Validation audit report — 2026-06 - progenax — differentiable initial conditions for N-body simulations

1. What was verified¶

Every row is backed by a committed test and a regenerated figure whose measured value is shown against the tested tolerance (no self-consistency-only checks).

Module	Page	Tests	Figures	Key anchor(s)	Date
profiles/Plummer	Plummer equilibrium	20	5	$a=0.7664\,r_h$ exact; $\mathrm{Beta}(3/2,9/2)$ speed law; $Q=0.50$	2026-06-08
profiles/King	King profile validation	32	5	$c(W_0)$ vs King 1966 Table II (Δ≤0.002); density vs DF oracle (1e-10)	2026-06-08
profiles/EFF	EFF profile validation	23	5	slope $\to-\gamma$ ; $\gamma=5\equiv$ Plummer (exact); $f(\mathcal{E})\ge0$	2026-06-08
kinematics/Michie	Michie-King anisotropy validation	12	5	$\beta(r)$ vs DF moment oracle; isotropic King limit (2.7e-3)	2026-06-08
kinematics/rotation + OM anisotropy	Rotation & anisotropy validation	10	5	$v_\phi=\Omega R$ (exact); OM $\beta=r^2/(r^2+r_a^2)$ exact	2026-06-08
substructure/CW04 Q + azimuthal	Substructure (CW04 Q) validation	14	8	CW04 2004 Table 1 (3D0/1/2); $\sigma_\Sigma$ –Q anti-correlation	2026-06-08
diagnostics/Λ_MSR + segregation	Mass segregation validation	8 + 13 + 27	9	Λ_MSR vs analytic ground truth; energy-sorted generator; differentiable observables (soft Λ_MSR / radial / Σ–m) vs exact soft→hard limit + Fisher identifiability	2026-06-09
cluster/two-component	Two-component populations (superseded)	✅	1	per-component mass + dynamical-state recovery	2026-06-08

All four velocity DFs are true equilibria with no external virial rescale (unscaled $Q=T/|V|$ : Plummer 0.50, King 0.505, EFF 0.502, Michie 0.501; 100% bound), and every released structural parameter ( $r_h, r_c, a, W_0, \gamma, M, \Omega, r_a$ ) passes an autodiff-vs-finite-difference gradient check (agreement 10^-6–10^-11).

2. Scientific trustworthiness¶

Verification rests on three independent legs:

Physics-anchored tests — each asserts a quantitative match to an analytic result, an independent oracle, or a published table — never self-consistency.
Independent oracles where no closed form exists (King density via direct velocity integral; Michie $\beta$ via DF 2nd moments) — these catch implementation bugs a self-referential test cannot.
Paper-grounding against held PDFs (King 1966; Cartwright & Whitworth 2004; Merritt 1985; Elson, Fall & Freeman 1987), not memory.

A useful trust grading:

Tier	Meaning	Examples
A — exact/analytic	matches a closed form to ~machine precision	Plummer $a$ , EFF $\rho(a)$ & $\gamma=5\equiv$ Plummer, rotation $v_\phi=\Omega R$ , OM $\beta$ , all gradients
B — published anchor	reproduces a published table within stated tolerance	King $c(W_0)$ vs Table II; CW04 Q vs Table 1
C — approximate / finite-N	statistically anchored or a documented surrogate	Monte-Carlo $Q$ /dispersions (±0.02–0.04); differentiable `q_approx` (kNN)

Limits stated honestly (not papered over)¶

Limitation	Status
EFF sharp truncation ( $\gamma=3$ ) is ~5–8% sub-virial	intrinsic to truncating an empirical (non-DF) profile; use King for a strict lowered-DF equilibrium, or mild EFF truncation. Documented on the page.
*Michie $\beta$ is suppressed below* pure Osipkov-Merritt**	real physics (lowering term breaks $f(Q)$ ); validated vs the DF’s own moment oracle, not textbook OM.
`q_approx` over-reads ~0.1 for concentrated configs ( $Q>0.85$ )	faithful in the substructure regime it is used for; gradient is a kNN soft-surrogate (median AD-FD 0.9%, cell-boundary worst-case ~20%).
King scalar $r_t$ not differentiable (argmax crossing)	deferred + documented; the profile shape in $W_0$ is differentiable.
No LIMEPY cross-validation	by design — validated against the original papers, not the external package.
Inference loop not validated end-to-end on data	gradients verified at the IC / forward-model level; structural-parameter recovery on real/mock observations is the research program, not a shipped result.

