/07 §3), claims → figures → ladder IDs. C1-C3 (discretization order,
  solver conditioning, conservation structure) are green as CI artifacts;
  C4-C8 (neoclassics, large-w limits, the York triangle, the Δ_pol(ω_E)
  reversal, the L23 electron-Δ_pol question) are gated on QUESTIONS Q2-Q4.
- docs/src/islands.md: M2 status (all seven submodules, gating summary).
- LOG.md: M2 session entry, including the solve-well-posedness lesson (generic
  y-diffusion without BCs is unstable; the mimetic degenerate form + far-field
  x-BCs are the correct structure).

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…docs missing_docs)

Documenter's checkdocs=:exports requires every exported docstring in the
package (including submodules) to appear in a page; the islands.md @autodocs
block only listed the top-level Islands module, so the PhaseSpace/Operators/
Verify exports failed the docs build. Add per-submodule @autodocs sections.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…iginal/:improved toggle (D7)

Second M2b derivation-lane chapter. Independent re-derivation of the orbit-
averaged magnetic drift frequency from the guiding-centre drift (I19 Eq.8) and
p_phi conservation:

  omega_hat_D = (sigma v_hat/(1+eps))[ (1/L_hat_q)<sqrt(1-yb)/b>
                - (1/2)<(1/L_hat_B)(2-yb)/(b sqrt(1-yb))> ]

Two physical pieces: (1) the drift-orbit radial width x_D = rho_hat_theta_i
(sigma v_hat/(1+eps)) sqrt(1-yb)/b coupled to the safety-factor shear (1/L_q);
(2) the grad-B drift from dB/dpsi (1/L_B). The :original/:improved toggle is the
treatment of dB/dpsi: D21 Eq.(A2) gives dB/dpsi = -(B_phi/R_0^2 B_theta) cos theta
(low beta), so the cos theta orbit-averages to O(eps) => :improved sets
L_hat_B^-1 = 0; :original keeps it finite. This toggle is the highest-impact
physics item (the T2 ~x6 threshold-width differential; absolute 8.73->1.46 rho_bi
is T4/audit-gated, per D9).

Cross-checks (all first-hand): I19 Eq.(32), D21 Eq.(B1) [with the identity
V_par + lambda V^2/(2 V_par) = V(2-lambda B)/(2 sqrt(1-lambda B))], D21 Eq.(A2).
No discrepancy (unlike the psi-tilde typo).

physics-verifier: PASS (algebra (a)-(d) verified by hand, D21-B1 reconciliation
confirmed, provenance clean, no src/ change; MagneticDrift.c_D stays gated).
Docs-only; local docs build green.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…le; drift-model toggle usable)

Human sign-off 2026-07-11 of the omega-hat-D derivation (D7). Records the
clearance (docs/01 §2.1: [CHECKED] -> [CLEARED]) and implements the cleared
coefficient:

- New src/Islands/coefficients/Coefficients.jl: the home for human-cleared L0
  physics coefficient builders (M2b fill-ins). magnetic_drift_frequency(...)
  computes omega-hat-D = (sigma v_hat/(1+eps))[inv_Lq A - (1/2) inv_LB G] via the
  orbit-average brackets A=<sqrt(1-yb)/b>, G=<(2-yb)/(b sqrt(1-yb))> (QuadGK;
  passing full-circuit, trapped bounce-integral with the integrable turning-point
  singularity). variant=:improved forces inv_LB=0 (D21 proxy). Wired into
  Islands.jl.
- 13 tests: eps->0 analytic limit (A->sqrt(1-y), G->(2-y)/sqrt(1-y)), the toggle
  (:improved == :original with L_B=0), sigma-odd / v_hat-linear prefactor,
  trapped finiteness, forbidden-y and bad-variant guards. 84 islands-solve tests
  green. The toggle is a large effect at the test point (grad-B nearly cancels
  the shear term) -- the mechanism behind the ~x6 threshold differential.

physics-verifier: PASS (formula matches signed-off derivation; clearance recorded;
trapped branch sound; Frames/Fields NaN-gates and Moments prefactors untouched).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…deflection frequency + nu_star (D7)

Third M2b derivation-lane chapter. Independent re-derivation of the Level-0
momentum-conserving pitch-angle (Lorentz) collision operator (I19 Eq.9), the
deflection frequency (I19 Eq.11), and the nu_star normalization (L23 Eq.2.3.40):

- The pitch operator is Lorentz scattering in self-adjoint form
  w^-1 d_lambda(P d_lambda) with diffusivity P(lambda)=lambda sqrt(1-lambda B)
  and measure w=B/sqrt(1-lambda B); the change of variables from the pitch
  cosine derives the exact 2 nu_jj prefactor. This is the same divergence form
  the mimetic PitchAngleDiffusion discretizes (conservation gate A4).
- Deflection frequency nu_jj = nu_tilde[phi(v_hat)-G(v_hat)]/v_hat^3
  (Chandrasekhar), with the :chandrasekhar/:vcubed sub-toggle. Low-v: phi-G ->
  (4/3 sqrt-pi) v_hat (LINEAR) so nu_jj ~ 1/v_hat^2 diverges -- the
  "nu_tilde ~ u^-2 divergence" of docs/01 §2.3 motivating the analytic average.
- nu_star = nu_jj Rq/(eps^3/2 v_th) normalization.

