The scientific throughline

Every demo in this section is the same inverse problem: given a present-day snapshot of a star cluster, which birth and structural parameters can you actually recover, and what are the fundamental limits? That is the inverse of what progenax does in the forward direction (parameters → initial conditions), run through one physics-direct differentiable inference engine: forward model → analytic predicted statistic → Gaussian/Poisson likelihood → MLE + Fisher information.

The deeper point is not “we can recover $X$.” It is that a differentiable forward model turns inference into information geometry. Because every initial condition is jax.grad-able, the Fisher matrix is computable, and it tells you — before you observe — what is measurable, what is degenerate, and what will bias you. And the recurring discovery across the demos is that the astrophysics lives in the degeneracies and systematics, not the point estimates. Most of the headline results are not “we recovered $X$” but “$X$ is degenerate with $Y$,” or “$X$ is biased if you assume the wrong $Z$.”

The demos sort onto three axes.

The shared method: physics-direct differentiable inference¶

The demos share one inference layer (scripts/_demo_inference.py). The design avoids both expensive likelihood-free / simulation-based inference and any non-differentiable resampling inside the loss. The recipe:

1. Forward-sample mock stars at truth. Draw $N$ positions+velocities from a progenax model (here $N = 3\times10^4$ to 10⁵). This is done once, to make the data; it is not repeated inside the optimizer.

2. Compress to binned kinematic summaries. Reduce the stars to per-group (or per-component) binned 1-D velocity dispersions $\hat\sigma_{1\mathrm D,j}(r)$ on a frozen set of radial bins, each carrying a finite-$N$ standard error. For an isotropic pooled estimator over the three velocity components,

   $$\hat\sigma_{1\mathrm D}^2 = \frac{1}{3}\,\overline{|\mathbf v|^2} = \frac{\sigma_r^2 + \sigma_t^2}{3}, \qquad \mathrm{SE}(\hat\sigma) = \frac{\hat\sigma}{\sqrt{6 n}},$$

   (1)

   where the $\sqrt{6n}$ (not $\sqrt{2n}$) comes from pooling all three one-D components into a $\chi^2_{3n}$ inner sum. This compression is what makes the cost $N$-independent: the likelihood sees a few hundred binned numbers, never the 10⁵ stars.

3. Likelihood against analytic predictions. Compare the binned data to the model’s analytic expectation with a Gaussian (χ²) likelihood,

   $$\ln \mathcal L(\theta) = -\tfrac12 \sum_{j,k} w_{jk} \left(\frac{\hat\sigma_{jk} - \sigma_{jk}^{\rm pred}(\theta)}{\mathrm{SE}_{jk}}\right)^2 ,$$

   (2)

   with $w_{jk}\in\{0,1\}$ masking under-occupied bins. Gradients flow only through the analytic prediction $\sigma^{\rm pred}(\theta)$ — the data side is a fixed constant — so $\nabla_\theta \ln\mathcal L$ is exact and cheap, and there is no reparametrization-gradient noise from resampling.

4. Optimize and quantify. Adam (mle_adam, a fixed-step lax.scan — deterministic and differentiable-friendly) from several dispersed starts; the Fisher information from the Gauss-Newton $J^{\mathsf T}J$ of the residual vector (fisher_information_gn via jax.jacrev, which is robust through the ODE/quadrature custom_vjps, unlike a full Hessian); and, where affordable, full posterior sampling with a vendored blackjax NUTS.

Measurable — clean recovery, then the catch¶

These establish that the machinery recovers truth, and then find what the point estimate hides.

Degenerate — the Fisher matrix is rank-deficient¶

The strongest results: the inverse problem is fundamentally (not statistically) ill-posed, and the differentiable Fisher exposes it exactly.

Birth-environment archaeology (B5) is the deepest. The environment-dependent IMF Marks et al., 2012Jeřábková et al., 2018 maps $(\,[\mathrm{Fe/H}],\,\log M_{\rm ecl},\,\mathrm{sfe})\to\alpha_3$, the high-mass slope. $\alpha_3$ is trivially measurable (a $\sim$40$\sigma$ top-heavy detection at $N=10^4$), but the map is three-to-one, so the environment-space Fisher is exactly rank 1 — two machine-precision-zero eigenvalues. A continuum of birth environments produces the identical mass function. You cannot read a cluster’s birth metallicity, mass, or star-formation efficiency off its IMF slope without an external constraint on two of the three. This is a caution for any claim that infers birth conditions from a present-day mass function.

Rotation + projection (B8) is the same rank-deficiency from geometry: a cluster rotating at rate $\omega$ viewed at inclination $i$ has a mean line-of-sight velocity $\propto \omega\sin i$, so $\langle v_{\rm los}\rangle$ constrains only the product — the $(\omega, i)$ Fisher is rank 1. Cluster rotation amplitudes from line-of-sight velocities are lower limits, degenerate with inclination; breaking it needs the projected flattening, which velocities cannot supply.

Biased — wrong-physics systematics¶

Unresolved-binary mass function (B4) is the headline. Unresolved binaries blend photometrically (their summed light inflates the inferred mass), distorting the observed mass function. A controlled experiment — shuffle the mass ratio $q$ against the period $P$ to preserve both marginals and change only the correlation — isolates the Moe & Di Stefano Moe & Di Stefano, 2017 $P$–$q$ coupling. Ignoring it biases the binary fraction low by $\sim$5% ($3.6\sigma$ at survey scale), because the blended (close) subset is more equal-mass under the real coupling than an independent model assumes. It is a systematic — fixed fractional bias whose significance grows as $\sqrt N$. Binary-fraction measurements from the photometric binary sequence that assume an independent $q$-distribution are biased; you need the realistic $P$–$q$–$e$ covariance.

The wrong-IMF curve (B2d) and the OM-vs-Michie misfit (B6) are the same lesson in other channels: assume the wrong generative physics and the recovered parameter shifts in a quantifiable, detectable way.

Honest diagnostics — what the observables actually carry¶

• Binary energy budget (B9): the internal binary binding energy is a separate reservoir that dwarfs the cluster potential (12–1900×) and is environment-dependent — the seed of binary-burning / core-collapse arrest. The naive resolved virial ratio is contaminated by orbital phase and is not the cluster’s $Q$; virialize binary ICs on the system centres-of-mass.

• Differentiable diagnostics (B10): the JAX surrogates for the CW04 substructure $Q$ Cartwright & Whitworth, 2004 and the Allison mass-segregation ratio Allison et al., 2009 track their non-differentiable MST counterparts — so “substructure” and “segregation” can enter a gradient-based loss they never could before.

• Tidal radius (B7): $93\%$ of the truncation-radius information lives in the few outermost (count-limited) bins, so the Poisson treatment is essential — and a cluster’s orbit (Galactocentric distance, via the Jacobi radius King, 1962) is encoded in its faint outer star counts.

• Cross-engine agreement (B1): two independent engines build the same cluster — the consistency check the recoveries rest on.

What this is, and what it is not¶

The implication¶

The reason to build differentiable cluster initial conditions is not faster point estimation. It is that the Fisher information of a differentiable forward model is a forecasting instrument: for a given observable and sample size, it tells you what a cluster survey can and cannot claim. These demos are a first map of that information content for star-cluster observables — and the most valuable entries on the map are the ones that say “this is degenerate” (birth environment $\not\leftrightarrow$ IMF slope; rotation $\not\leftrightarrow$ inclination) and “this is biased if you assume the wrong physics” (binary fraction under the wrong $P$–$q$ coupling; anisotropy under the wrong DF). Those are cautions an observer can carry to real data.