Bottom line: trustworthy for production ICs and a methods paper. Not a claim that the differentiable-inference pipeline is validated end-to-end against data.

3. Implementation completeness¶

Layer	State
Validated (publication tier: unit + integration + validation + figures)	Plummer, King, EFF, Michie; matched isotropic DFs + OM anisotropy + Michie anisotropy + rotation; CW04 Q + azimuthal variation; Λ_MSR + energy-sorted segregation; two-component clusters; imf-statistics (25 tests + 5 figures, 2026-06-08); binary-imf (24 orbital tests + 5 figures incl. the regenerated “confidently wrong” recovery, 2026-06-08); environment-imf (Marks+2012/Jeřábková IGIMF: new validation tier of 12 published-table tests + 5 figures, 2026-06-08); analytical-test-cases (two-body Kepler, Chenciner–Montgomery figure-eight, solar-system Kepler III, harmonic oscillator: 12 tests + 5 figures, 2026-06-09); tidal-truncation (Jacobi radius vs the restricted-3-body L1 point + analytic Plummer truncation: new validation tier of 9 tests + 5 figures, 2026-06-09).
Released, tested, not yet figure-validated ⚠️	(none — analytical-test-cases figure-validated 2026-06-09; see the validated row).
Released, unit-tested only ⚠️	(none — tidal-truncation hardened-and-validated 2026-06-09: new `tests/validation/test_tidal_physics.py`, 9 tests vs the L1 Lagrange point + analytic Plummer, + 5 figures; see the validated row).
Experimental (repo-only, not in the wheel) 🔬	`gravoturb` turbulent/fractal density-field subsystem (`src/experimental/`, AC1–AC17).
Deferred / not implemented ⏳ 🚧	differentiable King $r_t$ (IFT, plan written); unified differentiable lowered-model family (Wilson/Woolley/multi-mass); multi-mass equipartition; self-consistent rotating equilibria.

4. What it lets us model¶

The architecture is compositional and differentiable end-to-end: any SpatialProfile pairs with any VelocityDF, plus IMF, binaries, anisotropy, rotation, and substructure diagnostics — flowing into gravax (dynamics) and fluxax (photometry). You can generate, as differentiable initial conditions:

Equilibrium clusters of chosen concentration (Plummer / King $W_0$ / EFF $\gamma$ ) in true virial balance;
radially anisotropic clusters (Michie self-consistent, or OM via $r_a$ ) and rotating clusters (solid-body or differential);
realistic stellar populations — IMF-sampled masses with a faithful binary population (Moe & Di Stefano $P$ – $q$ – $e$ coupling), split across components;
tidally truncated clusters tied to a Jacobi radius;
and quantify substructure (Q, azimuthal variation, mass segregation) on any of them.

5. Example science questions¶

Scoped to the validated, differentiable capability:

Structural inference — jointly infer $(W_0\text{ or }\gamma, r_c, M, r_a, \Omega)$ from projected density + kinematics by HMC, with calibrated uncertainties.
Anisotropy-vs-rotation degeneracy — two distinct anisotropy routes (exact-OM stretch vs suppressed self-consistent Michie) + rotation transforms: can projected data distinguish radial anisotropy from rotation?
Primordial vs dynamical substructure — generate clumpy/smooth ICs, quantify with the validated Q / azimuthal metrics, evolve in gravax: how fast does measurable substructure erase, and does mass segregation build dynamically vs primordially?
Binary imprint on the IMF — with the Moe & Di Stefano engine + binary-aware likelihood: how biased is a single-star IMF fit with unresolved binaries, and at what $N$ is it “confidently wrong”?
Tidal-field coupling (once $r_t$ is differentiable — see roadmap) — can a population of cluster limiting radii constrain the Galactic potential?