Two sub-items honestly DEFERRED (not asserted, policy rule 4): the analytic
<nu_hat_ii>_u constant (L23 Eq.4.1.6, its own short derivation) and the
discretized diffusivity profile (numerics, ties to A4).

Review caught two errors, both fixed before sign-off: the §3 low-v limit
(phi-G is linear, not cubic -> nu_jj diverges, not finite) -- caught
independently by me and the physics-verifier -- and a §2 factor-of-2 in the
change of variables (now derives the exact 2 nu_jj). Docs-only; local docs build
green. Awaiting human sign-off.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

… src (P(lambda), deflection frequency)

Human sign-off 2026-07-11 of the collision-operator derivation (D7). Records the
clearance (docs/01 §2.3: [CHECKED] -> [CLEARED] for the operator structure,
deflection frequency, and nu_star; the <nu_hat_ii>_u constant stays [CHECKED]-
gated as a deferred sub-item) and un-gates the cleared building blocks:

- Coefficients.pitch_diffusivity(lambda, B) = lambda sqrt(1-lambda B): the
  Lorentz operator's self-adjoint diffusivity P (feeds conservative_pitch_operator;
  vanishes at both endpoints, preserving the A4 gate).
- Coefficients.deflection_frequency(v_hat; model): nu_jj = nu_tilde[phi-G]/v_hat^3
  (:chandrasekhar) or nu_tilde/v_hat^3 (:vcubed), phi via QuadGK, G Chandrasekhar.
- 11 tests: P endpoints/positivity/domain; deflection high-v (-> nu_tilde/v_hat^3,
  with the slow 1-1/2v_hat^2 approach) and low-v (-> (4/3 sqrt-pi)/v_hat^2
  divergence, the derived linear phi-G limit); chandrasekhar < vcubed at low v;
  model/domain guards. 96 islands-solve tests green.

The full discretized/orbit-averaged PitchAngleDiffusion assembly and the
<nu_hat_ii>_u momentum-restoring constant remain deferred (docs/01 §2.3 note).

Derivation physics-verifier: PASS (after review caught + fixed a low-v limit
error and a change-of-variables factor). Un-gate implements the just-verified
cleared forms directly, validated by the eps/velocity-limit tests.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…a) + amplitude (D7); fix Coefficients docs coverage

Fourth M2b derivation-lane chapter (awaiting sign-off). Independent re-derivation
of the flattened-electron profile function and its amplitude:

- The closure constraint <d2h/dx2>_Omega = 0 (the A7 identity) gives the ODE
  2 h'' <Omega+cos xi>_Omega + h' = 0; with <Omega+cos xi>_Omega = Q/(2Q')
  (Q=(1/2pi)∮sqrt(Omega+cos xi)dxi) this integrates to h'(Omega) = C/Q(Omega) --
  i.e. h = Theta(Omega-1) C ∫_1^Omega dOmega'/Q. The already-green A7 gate IS this
  closure constraint.
- Far-field matching h -> x (Q -> sqrt(Omega)) fixes the amplitude
  C = w_psi/(2 sqrt2), matching I19 Eq. 18 exactly.
- The I19 Eq. 22 flow-relation structure is presented; the two constants
  k ~= -1.173 (Hirshman-Sigmar) and the 1.46 in f_p = 1 - 1.46 sqrt(eps) are
  honestly DEFERRED (their own short derivations), not asserted.

physics-verifier: PASS (algebra (a)-(e) verified first-hand incl. h'=C/Q and the
C=w_psi/2sqrt2 matching; provenance clean; deferred constants not asserted).

Also FIXES the docs build (was red since the omega-D commit): the new
Coefficients module's exported docstrings were not in the manual; added its
@autodocs section to islands.md (checkdocs=:exports). Local docs build green.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…ts.h_amplitude)

Human sign-off 2026-07-11 of the electron-closure derivation (D7). Records the
clearance (docs/01 §2.4: h(Omega) form + amplitude [CLEARED]; the flow constants
k ~= -1.173 and f_p's 1.46 stay [CHECKED]-gated as deferred sub-items) and
un-gates the cleared amplitude:

- Coefficients.h_amplitude(w_psi) = w_psi/(2 sqrt2): the flattened-electron
  profile prefactor C, derived from the flattening constraint (h'=C/Q) + far-
  field matching h->x. Feeds Fields.h_profile's prefactor.
- 3 tests: the amplitude value; the far-field asymptote
  h_profile(Omega; C) -> (w_psi/sqrt2)(sqrt(Omega)-1) (a real check of the §3
  derivation + the Q->sqrt(Omega) asymptotics); ElectronClosure k/f_p stay NaN.
  98 islands-solve tests green.

