Gravoturbulent + PP20 validation - progenax — differentiable initial conditions for N-body simulations

Current test + acceptance files:

gravoturb/validation/acceptance.py — AC3/AC4 anchor ζ(p) on PP20 Eq. 6 + analytic + the direct-field estimator; AC1/AC2 anchor the BM19 scalars + mass conservation. These print their numbers (see gravoturb/VALIDATION_SUMMARY.md).
tests/experimental/unit/test_pp20.py — PP20 ζ(p) unit tests (anchors, no-spurious-pole regression, divergence at $p=2$ ).
tests/experimental/unit/test_bm19.py, tests/experimental/unit/test_pdf.py — BM19 scalar + density-PDF/iCDF coverage.
tests/experimental/validation/test_acceptance.py — asserts the AC PASS verdicts so “validated” is backed by fresh output.

What is verified (PP20 ζ(p))¶

Property	Tolerance	Anchor
$\zeta(p=0) = 1$	10^-10 abs	Top-hat reference
$\zeta(p=1) = 2 \cdot 2^{3/2}/3^{3/2}$	10^-12 rel	Exact analytic
$\zeta(p=1.5) = \sqrt{2}$	10^-12 rel	Exact analytic
$\zeta(p=1.67) \approx 1.789$	0.02 abs	Kainulainen et al. (2014) median
PP20 Eq. 6 vs analytic at 10 $p$ values	0.2% rel	2.6 ≈ $3^{3/2}/2$ rounding
Monotonic in $p$ over $[0, 1.9]$	Strict	No spurious singularity at $p = 1.3$
Smooth across $p = 1.3$	$< 1\%$ change	Refutes the bug-induced “domain limit”
Diverges as $p \to 2$	$\zeta(1.95) > 5\,\zeta(1)$	Singular isothermal limit
$P_{\max} = 1.95$ clipping	Identity above $P_{\max}$	Numerical safety
JIT, vmap, grad compatibility	Round-trips	JAX-native
No buggy form in source tree	Source-text scan	Regression trap

These regression anchors are exercised by tests/experimental/unit/test_pp20.py and printed by AC3/AC4; run the commands below for fresh status.

What is verified (BM19 unit coverage)¶

Property	Tolerance	Anchor
$\sigma_s^2 = \ln(1 + b^2 \mathcal{M}^2)$	10^-12 rel	Closed form
Transition density $s_t$ continuity	10^-12 rel at lognormal-power-law match	Analytic matching condition
$f_{\mathrm{dense}}$ at canonical params	1% rel	Numerical integration cross-check
α↔p mapping	$p = 3/\alpha$ exactly	Definition
End-to-end forward chain (1D → 3D realization)	AC6 cornerstone: realized vs BM19 $f_{\mathrm{dense}}$ , ensemble bias $<1\%$	`gravoturb` AC6 (128³); see its `VALIDATION_SUMMARY.md`

Spot ζ values (canonical anchors)¶

$p$	$\zeta(p)$	Comment
0	1.0 (exact)	Top-hat
0.5	1.014	Pre-2026-04-28 buggy formula returned 1.235
1.0	1.0887	Exact analytic anchor
1.5	$\sqrt{2}$ ≈ 1.4142	Exact analytic anchor
1.67	1.789	Kainulainen+14 observational
1.95	8.28	Approaching $P_{\max}$

How to run¶

# Unit tests (repo-only; needs src/experimental on the path)
PYTHONPATH=src:src/experimental pytest tests/experimental/unit/test_pp20.py -v
PYTHONPATH=src:src/experimental pytest tests/experimental/unit/test_bm19.py tests/experimental/unit/test_pdf.py -v

# Acceptance suite — prints the ζ/BM19 anchor numbers (AC1–AC4)
PYTHONPATH=src:src/experimental python -m gravoturb.validation.acceptance

A few seconds for the unit tests; the full AC suite (incl. the 128³ AC6 cornerstone) takes a couple of minutes.

The 2026-04-28 fix¶

The ζ(p) regression suite was added simultaneously with the bug fix on 2026-04-28 (those tests now live in the experimental gravoturb package; see the test dashboard for current counts). Pre-fix, the buggy magnification_factor returned values like $\zeta(0.5) = 1.235$ that all prior tests accepted (because the prior tests anchored on the buggy formula’s output). The fix re-anchored every test on either PP20 Eq. 6, the integral-derived analytic form, or the Kainulainen+14 observation. Full history at PP20 ζ(p) transcription fix.

References¶

Parmentier & Pasquali (2020) for ζ(p); Kainulainen et al. (2014) for the observational anchor; Burkhart (2018)Burkhart & Mocz (2019) for the BM19 framework. Theory at The magnification factor ζ — three ways to compute it and BM19 dense-gas SFR framework.

References¶

Kainulainen, J., Federrath, C., & Henning, T. (2014). Unfolding the laws of star formation: The density distribution of molecular clouds. Science, 344, 183–185. 10.1126/science.1248724
Parmentier, G., & Pasquali, A. (2020). A new parameterization of the star formation rate–dense gas mass relation: Embracing gas density gradients. The Astrophysical Journal, 903, 56. 10.3847/1538-4357/abb8d3
Burkhart, B. (2018). The Star Formation Rate in the Gravoturbulent Interstellar Medium. The Astrophysical Journal, 863, 118. 10.3847/1538-4357/aad002
Burkhart, B., & Mocz, P. (2019). The self-gravitating gas fraction and the critical density for star formation. The Astrophysical Journal, 879, 129. 10.3847/1538-4357/ab25ed