Substructure (CW04 Q) validation - progenax — differentiable initial conditions for N-body simulations

The Cartwright & Whitworth (2004) Q = m̄/s̄ parameter separates centrally concentrated clusters ( $Q>0.8$ ) from fractal/substructured ones ( $Q<0.8$ ). Test file: tests/validation/test_substructure_q_physics.py (see the test dashboard for the live per-suite count); figures: scripts/validate_substructure_q.py.

What is verified¶

Each row maps to assertions in test_substructure_q_physics.py. Measured values are regenerated by scripts/validate_substructure_q.py. The exact estimator uses the CW04 area convention $A=\pi R^2$ (convex-hull area biases $Q$ high by $\sim0.1$ ).

Property	Tolerance (as tested)	Measured	Anchor
Uniform sphere $Q$ (CW04 ‘3D0’)	$0.75<Q<0.85$	0.78	Cartwright & Whitworth (2004) Table 1 (0.79)
$r^{-1}$ profile (3D1)	$Q>$ uniform	0.82	CW04 Table 1 (0.84)
$r^{-2}$ profile (3D2)	concentrated, $Q>0.8$	0.92	CW04 Table 1 (0.93)
Substructure lowers $Q$ (clumpy)	$Q<0.80$	0.42 (clumpy), 0.25 (clumpier)	fractal/clumpy regime
Regime ordering	clumpy $<$ uniform $<$ concentrated	$0.25<0.42<0.78<0.82<0.92$	monotone $Q$
$Q$ is $N$ -independent ( $N>100$ )	spread $<25\%$	$0.9\%$ (uniform), $10\%$ (clumpy)	CW04 §3
Differentiable $q_{\rm approx}$ vs exact (substructure regime $Q\le0.85$ )	$<0.06$	clumpy +0.004, uniform +0.045	kNN approximation
$q_{\rm approx}$ ordering preserved (all regimes)	monotone	clumpy $<$ uniform $<$ concentrated	monotone surrogate
$q_{\rm approx}$ differentiability	finite, correct sign, median AD $=$ FD few %	median rel $0.9\%$ (all $\partial q/\partial p>0$ )	kNN soft surrogate

Figures¶

Generated by scripts/validate_substructure_q.py (PASS/FAIL per panel).

The CW04 (\bar s,\bar m) plane. The Q=0.8 line divides centrally
concentrated (r^{-1}, r^{-2}; above) from substructured (clumpy, clumpier;
below) configurations, with the uniform sphere on the divide (Q=0.78). This is
the diagnostic’s defining diagram. — Figure 1:**The CW04 $(\bar s,\bar m)$ plane.** The $Q=0.8$ line divides centrally concentrated ( $r^{-1}$ , $r^{-2}$ ; above) from substructured (clumpy, clumpier; below) configurations, with the uniform sphere on the divide ( $Q=0.78$ ). This is the diagnostic’s defining diagram.

Q ladder vs CW04 Table 1. The exact estimator reproduces CW04 ‘3D0/3D1/3D2’
(stars: 0.79/0.84/0.93; measured 0.78/0.82/0.92) and places clumpy/clumpier
well below the Q=0.8 divide. — Figure 2:**Q ladder vs CW04 Table 1.** The exact estimator reproduces CW04 ‘3D0/3D1/3D2’ (stars: $0.79/0.84/0.93$ ; measured $0.78/0.82/0.92$ ) and places clumpy/clumpier well below the $Q=0.8$ divide.

Differentiable q_{\rm approx} vs exact Q. In the substructure regime
(Q\le0.85, shaded) — where q_{\rm approx} is used for inference — the JAX kNN
estimator tracks the exact scipy MST Q on the 1:1 line (clumpy +0.004, uniform
+0.045). For concentrated profiles (Q>0.85) it over-reads by \sim0.1 (shown
honestly); ordering is preserved throughout. — Figure 3:**Differentiable $q_{\rm approx}$ vs exact $Q$ .** In the substructure regime ( $Q\le0.85$ , shaded) — where $q_{\rm approx}$ is used for inference — the JAX kNN estimator tracks the exact scipy MST $Q$ on the 1:1 line (clumpy +0.004, uniform +0.045). For concentrated profiles ( $Q>0.85$ ) it over-reads by $\sim0.1$ (shown honestly); ordering is preserved throughout.

N-independence. Q is approximately independent of N for N>100 (CW04):
the uniform sphere varies by 0.9\%, the clumpy distribution by 10\% across
N=150–2400. — Figure 4: **$N$ -independence.** $Q$ is approximately independent of $N$ for $N>100$ (CW04): the uniform sphere varies by $0.9\%$ , the clumpy distribution by $10\%$ across $N=150$ –2400.

Differentiability¶

q_{\rm approx} gradient. \partial q_{\rm approx}/\partial p for a radial
concentration parameter p (r=u^p): autodiff and finite difference agree to a
median 0.9\%. As a kNN estimator, the gradient is exact within a neighbour-set
cell and the finite difference picks up cell-boundary truncation (worst-case
\sim20\% at a boundary); the autodiff gradient is finite and correct-signed
everywhere — usable for gradient-based substructure inference (the reason
q_approx exists alongside the non-differentiable exact compute_q_parameter). — Figure 5: **$q_{\rm approx}$ gradient.** $\partial q_{\rm approx}/\partial p$ for a radial concentration parameter $p$ ( $r=u^p$ ): autodiff and finite difference agree to a median $0.9\%$ . As a **kNN estimator**, the gradient is exact within a neighbour-set cell and the finite difference picks up cell-boundary truncation (worst-case $\sim20\%$ at a boundary); the autodiff gradient is finite and correct-signed everywhere — usable for gradient-based substructure inference (the reason `q_approx` exists alongside the non-differentiable exact `compute_q_parameter`).