Derivation physics-verifier: PASS (h'=C/Q ODE and C=w_psi/2sqrt2 matching
verified first-hand). Un-gate is a one-line cleared relation; far-field test
validates it physically.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…), arbitrary tau (D7)

Fifth M2b derivation-lane chapter (awaiting sign-off). Independent re-derivation
of the Level-0 quasineutrality closure from the ion/electron density moments:

- Ion moment n_i = n0(1-ePhi/T_i) + n0'(psi-psi_s) + dn_bar_i (from f_i, I19 Eq.23).
- Electron moment n_e = n0(1+ePhi/T_e) + n0' h(Omega) (flattened closure; the
  v_par-odd term vanishes).
- n_i=n_e => Phi_hat = tau/(tau+1)[dn_bar_i/n0 + L_n0^-1(x - h_hat)]. The
  denominator (tau+1)/tau is the SUM of ion+electron adiabatic shielding
  responses (1/T_i + 1/T_e) -- the origin of the "2" in 1/(2 L_n0). At tau=1 this
  reduces exactly to I19 Eq. A.11 [dn_i/n0 + x - h_hat]/(2 L_n0), and the
  derivation adds the arbitrary-tau generalization docs/01 §3 asks for.

Initial pass surfaced a structural question (dn_i inside vs outside the 1/L_n0
factor); read I19 Eq. A.11 first-hand (print p.11) and resolved it: no
discrepancy -- I19's normalized dn_i = L_n0 * (raw moment), a definitional
convention, flagged openly. The invariant x-h_hat structure and the tau=1
coefficient match I19 exactly with nothing tuned.

physics-verifier: PASS (algebra a-e verified first-hand incl. the (tau+1)/tau
factor and tau=1 reduction; provenance clean, no src/ change; dn_i normalization
honestly a convention). Local docs build green.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…tau+1)); triple-checked dn convention

Human sign-off 2026-07-11 of the quasineutrality-closure derivation (D7), after
an independent triple-check of the dn normalization. Records the clearance
(docs/01 §3) and un-gates the coefficient:

- Coefficients.quasineutrality_coefficient(tau) = tau/(tau+1): the closure
  coefficient (sum of ion+electron adiabatic responses; -> 1/2 at tau=1).
- 4 tests: tau=1 -> 1/2, tau=2 -> 2/3, tau->inf -> 1, positivity guard.
  102 islands-solve tests green.

Triple-check resolution: the (x - h_hat) structure and the coefficient match
I19 Eq. A.11 exactly; the only difference from I19's delta_n_i is a scaling by
L_n0, which is PHYSICALLY FORCED (the raw moment is gradient-driven,
delta_n_bar_i ~ L_n0^-1). Decisively, the code uses the RAW-MOMENT form
(delta_n_bar_i = velocity_moment! output), so I19's delta_n_i normalization is a
cross-check nuance only and does not enter src -- the un-gate is unambiguous.
Also derived the arbitrary-tau generalization docs/01 §3 asked for.

physics-verifier: PASS (derivation algebra a-e verified first-hand). Un-gate is a
one-line cleared coefficient.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…tors (D7; sin-normalization pin)

Sixth and final M2b main-derivation chapter (awaiting sign-off). Derives the
output-moment prefactors:

- Delta_cos = Delta_neo = -(mu0 R/2 psi_tilde) ∫∫ Jbar_par cos xi: the growth
  moment matched to the outer Delta' (stationarity Delta' + Delta_neo = 0). The
  prefactor is sourced piece by piece -- mu0 R from parallel Ampere, psi_tilde
  the ALREADY-CLEARED island_flux_amplitude, the 1/2 the adopted Diss19 Eq. 4.12
  Rutherford matching convention (not re-derived from scratch, stated honestly).
- Delta_sin = +(mu0 R/2 psi_tilde) ∫∫ Jbar_par sin xi: the sin normalization has
  no external Delta' to match, so it is a [DERIVED] CHOICE (docs/01 §4 flags it
  as such). Pinned SYMMETRICALLY to cos so Delta_cos + i Delta_sin is the natural
  complex growth+torque moment mapping to the linear-layer Delta(Q) (D1), and the
  A3 parity gate (cos even/sin odd) holds with matched normalization.

physics-verifier: PASS (provenance clean, no src/ change, delta_moments prefactors
stay required/gated; the cos attribution is honest -- adopted Diss19 convention,
not overclaimed; the sin pin is honestly a [DERIVED] decision, consistent with the
green A3 gate). Local docs build green.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…ation lane complete (6/6 cleared)

Human sign-off 2026-07-11 of the Delta-prefactor derivation (D7). Records the
clearance (docs/01 §4) and un-gates the prefactors:
- Coefficients.delta_moment_prefactors returns (cos=-mu0R/2psi_tilde,
  sin=+mu0R/2psi_tilde); psi_tilde the cleared island_flux_amplitude; feeds
  Moments.delta_moments. Symmetric sin pin ([DERIVED]). 105 tests green.

