[rogeriojorge]

Integration branch. Do not merge. This PR is a pin target, not a feature.

This branch combines the unchanged commits of four open PRs:

• Differentiable simulation: Diffrax adjoint pass-through, tests, and an Orszag-Tang inverse-design example #44 (12201a6): the differentiable driver, with adjoint, checkpoint, step-budget and progress controls.
• Stabilize high-order Maxwellian initialization #24 (6f3e7cd): the stable high-order Maxwellian initializer.
• Fix multispecies perturbation bookkeeping #12 (e428bb3): perturbation bookkeeping for all populations.
• Add Hermite boundary-resolution diagnostic #17 (bc90950), which carries Fix odd-grid RFFT energy weights; add a Hermite boundary diagnostic #9 (c3a6598): the Hermite-tail diagnostic and the odd-grid energy weights.

The merge commit a7ca601 (tree 4e1ad06) is the SPECTRAX version pinned by the Parker Solar Probe chirp study. That study now lives in a separate repository.

Why both #44 and #24 are needed:

• Differentiable simulation: Diffrax adjoint pass-through, tests, and an Orszag-Tang inverse-design example #44 alone fails: its initializer overflows to NaN at 64 or more Hermite modes with physical proton thermal widths, and the study's linear benchmark needs 128 or more.
• Stabilize high-order Maxwellian initialization #24 alone is not enough: it lacks the adjoint and step-budget controls.

No other open PR contains both.

Checks at a7ca601:

• All 17 existing tests pass.

• Linear three-Maxwellian ion-acoustic growth rates match independent full-kinetic roots when the high-Hermite sink is ν ≈ 1 ω_pe:

  Hermite modes	Growth-rate error
  128	within 0.7%
  256	within 2×10⁻⁶ ω_pc

The last commit on this branch only removes the study files, so the branch tree is identical to a7ca601. When #44 and #24 are in main, close this PR and move the study's pin to main.

[rogeriojorge]

Collisionless PIC cross-check with JAX-in-Cell: review and plan (no runs yet)

Why this is worth doing

SPECTRAX matches the independent full-kinetic linear roots only with a strong high-Hermite sink, ν ≈ 1 ω_pe; see the G2 update in the description. In linear theory this is harmless, because the sink stands in for the electron Landau resonance, which sits deep inside the truncated electron Hermite spectrum (ζ_e ≈ 0.02). In the nonlinear phase it may not be harmless.

The concern. SPECTRAX has one ν for all populations. The damping it applies to Hermite mode n is ν n³/H³. For protons at H = 128 and n ≈ 40, that is about 0.03 ω_pe, or about 1.4 ω_pc, far above the growth rates of about 0.016 ω_pc.

Why it matters. Proton trapping at saturation creates velocity structure of width v_tr/σ_c = 2(eφ/T_c)^½. That width maps onto exactly those Hermite orders. Saturation level, trapping, phase-space holes and the late-time chirp could therefore be set by the numerical closure rather than by the physics.

The test. A collisionless particle-in-cell (PIC) run has no Hermite truncation and no sink. It is the right independent check of the nonlinear (M2) results, which the kinetic roots cannot supply.

Review of JAX-in-Cell at research-release f42e265 (PR uwplasma/JAX-in-Cell#42, unmerged)