6. Improvement recommendations¶

¶	Recommendation	Why / payoff
R1	Differentiable King $r_t$ (Approach B / implicit function theorem)	unlocks tidal-field inference (science Q5); plan already written, ~15–20 LOC + a 6th gradient panel.
R2	Figure-validate IMF / binary / analytical (61 tests already pass)	closes the three ⚠️ released gaps; `scripts/validate_imfs.py` already exists — needs pub-figure rewrite + measured tables + pages.
R3 ✅	Tidal-truncation validation tier — DONE 2026-06-09	`tests/validation/test_tidal_physics.py` validates `jacobi_radius` / `apply_tidal_truncation` against the restricted-3-body L1 point + analytic Plummer; 5 figures.
R4	`q_approx`: extend calibration above $Q\sim0.85$ or hard-document scope; consider a soft-rank neighbour surrogate	removes the concentrated-regime over-read and smooths the gradient cell-boundary noise.
R5	EFF sharp-truncation: optional virial rescale or more prominent guidance	lets users get a virial $\gamma=3$ IC without silently inheriting the ~5–8% sub-virial offset.
R6	End-to-end inference validation (SBC-style recovery on mocks through gravax/fluxax)	this is the real outstanding gap: gradients are verified at the IC level, not against data.
R7	Retrofit a Measured column onto the older figured pages (two-component; mass-segregation done 2026-06-09)	consistency with the new pages’ tolerance/measured convention.

7. Remaining modules to validate + plot¶

Released, tested, missing only figures/pages (do these next):

✅ imf-statistics — DONE 2026-06-08: recovered slopes (sample MLE), PDF/CDF coverage + KS, mean_mass() vs 200k log-grid ref (with the linear-grid failure mode), and inference-layer gradient validation (5 figures, scripts/validate_imfs.py).
✅ binary-imf — DONE 2026-06-08: Kepler/orbit oracles, Moe+17 q-distribution (mass-dependent twin excess), the regenerated “confidently wrong” recovery (naive single-star fit 18σ-wrong at N=10⁵ vs binary-aware 0.4σ, via fast MLE — no MCMC), bias-vs-f_b mechanism, and grad validation (5 figures, scripts/validate_binaries.py).
✅ analytical-test-cases — DONE 2026-06-09: two-body Kepler (E=−Gm₁m₂/2a, closes), Chenciner–Montgomery figure-eight (L=0, closes), Kepler III across the eight planets + IAU mass ratios, harmonic oscillator, and an adversarial spun-figure-eight (5 figures, scripts/validate_analytical.py).
✅ tidal-truncation — DONE 2026-06-09 (harden-then-validate): new tests/validation/test_tidal_physics.py (9 tests vs the restricted-3-body L1 Lagrange point, the tidal force balance, the (3/2)^(1/3) Keplerian-vs-isothermal factor, and analytic Plummer enclosed mass) + 5 figures (scripts/validate_tidal.py).

8. Incomplete — harden first, then validate + plot¶

These are not merely unplotted; they need implementation/hardening before a validation tier is meaningful:

gravoturb (experimental, repo-only) → harden to release quality, then validate turbulent ICs + the Küpper $\sigma_\Sigma$ – $D$ slope (which the released azimuthal test deliberately does not claim, pending fractal- $D$ models).
Differentiable King $r_t$ → implement (R1), then add the gradient panel.
Unified differentiable lowered-model family (Wilson/Woolley/multi-mass) → implement per the roadmap, then validate against the original models.
Self-consistent rotating equilibria → implement (current rotation is a streaming transform on a non-rotating equilibrium), then validate.

9. Changelog¶

Date	Milestone
2026-06-08	Plummer, King (retrofit), EFF, Michie, rotation+OM-anisotropy, substructure/CW04 Q, azimuthal-variation — all to publication standard (5 figures each; +measured tables). Shared `scripts/_plotstyle.py`; LIMEPY reframed across 13 pages + lowered-model-family roadmap; `validate_profiles.py` retired. Released-core 830 → 866 (+36 validation tests). This report authored.
(prior)	Λ_MSR + energy-sorted segregation, two-component clusters, IMF + Moe binary engine, tidal machinery — implemented and tested.