Completes the M2b derivation lane: all six main L0 coefficient families derived
(D7), signed off, cleared. Deferred sub-constants (<nu_hat_ii>_u, k, f_p) stay
[CHECKED]-gated.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…dit + docs infra)

Next milestone after the completed M2b derivation lane: assemble the six cleared
coefficient families into a runnable L0 configuration, produce the D9 input-
completeness audit (docs/09), stand up docs/07 infra (STATE dashboard +
anchor-sync), and scaffold the B-ladder to the assembly. Autonomous-mode: un-gate
nothing new (deferred sub-constants stay gated), draft deferred derivations only
if rigorous else escalate to QUESTIONS, never guess.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…9 deliverable)

Per-source input manifests (I19, D21, D23a/b, L23) auditing which published
NTM-threshold configurations pin down every required input. Type specimen: I19's
run collisionality is internally contradictory (0.01 vs 10^-3) and its Delta'
value is unspecified, so B5a's absolute threshold downgrades to T3 (existence).
L23 (thesis) is the only clean T4 candidate. This is itself a reproducibility
result for Paper I C9. Added to docs nav.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…coeffs wired, gated families escalated)

configure_level0 (src/Islands/configure/Configure.jl) assembles the Level-0
named configuration: it wires the CLEARED coefficient builders onto the operator
stack and takes every uncleared family as a supplied gated input (never guessed).

Cleared wiring (verified faithful):
- c_D from magnetic_drift_frequency, node-for-node (Δ=0.0), with the
  :original/:improved toggle and forbidden-region (y >= 1/b_min) zeroing;
- the pitch-collision shapes from pitch_diffusivity + deflection_frequency;
- the Δ-moment prefactors from delta_moment_prefactors.

Gated/supplied (GatedLevel0Inputs, escalated to QUESTIONS Q5): parallel
streaming, E×B c_E, gradient drive, quasineutrality α, collision magnitude
(⟨ν̂_ii⟩_u/ν_star), orbit-averaged pitch measure, neoclassical far field. The
assembly surfaced a structural gap — the quasineutrality operator lacks the
L̂_n0⁻¹(x−ĥ) field source the cleared closure requires — so no Level-0 physics
run is possible until that lands; M2c delivers the assembly scaffold, verified
STRUCTURALLY (level0_placeholders → Newton-Krylov converges, 5 iters ‖F‖=1.3e-9).

184 islands tests green (new runtests_islands_configure.jl). The y=0 orbit-average
guard was relaxed (y>0 → y>=0): a domain-boundary fix, no y>0 value changes.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…ding wired to the assembly

#3a STATE dashboard (docs/07 §1.3): Verify.write_state_dashboard + ladder_status
generate docs/src/islands/state/STATE.md (auto-generated, do-not-hand-edit) — the
docs/05 ladder as a status table (8 A-ladder rows green via the test suite; B/C
physics rows gated on QUESTIONS). Nav-wired.

#4 B-ladder scaffolding: benchmarks/islands/benchmark_B{2,4,5}*.jl wired to
Islands.Configure.configure_level0 with a one-line un-skip (const UNGATED = true),
kept skipped on QUESTIONS Q3/Q5. B5 carries the full T2 :original/:improved toggle
scaffold; the physics w_c extraction stays gated on the Q5 far field + QN source.

Also fixes a latent docs-CI breakage: cross-module @refs in the Coefficients
delta_moment_prefactors docstring (island_flux_amplitude, Moments.delta_moments)
do not resolve from the Coefficients module context — converted to plain code
spans. build_docs_local.jl now green. 184 islands tests green.

Remaining M2c: the anchor-sync CI check (#3b) is deferred — it needs the
as-implemented Physics Book chapters + operator anchors (a larger docs deliverable).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…d assembly docs

#3b anchor-sync (docs/07 §1.1): Verify.check_anchor_sync enforces the
bidirectional operator<->docs sync — forward (every AbstractTerm operator is
named by an 'Implemented by:' marker in numerics.md) and reverse (every marker
symbol resolves to a real Islands binding). A new operator without docs, or a
doc naming a deleted symbol, fails the check. Tested with negative controls.

#6 as-implemented docs: numerics.md gains the Level-0 configuration-assembly
section (configure_level0, the cleared wiring vs the Q5-gated families) and
'Implemented by:' markers over the operator stack.

This completes the autonomous portion of M2c: #1 assembly, #2 audit, #3 docs
infra (STATE dashboard + anchor-sync), #4 B-ladder scaffolding, #6 numerics.md.
Remaining M2c is human-gated by design: Q5 (the un-cleared coefficient families
+ the QN operator structural fix) gates any Level-0 physics run and the B-ladder
physics gates; the deferred sub-constants (#5) need a sign-off session.

189 islands tests green; build_docs_local.jl green.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…tion (awaiting sign-off)

derivations/passing-fraction.md [DERIVED] derives the electron-closure passing
fraction f_p ~= 1 - 1.46*sqrt(eps) from the effective trapped-fraction integral
(Lin-Liu-Miller / Wesson) in the pinned b(theta) convention, shows the eps->0
all-passing limit analytically (B(2,1/2)=4/3), and NUMERICALLY confirms the
leading coefficient c1 = 1.4624 — matching the sources' quoted 1.46 to 3 s.f.

DRAFTED, NOT CLEARED: f_p stays NaN-gated in Fields.ElectronClosure (no src
change); awaits human sign-off with one open reviewer item (confirm I19 Eq. 22
uses the same effective-trapped-fraction definition). The companion deferred
constants <nu_ii>_u and Hirshman-Sigmar k are left escalated (QUESTIONS Q3/Q5) —
each needs its specific source integrand, not derived speculatively.