• Fit for this study:
  • The code is 1D3V, electrostatic or electromagnetic, with any number of Species and SI units.
  • Its vth = sqrt(2kT/m) equals SPECTRAX's α·c exactly.
  • Species(x=..., v=...) accepts arbitrary (n, 3) phase-space arrays. The free-streamed Gaussian pulse can therefore be loaded exactly: sample the age-0 Gaussian in (x, v), then set x ← x + v·a. This needs no library change, and the per-species particle count sets the resolution of the weak beam independently of the core.
  • Collisions are off unless configured.
  • Charge-conserving current deposition keeps the discrete Gauss law.
  • quiet_start and displacement seeding give a controllable noise floor.
  • run(store_every=k, store_particles=False) stores gridded E only, so a probe trace is cheap.
  • jax.vmap over seed gives an ensemble in one compilation.
• Conventions to match or record:
  • Mean field. Solver(model="electrostatic") forces ⟨E⟩ = 0 in a periodic box. The electromagnetic path with field_solver="ampere" evolves ⟨E⟩ from the mean particle current, as SPECTRAX does (dE₀/dt = −J₀). Use the Ampère path, and record E₀ in both codes.
  • Resolution. The quadratic shape needs Δx ≲ λ_De to avoid finite-grid heating. The time step needs ω_pe Δt ≤ 0.2, and in the electrostatic limit a Courant number above 1 is allowed.
  • Filtering. Use filter_passes=0 for growth rates, or report the compensated-filter response at the resonant k.
  • Noise. PIC noise acts as an extra seed. The seed must dominate the quiet-start noise floor, and several seed amplitudes must bracket the linear window.
• Not in scope:
  • PIC gradients over long, chaotic runs; the gradient checks (G4) stay with SPECTRAX.
  • Any change to either repository. The PIC driver scripts live in ~/local/spectrax-psp-chirp/scripts/, pinned to f42e265. A JAX-in-Cell example PR can be considered separately later.

Proposed gate G3-PIC, after G2 and before any M2 claim

1. P1, homogeneous benchmark, linear. The same three plan cases plus the T_b/T_c = 16 stable control, at M = 1836 and a cheaper M = 100.
   • Settings: L = 60 λ_Dc, Nx = 64 (Δx ≈ 0.3 λ_De), ω_pe Δt = 0.2, quiet start, 10⁶ particles per species, and a beam-drift or displacement seed at k₁ with 3 amplitudes.
   • Pass criterion: γ within 1% of the full-kinetic root (and of SPECTRAX at ν = 1, H = 128), fitted over an explicitly chosen linear window.
2. P2, homogeneous benchmark, nonlinear. This is the decisive test. Run to saturation and past it.
   • Compare with SPECTRAX at ν ∈ {0.3, 1, 3} and H ∈ {128, 256}: the saturation amplitude |E_k1|max, the time of saturation, the bounce frequency, the phase-space trapping width, the proton parallel-variance change including the w(1−w)ΔU² term, and the energy budget.
   • Pass criterion: SPECTRAX within the PIC ensemble spread, from 4 seeds, and independent of ν in its plateau. If it fails, the M2 claims must be limited to the linear or early nonlinear window, or ν must be lowered with H raised. That is a stop condition, not a tuning knob.
3. P3, the Gaussian-pulse pilot. Use the same initial state as the SPECTRAX M2 pilot from next step 4: exact particle sampling and the same seed spectrum through displacement.
   • Compare the synthetic probe trace, the spectrogram ridge (slope and frequency range), the envelope and the growth history along the trajectory.
   • Pass criterion: ridge slope and frequency within the observational uncertainty (about 7% on the slope), and the envelope within the PIC ensemble spread.

Cost, estimated only (office A4000, float64, about 11 ns per particle-step from the JAX-in-Cell #42 profile)

Run	Particles	Steps	Estimated time per seed
P1/P2, M = 1836 (to 2× saturation)	3×10⁶	about 3×10⁵	about 3 h
P1/P2, M = 100	3×10⁶	about 7×10⁴	about 40 min
P3 pilot (L ≈ 2 km, Nx = 512)	about 1.5×10⁶	about 10⁵	about 30 min

That is roughly one to two days of GPU time for the full gate with 4-seed ensembles. Run M = 100 first. If P2 already shows the sink bias there, M = 1836 is confirmation only. Wait for an idle card, because other users' jobs distort timing, and check the office disk (it was nearly full on 2026-09-13).

Deliverables, local only

• scripts/pic_benchmark.py: P1 and P2.
• scripts/pic_pulse.py: P3.
• runs/pic_*: manifests with the commit, seed, particle counts, Δt, Δx, filter settings, E₀ history and energy error.
• One comparison figure for the research note: the growth and saturation history, SPECTRAX at three values of ν against the PIC ensemble band, plus phase-space snapshots.

[rogeriojorge]

The study code moved out of SPECTRAX; this branch now only integrates #44, #24, #12 and #17 (+#9). The earlier description and comments above describe the study as it was at the pause; its handoff continues in the separate study repository.

[rogeriojorge]

Integration pin with no new code; its parts are reviewed individually in #9, #12, #51 (replaces #24) and #44.