Azimuthal-variation diagnostic (Küpper+2011)¶

compute_azimuthal_variation returns $\sigma_\Sigma/\langle\Sigma\rangle$ , the relative scatter of star counts in azimuthal sectors — a cheap $O(N)$ substructure proxy. Test file: tests/validation/test_azimuthal_variation_physics.py (see the test dashboard for the live per-suite count); figures: scripts/validate_azimuthal_variation.py.

Property	Tolerance (as tested)	Measured	Anchor
Smooth axisymmetric → Poisson floor	$< 3\sqrt{n_b/N}$	0.059 (floor 0.069)	Poisson counts
Monotonic rise with clump fraction	monotone	$0.06\to1.25$	more clumps ⇒ larger $\sigma_\Sigma$
Spans the Küpper range	smooth $<0.2$ , clumpy $>0.5$	0.06 / 0.9	Küpper et al. (2011) $[0.07,0.76]$
Anti-correlates with CW04 $Q$	$\rho < -0.5$	$\rho = -0.83$	both detect substructure

Sector-count histograms. A smooth axisymmetric cluster (left) has nearly flat
azimuthal counts (\sigma_\Sigma/\langle\Sigma\rangle=0.06, the Poisson floor);
azimuthal clumps (right) give spiky sectors (\sigma_\Sigma/\langle\Sigma\rangle=1.04). — Figure 6:**Sector-count histograms.** A smooth axisymmetric cluster (left) has nearly flat azimuthal counts ( $\sigma_\Sigma/\langle\Sigma\rangle=0.06$ , the Poisson floor); azimuthal clumps (right) give spiky sectors ( $\sigma_\Sigma/\langle\Sigma\rangle=1.04$ ).

Response to clumpiness. \sigma_\Sigma/\langle\Sigma\rangle rises monotonically
from the Poisson floor through the range ([0.07,0.76], shaded)
as the azimuthal clump fraction increases. The fractal generator was removed, so the
specific \sigma_\Sigma\!-\!D slope is not reproduced here — only the floor, span, and
monotonic behaviour. — Figure 7:**Response to clumpiness.** $\sigma_\Sigma/\langle\Sigma\rangle$ rises monotonically from the Poisson floor through the Küpper *et al.* (2011) range ( $[0.07,0.76]$ , shaded) as the azimuthal clump fraction increases. The fractal *generator* was removed, so the specific $\sigma_\Sigma\!-\!D$ slope is not reproduced here — only the floor, span, and monotonic behaviour.

Cross-check vs CW04 Q. Across a smooth→clumpy sequence the two independent
substructure diagnostics anti-correlate (\rho=-0.83): higher
\sigma_\Sigma/\langle\Sigma\rangle tracks lower Q. — Figure 8:**Cross-check vs CW04 $Q$ .** Across a smooth→clumpy sequence the two independent substructure diagnostics anti-correlate ( $\rho=-0.83$ ): higher $\sigma_\Sigma/\langle\Sigma\rangle$ tracks lower $Q$ .

How to run¶

pytest tests/validation/test_substructure_q_physics.py -q          # ~5 s
pytest tests/validation/test_azimuthal_variation_physics.py -q     # ~1 s
python scripts/validate_substructure_q.py                          # 5 Q figures
python scripts/validate_azimuthal_variation.py                     # 3 azimuthal figures

What this suite does not test¶

Fractal IC generation — removed in the 2026-06 rewrite (no released successor); the turbulent-density method is experimental gravoturb (src/experimental/, AC6/AC7 — see its VALIDATION_SUMMARY.md).
The Küpper $\sigma_\Sigma\!-\!D$ slope — requires fractal-dimension models (generator removed); only the metric’s floor/span/monotonicity is validated.

References¶

The Q diagnostic is Cartwright & Whitworth (2004); the GW04 fractal baseline is Goodwin & Whitworth (2004); the dynamical context is Allison et al. (2009). Theory at Fractal substructure and JAX-native CW04 substructure Q parameter.

References¶

Cartwright, A., & Whitworth, A. P. (2004). The statistical analysis of star clusters. Monthly Notices of the Royal Astronomical Society, 348, 589–598. 10.1111/j.1365-2966.2004.07360.x
Küpper, A. H. W., Maschberger, T., Kroupa, P., & Baumgardt, H. (2011). Mass segregation and fractal substructure in young massive clusters. Monthly Notices of the Royal Astronomical Society, 417, 2300–2317. 10.1111/j.1365-2966.2011.19412.x
Goodwin, S. P., & Whitworth, A. P. (2004). The dynamical evolution of fractal star clusters: The survival of substructure. Astronomy and Astrophysics, 413, 929–937. 10.1051/0004-6361:20031529
Allison, R. J., Goodwin, S. P., Parker, R. J., Portegies Zwart, S. F., de Grijs, R., & Kouwenhoven, M. B. N. (2009). Using the minimum spanning tree to trace mass segregation. Monthly Notices of the Royal Astronomical Society, 395, 1449–1454. 10.1111/j.1365-2966.2009.14508.x