This is the M2c #5 deliverable (draft where rigorously possible; escalate the
rest). Wired into the derivations index + nav + QUESTIONS Q5. build_docs_local.jl
green.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…y closure (Phi now driven)

Closes the M2c-surfaced quasineutrality structural gap (QUESTIONS Q5). The
closure was signed off in M2b, but Operators.Quasineutrality carried only
R_Phi = M[g] - alpha*Phi (no drive), so the Level-0 potential collapsed to zero.

Implemented the full cleared closure (physics-verifier PASS):
- Operators.Quasineutrality gains an optional source field; apply! adds it
  (R_Phi = M[g] - alpha*Phi + S). Backward-compatible Quasineutrality(alpha).
- Configure.configure_level0 builds alpha = (tau+1)/tau (= 1/quasineutrality_
  coefficient(tau), the reciprocal, = 2 at tau=1) and the drive
  S = inv_Ln0*(x - h(Omega)) via the new Configure.quasineutrality_source, from
  the cleared h_amplitude/h_profile with one width w=w_psi for both Omega and the
  h prefactor (electron-closure.md §3 far-field matching).
- Added inv_Ln0 to Level0Physics; removed alpha from GatedLevel0Inputs and
  level0_placeholders; :quasineutrality moved from gated to cleared.

Verified: max|Phi| ~= 5.7 after the placeholder solve (was ~0). 194 islands
tests green (new testset asserts the reciprocal alpha, the wired source, and a
nonzero solved Phi). Doc-first: docs/01 §3, quasineutrality-closure.md §6,
numerics.md, QUESTIONS Q5 amended. build_docs_local.jl green.

The kinetic Q5 families (streaming, E×B, gradient drive, <nu_ii>_u, pitch
measure, far field) remain gated; a physics threshold still needs those.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…fficients

Re-derived the island-streaming channel from the master DKE (I19 Eq. 32, L23
§2.6-amended trapped mask) — derivations/parallel-streaming.md, human sign-off.
The two coefficients factor EXACTLY into {Omega, g} flux-surface advection:
  a_xi = (inv_Lq/rho_hat_theta_i)*x*Theta(y_c-y)
  a_x  = -(inv_Lq*w^2/4rho_hat_theta_i)*sin(xi)*Theta(y_c-y)
= (inv_Lq*w^2/4rho_hat_theta_i)*Theta*{Omega, .}  -- a coefficient-free structural
check that leaves no freedom (streaming must advect along Omega-contours).
Passing-only via Theta(y_c-y). Normalization (divide by -m*rho_hat_theta_i) chosen
so the already-cleared c_D = omega_hat_D is UNCHANGED.

Implemented as Configure.streaming_coefficients with a new
Level0Physics.rho_hat_theta_i; :streaming moved gated -> cleared; a_xi/a_x removed
from GatedLevel0Inputs and level0_placeholders. physics-verifier PASS (forms,
signs, {Omega,g} structure, c_D unchanged, no guessed coefficient all checked
vs the signed-off derivation). 1494 islands assertions green (new per-node
{Omega,g} structure testset). Doc-first: docs/01 §2, QUESTIONS Q5, numerics.md
§2/§8, derivations index + nav. build_docs_local.jl green.

Cleared now: streaming, drift, collision shapes, quasineutrality field, Delta
prefactors. Still gated (Q5): E×B, gradient drive, <nu_ii>_u, pitch measure,
far field.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…= far-field BC)

Read I19 first-hand (Eqs. 8, 23-32) to derive the Level-0 gradient drive.
Finding (derivations/gradient-drive.md, draft): the master orbit-averaged
equation (I19 Eq. 32) is HOMOGENEOUS in G0 -- no interior source term. The
gradient drive enters as the FAR-FIELD BOUNDARY CONDITION: far from the island
G0 -> p_phi*(omega_si^T/omega_ci)*(n'/n)*F_Mi (Eq. 29), the diamagnetic-driven
distribution.

Consequence: Operators.GradientDrive is ZERO at Level 0, and the Q5
'gradient_drive' and 'far_field' items merge into one physical object -- the
diamagnetic far field g_drive = D_dia*x*[1+(vhat^2-3/2)*eta_i]*F_Mi. The
x-linearity, the (vhat^2-3/2)eta_i temperature correction, and the Maxwellian
are cleared structure; the normalized amplitude D_dia bundles the frame
convention Frames.C_dia (still NaN-gated) -- clearing the drive = clearing C_dia
(the ion omega_dia normalization) in the same sign-off.

Nothing entered src/ (docs-only draft). QUESTIONS Q5, derivations index + nav,
LOG updated. build_docs_local.jl green.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

Signed off derivations/passing-fraction.md and cleared
Coefficients.passing_fraction(eps) = 1 - 1.4624*sqrt(eps) -- the effective
trapped-fraction coefficient (Lin-Liu-Miller / Wesson), derived analytically
(eps->0 all-passing limit) and numerically confirmed (c1 = 1.4624, matching the
sources' quoted 1.46 to 3 s.f.). Authorizes Fields.ElectronClosure.f_p.

physics-verifier PASS (constant matches the derivation's converged value, form
matches I19 Eq. 22, k correctly stays gated). 1499 islands assertions green
(limits + monotonicity + the 1.46 match). Doc-first: docs/01 §2.4 [CLEARED],
QUESTIONS Q3/Q5 RESOLVED, derivations index/nav, Fields.ElectronClosure
docstring. build_docs_local.jl green.

