[rogeriojorge]

Stacked on #60 (adaptive/p5-hypercollision, d3379cc). Lands on main after the foundation PRs and #56–#60.

What

field_scaled_closure_rate(F, alpha_s, qs, Omega_cs, Nn, Nm, Np, Ns, c) returns a per-species rate
nu_s = c |q/m|_s max_i sqrt(2 N_i) max_x |E_i - <E_i>| / a_{s,i} shaped (Ns,1,1,1,1,1,1), so it can be passed as
nu to Hermite_Fourier_system (whose Col = -nu * col * Ck already broadcasts) together with
hypercollision_spectrum(order>=2). The docstring of nu now states it may be per species. No default changes:
nothing calls it unless the user does. 2 files, +93/-3 (_model.py +42/-3, new tests/test_field_closure.py).

Why (defect it addresses)

The truncated asymmetric-Hermite (AW) Fourier–Hermite operator -v d/dx + (q/m) E d/dv is not anti-self-adjoint.
In a non-uniform field it has spurious growing modes in the top third of the Hermite ladder at the largest kept |k|:

• frozen field E1 cos(k1 x), a = sqrt(2) v_te, Nn = 64: max Re(lambda) = 0.26 (|k| <= 2), 0.42 (|k| <= 5), 0.58 (|k| <= 10)
  at E1 = 4e-3; the symmetric (SW) basis gives 1e-15;
• oscillating field (omega = 1): Floquet exponent 0.044 (|k| <= 2) and 0.146 (|k| <= 5) at E1 = 0.02, zero below
  E1 ~ 8e-3; SW zero throughout.

The rate grows with the Fourier cut-off and with Nn, which is why, in the pump-frame strong-drive H05 runs,
x-refinement failed earlier (Nx 16 at t = 476 vs Nx 8 at 632) and Nn-refinement did not help. Aliasing
(padded-convolution check: 1e-16), step size (fixed dt 0.01: same failure time) and segment remaps (disabled:
same failure) were ruled out. A fixed nu is overtaken because the self-consistent field grows secularly; a rate
that follows max|E| is not.

Evidence (Dark-SPECTRAX branch highk, studies/highk_*.py, macOS, py3.12, JAX 0.11.2, CPU x64)

H05 pump frame, v_q/v_te = 0.1, c = 1, order-2 spectrum:

• Nx 16: reaches t = 700 (baseline failed at 476); c = 0.5 and 2 and Nn 128 agree with it to t = 700 (10% rule) on
  electron dK and W_ext; no negative kinetic energy or k=0 temperature; mobile-ion grid agreement on dK_e to 691.5
  and W_ext to 700. Before 550 the closure run equals the unregularized run to 5 digits.
• v_q/v_te = 0.03, Nx 16: reaches t = 1000; c halved/doubled agree to 1000; Nn 128 agrees to 980.5 (dK_e).
• Nx 32 (with the noise floor of Dark-SPECTRAX main): agrees with Nx 16 on dK_e and W_ext to 680 (v_q 0.1) and
  952 (0.03), then loses positivity at 672 / 942. That late loss is a physical broadband finite-k instability of the
  quivering plasma (growth rising with k up to k lambda_D ~ 0.1; exact Volterra, linear Hermite and the grid agree on
  it), which the closure correctly does not suppress; the grid at Nx 32 blows up at the same time.
  The study wrapper used there equals this helper used as nu to 4e-16 relative.

Tests

tests/test_field_closure.py: c = 0 gives exactly zero; analytic value (uniform field excluded, axes with N <= 3
inactive); per-species nu in the RHS gives exactly -nu_s col C and zero rate of n, M_i, M_ii; jit.
Local: python -m pytest -q -n 2 tests -> 187 passed (tested SHA d0f371f). GitHub CI does not run on stacked PRs.

Open questions

• c = 1 is a choice; the H05 results are insensitive to c in [0.5, 2]. It is a regularization: report it.
• An L2-stable alternative is the symmetric (SW) basis, which trades exact momentum/energy conservation.