The companion Hirshman-Sigmar k ~= -1.173 remains deferred (own derivation).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…o frame convention)

Completed the gradient drive by re-reading I19 Eq. 29 first-hand. CORRECTION:
the earlier draft misread the ratio as omega_si^T/omega_ci (=> would need the
frame convention); it is omega_si^T/omega_si = 1 + (vhat^2-3/2)*eta_i, a
dimensionless TEMPERATURE factor, not a frequency ratio. So NO frame convention
is needed. The drive is the standard neoclassical p_phi*F'_Mi, imposed as the
FAR-FIELD boundary condition (master eq is homogeneous -- I19 Formulation A), not
an interior source.

Cleared: Operators.GradientDrive = 0 (zero interior source) and
Configure.gradient_far_field builds g_far = x*inv_Ln0*[1+(E-3/2)*eta_i]
(Phi_far = 0 at omega_E=0), with a new Level0Physics.eta_i. Both :gradient_drive
and :far_field moved gated->cleared; drive/bc dropped from GatedLevel0Inputs.

physics-verifier PASS -- re-read the PDF first-hand and CONFIRMED Eq. 29 is
omega_si^T/omega_si (not omega_ci), validating the correction; far-field form,
zero source, Phi_far=0, no frame convention, no guessed literal all checked. 1511
islands assertions green (the assembly now solves with the physical far field).
Doc-first: gradient-drive.md signed off, docs/01 §2, QUESTIONS Q5, numerics.md,
derivations index/nav. build_docs_local.jl green.

Remaining gated (Q5): E×B coupling, collision magnitude <nu_ii>_u, orbit-averaged
pitch measure. Plus the deferred k ~= -1.173.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…STIONS Q5)

Captured the E×B derivation core for the next session: c_E = (1/2)<1/vhat_par>_theta
(the orbit transit-time weight, NOT frame-entangled), plus the three wrinkles to
finish carefully -- velocity-dependent coefficient (operator change), the new
<1/sqrt(1-yb)> orbit bracket, and the passing-vs-trapped sigma-parity. Paused
before implementing to give the sigma subtlety a fresh look rather than
end-of-session attention.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…, trapped ≡ 0)

Derive and clear the E×B coupling c_E = ½⟨1/v̂_∥⟩_θ (human sign-off,
derivation exb-coupling.md). Matching the master-eq E×B braces to the
ExBDrift Poisson bracket in the c_D=ω̂_D normalization (÷ −m ρ̂_θi, ρ̂_θi
cancels — no new physics parameter) fixes the coefficient with no sign
freedom.

σ-parity (the crux): passing (y<y_c) is σ-odd c_E = (σ/2√E)B₁(y) with a new
orbit bracket B₁(y)=⟨1/√(1−yb)⟩_θ; trapped (y>y_c) is identically 0 — the
σ-odd 1/v̂_∥ cancels between the two banana legs under Σ_σ. Required (not
chosen) by the published drift-island label S ∝ p̂ for trapped (docs/01 §2.2).
E×B is a passing-particle effect, like island-streaming.

- Coefficients.orbit_average_exb_bracket — B₁(y), passing-only, same y_c-layer
  handling as the drift G bracket.
- Configure.exb_coupling_table — c_E[ix,iξ,iy,iE,iσ], passing-only Θ(y_c−y).
- Operators.ExBDrift generalized to a velocity-dependent array coefficient
  (scalar path retained for manufactured tests).
- :exb moved gated→cleared; c_E removed from GatedLevel0Inputs.

physics-verifier PASS; 1566 islands assertions green; build_docs_local.jl
green. The M2c structural-solve assertion switched to the grid-independent
max-norm (04 §5 criterion): the now-active E×B spreads a ~1e-8 residual across
~N unknowns, inflating the √N L2 norm past 1e-7 (per-equation residual 4.5e-8;
not a physics change). Doc-first: docs/01 §2 (+ fixed the §2 line-107
orbit-average convention to match both cleared code paths), QUESTIONS Q5,
numerics.md §2/§8, derivations index/nav.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…ntum-restoring ⟨ν̂_ii⟩_u)

Derive and clear the two remaining pieces of the Level-0 collision magnitude
(human sign-off, derivation collision-magnitude.md), completing the deferred
sub-item of the signed-off collision-operator derivation (§7).

Read L23 Eqs. 4.1.4–4.1.6 (p. 87–88) first-hand and derived the
momentum-restoring speed average

  ⟨ν̂_ii⟩_u = (4ε^{3/2}ν_★/3√π)(√2 − ln(1+√2))

from L23's reduced integrand: the reduction u⁴ν̃_jj = u·erf − erf/2u + e^{−u²}/√π
plus three standard integrals (∫u e^{−u²}erf = 1/2√2, ∫e^{−2u²} = ½√(π/2),
∫u⁻¹e^{−u²}erf = ln(1+√2) = arcsinh(1)). Verified independently (QuadGK): all
three integrals and the u⁴ν̃ reduction exact to 15 digits, ⟨1⟩_u = 1
(normalized), and the full value reproduces L23's own unit-test 1.267537e-4
(ε=0.1, ν_★=0.01) to all 7 digits.

- Coefficients.momentum_restoring_average — the ⟨ν̂_ii⟩_u constant (the
  restoring operator term is a separate future addition; this clears its
  magnitude).
- Level0Physics.nu_star — the banana-regime collisionality ν_★, a scenario scan
  input (Decision D7).
- Configure builds nu_tilde = ε^{3/2}ν_★ (the already-signed-off §4
  normalization) and wires it into the collision coefficient; :nu_tilde moved
  gated→cleared; nu_tilde removed from GatedLevel0Inputs (now B_profile-only).

This un-gates the collision operator magnitude — only the orbit-averaged pitch
measure B_profile remains gated at Level 0. physics-verifier PASS; 1577 islands
assertions green; build_docs_local.jl green. Doc-first: docs/01 §2.3,
QUESTIONS Q5, numerics.md §2/§8, collision-operator.md §7, derivations
index/nav.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…or (5/6 terms, last operator gate)

Clear the full orbit-averaged collision operator (human sign-off, derivation
orbit-averaged-collision.md) — the last Level-0 operator gate. Reading L23
Eq. 2.3.47 / appendix 8.3.2 first-hand showed "B_profile" is the tip of a
six-term operator. Implement the five differential terms in the code
normalization ÷(−m ρ̂_θi):

- σ-odd mimetic PITCH DIFFUSION D+E — exact y-divergence ∂_y(P_oa ∂_y) with
  P_oa = y⟨√(1−yb)⟩_θ (orbit-averaged, replacing the local single-B
  placeholder), flat measure (divergence identity verified to 1e-11);
- σ-even ∂_x DRAG (A); σ-odd ∂²_x NEOCLASSICAL diffusion (B, uses ⟨1/√(1−yb)⟩_θ);
  σ-even ∂²_xy CROSS (C).

Key fidelity fixes: the pitch diffusion is σ-ODD (the 1/v̂_∥ transit weight, like
ω̂_D/c_E) — a correction of the former σ-even placeholder; and the collision
coefficients are POSITIVE (÷(−m ρ̂_θi) of L23's leading −, matching the streaming
anchor a_ξ=+(x/L̂q)Θ/ρ̂θ) — the physics-verifier caught an initial sign error and
confirmed the fix. The collision terms carry an explicit 1/(m ρ̂_θi) (m does not
cancel as it does for the drift), so m is a new Level0Physics scenario input.

- Coefficients.orbit_average_pitch_brackets — (S=⟨√(1−yb)⟩, T=⟨1/√(1−yb)⟩).
- Configure.{pitch_diffusivity_profile, pitch_collision_coefficient,
  collisional_drag_coefficient, neoclassical_diffusion_coefficient,
  collisional_cross_coefficient}.
- Operators.{CollisionalDrag, NeoclassicalDiffusion, CollisionalCross} + kernels;
  FarFieldConditions.forbidden_y pins g=0 in the no-particle region.
- GatedLevel0Inputs / level0_placeholders removed — no gated kinetic inputs
  remain; configure_level0 takes no gated argument (Q5 fully cleared).

physics-verifier PASS (after the sign fix); 1595 islands assertions green
(node-for-node coefficient/σ-parity checks + an A4 conservation/entropy gate on
the shipped P_oa); build_docs_local.jl green. Doc-first: docs/01 §2.3,
QUESTIONS Q5, numerics.md §2/§8, derivations index/nav; B5 benchmark de-gated.

Two collision follow-ups remain (not gates on the differential solve): the
nonlocal momentum-restoring term F (magnitude ⟨ν̂_ii⟩_u already cleared), and
repointing Verify.jl's assembled-MMS harness to the new operators.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

… on the gated parallel-flow moment weight)

Investigating the sixth collision term (momentum-restoring F,
2ν̂_ii(1+ε)Ū_∥ᵢ(ĝ+pF̂′)F̂_M) surfaced that it is blocked on a gated
normalization, so it is escalated rather than guessed (per the [VERIFY]
policy).

The structure is clear (L23 Eq. 2.3.47/8.3.17): a nonlocal operator forming the
parallel-flow moment Ū_∥ᵢ(x,ξ) = (1/√π⟨ν̂_ii⟩_u) Σ_σ σ ∫du u³ν̂_ii ∫dy ĝ, then
redistributing 2ν̂_ii(1+ε)Ū F̂_M (σ-even, positive sign per §3). Its magnitude
⟨ν̂_ii⟩_u is already cleared (collision-magnitude.md).

The blocker: Ū is exactly the parallel-flow velocity moment whose weight
W(y,E,σ) (the v̂_∥-structure) is [VERIFY]-gated (QUESTIONS Q3) —
Moments.parallel_current! and Operators.jl:185 already flag the field-particle
piece as Q3-gated. Compounding: the energy grid is Gauss-Laguerre
(Σ wₑ f ≈ ∫ f e^{−E} dE, Maxwellian folded in), but Ū's moment is a plain
∫du u³ν̂_ii, so the g↔F̂_M convention must be pinned first. Escalated to
QUESTIONS Q5; nothing entered src/ for F.

Doc-only: QUESTIONS Q5 (new F item), orbit-averaged-collision.md §9 status.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

… + parallel-flow weight W

Clearing Q3's parallel-flow weight W surfaced (escalated as Q6) that the code's
velocity_moment!/weighted_moment! used a flat measure missing the physical ∫d³v
Jacobians (√E/2 speed, 1/√(1−yb) pitch) — and that the already-cleared QN
density δn̄_i used that flat moment. The user chose the physical ∫d³v with the
flux-surface b. Derived and cleared (human sign-off, velocity-moment-measure.md):

- physical_velocity_weights: the √E/2 speed Jacobian (folded into
  Gauss-Laguerre) and the 1/√(1−y·b_min) pitch Jacobian (b_min=(1−ε)/(1+ε), an
  exact singular-weight quadrature = the IinvB edge; forbidden region zeroed,
  Σwy_phys = 2/b_min exactly).
- parallel_flow_weight: W = v̂_∥ = σ√E√(1−y·b_min) (clears Q3's W); its √(1−yb)
  cancels the pitch Jacobian so J̄_∥ is regular.
- velocity_moment!/weighted_moment! gained wy/wE kwargs (default flat — M1/M2
  manufactured tests untouched); Operators.Quasineutrality carries the physical
  weights, so δn̄_i is now the physical moment (max|Φ| shifted 5.7→4.5, the
  approved QN revision; closure algebra α/S unchanged).

This resolves Q6, clears Q3's W (un-gating Moments.parallel_current! → the Δ
outputs), and UNBLOCKS term F (Q5) — Ū is now a bounded physical moment.

physics-verifier PASS; 1596 islands assertions green (Σwy=2/b_min exact,
forbidden zeroed, W σ-odd, QN wired); build_docs_local.jl green. Doc-first:
docs/01 §3/§4, QUESTIONS Q6/Q3/Q5, Moments docstrings, derivations index/nav.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

… Level-0 collision operator is complete (6/6)

Implement the sixth and final collision term — the momentum-restoring
field-particle integral (F), the one nonlocal Level-0 term (human sign-off;
orbit-averaged-collision.md §6, unblocked by the Q6 physical ∫d³v measure).

- Operators.MomentumRestoring + apply!: forms the parallel-flow moment
  Ū(x,ξ) = (1/√π⟨ν̂_ii⟩_u){ν̂_ii v̂_∥ g}_v (physical measure; moment weight
  W = ν̂_ii·v̂_∥, σ-odd) into a new cache.Ubar scratch, then adds the σ-even
  redistribution +2ν̂_ii(1+ε)Ū/(m ρ̂_θi) (positive — ÷−m ρ̂_θi of the RHS; the
  F̂_M=e^{−E} cancels in the g=shape convention).
- Configure.momentum_restoring_term builds W, the redistribution, the physical
  weights, and inv_norm = 1/(√π⟨ν̂_ii⟩_u) from the cleared momentum_restoring_average.
- Wired into configure_level0 (9-term kinetic stack); IslandCache gained Ubar.

Linear in g (Ū is a moment), allocation-free (cache.Ubar), AD-transparent; uses
only cleared inputs. All six collision terms are now complete — no gated kinetic
physics remains.

physics-verifier PASS (traced Ū vs L23 Eq. 8.3.17 line-by-line, positive sign,
linearity, no guessed number); 1690 islands assertions green (W σ-odd,
redistribute positive, F(2g)=2F(g), nu_star=0 guard); build_docs_local.jl green.
Doc-first: docs/01 §2.3, QUESTIONS Q5, numerics.md §2/§8,
orbit-averaged-collision.md (6/6), anchor-sync marker.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr

…rrent Δ outputs

Wire the physical Level-0 solve into the Δ growth/torque moments — the first
physics deliverable off the converged state (docs/01 §4). No new [VERIFY]
coefficient: assembly + diagnostics only, physics-verifier PASS.

- Moments.parallel_current! gains wy/wE kwargs (forwarded to weighted_moment!)
  so J̄_∥ carries the physical ∫d³v measure; W = v̂_∥'s √(1−y b_min) cancels the
  pitch Jacobian and J̄_∥ is regular. Defaults to the flat quadrature.
- Configure.delta_outputs(grid, phys, species, Usol, cfg): builds cleared W +
  physical measure, forms J̄_∥ from the solved bulk-ion g, projects to
  Δ_neo ≡ Δ_cos and Δ_sin with the cleared ∓μ₀R/2ψ̃ prefactors.
- Moments.channel_decomposition + grid_interpolant: the docs/01 §4 flux-surface
  split — ⟨J̄_∥⟩_Ω bootstrap+curvature channel and the Δ_pol polarization
  residual, plus the ⟨J̄_∥⟩_Ω profile (approximate diagnostic, L23 Eq. 2.5.3).

Doc-first: numerics.md §7/§8. Tests: 138 solve + 1540 configure + 5 anchor-sync
green (nodal-exact interpolant, flux-function Δ_pol→0, profile recovers f(Ω),
additive split, single-bulk contract); build_docs_local.jl green.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_016pUwoFRo9a5AjF8HuwcZMr