Mass segregation validation

1. $\Lambda_{\mathrm{MSR}}$ diagnostic — validated against analytic ground truth¶

progenax.diagnostics.compute_lambda_msr (Allison et al. 2009; $\Lambda_{\mathrm{MSR}}=\langle\ell_{\mathrm{random}}\rangle/\ell_{\mathrm{massive}}$, definition verified against the held ApJ 700 L99 PDF — see Allison et al. (2009)). Tests in tests/validation/test_mass_segregation_physics.py:

The three regimes below are rearrangements of one shared, realistic baseline cluster (a centrally concentrated Plummer sphere): only the massive-star positions change, so the panels are an honest like-for-like comparison, the massive stars are spatially resolved, and the segregated $\Lambda$ sits in the observed range (Allison et al. 2009 measure $\Lambda \sim$ a few for segregated young clusters such as the ONC). The unbounded behaviour of $\Lambda$ — useful as a mathematical limit but not a physical configuration — is checked separately (the delta-core row), so it never sets the headline number.

2. Primordial (energy-ordered) generator — physics suite + end-to-end check¶

progenax.cluster.mass_segregation.energy_sorted_segregation is the PRIMORDIAL segregation generator: it assigns the most massive stars to the most bound orbits of an equilibrium pool, in the spirit of Baumgardt et al. (2008) and McLuster (Küpper et al. 2011). One deliberate, documented departure from the published recipe: Baumgardt+2008 draw orbits randomly within cumulative-mass bins; progenax uses a deterministic isotonic rounding of the bin-centre targets, which guarantees a distinct orbit per star for ANY mass spectrum — the random per-bin sampler collapsed below one orbit per bin for steep IMFs, producing coincident stars and $V = -\infty$ (the orbit-reuse bug, fixed in this arc and regression-locked below). Realisation variety comes from re-drawing the random pool.

Physics suite tests/validation/test_segregation_equilibrium_physics.py (ported in the 2026-06 redesign from the retired blend’s suite onto the protocol-API composition; see the test dashboard for the live count):

End-to-end via scripts/validate_cluster_ic.py §3 (re-run 2026-06-10, PASS): applying the generator to an unsegregated Kroupa Plummer pool drives the independently validated $\Lambda_{\mathrm{MSR}}$ from $1.40 \to 4.74$ and yields Spearman $\rho(m,E) = -0.92$ (massive stars in the most-bound orbits). (Earlier versions of this page quoted $1.40 \to 14.3$, $\rho = -0.84$ — numbers from before the orbit-reuse fix changed the deterministic assignment.)

```{admonition} The lambda_seg blend is retired — the equilibrium route is Engine A :class: important The historical partial-segregation knob lambda_seg (a linear phase-space blend between the unsegregated and fully-ordered catalogs) was retired in the 2026-06 unified redesign: its intermediate states drift from per-mass-group virial balance (drift peaking $\sim 0.05$ at $\lambda = 0.5$ in the Phase-0 error-budget study). The differentiable segregation knob is now MultiComponentCluster.from_mass_segregation(delta) — the multi-mass lowered-isothermal family with $w_j = \mu_j^{-\delta}$, a true shared-potential equilibrium at every $\delta$ (theory $Q_j = 0.5$ exactly; segregation ratio $1.0 \to 2.6$ over $\delta \in [0, 0.6]$). Validation: Multi-mass LIMEPY equilibrium (Engine A). energy_sorted_segregation survives as the labeled primordial (non-equilibrium, non-differentiable) generator above.

## 3. Mass-weighted substructure metric (experimental, repo-only)

`gravoturb.diagnostics.mass_density` — Maschberger & Clarke (2011) local surface density
$\Sigma=(k-1)/(\pi r_k^2)$, $k=6$ (Eq. 4, verified vs the held PDF —
[](../99-bibliography/per-paper/maschberger-clarke-2011.md)) + the $m$–$\Sigma$ plane. Robust to
substructure (a *local* density), unlike CW04 $\mathcal{Q}$ on small massive subsets. Validated in
`tests/experimental/unit/test_mass_density.py` (6 tests: exact Eq. 4 formula; uniform-density
recovery; dense $>$ sparse; random masses $\to \rho\approx 0$; **primordial correlation detected**
$\rho(m,\Sigma)>0.5$ when massive stars are placed in dense clumps via
`gravoturb.realization.mass_assignment.correlated_mass_assignment`; reproducible).

## 4. Differentiability status — generator parameter vs. observable

There are **two distinct** "differentiable segregation" questions, and conflating them is
easy:

```{list-table}
:header-rows: 1

* - Quantity
  - Kind
  - Differentiable?
  - Use
* - Equipartition $\delta$ (`MultiComponentCluster.from_mass_segregation`)
  - forward-model **parameter**
  - **Yes** (AD $=$ FD through the table-backed solve, $2.15\times10^{-4}$)
  - dial *equilibrium* segregation strength into a model;
    $\partial(\text{model})/\partial\delta$ — see [](multimass-equilibrium.md)
* - Generator `energy_sorted_segregation` (primordial)
  - forward-model generator
  - No (`argsort`, floor — discrete assignment, by design)
  - labeled primordial route (§2); use $\delta$ for gradients
* - `compute_lambda_msr` (Allison)
  - **observable** (measurement)
  - No (SciPy MST + `argsort`)
  - validated diagnostic / oracle
* - $m$–$\Sigma$ metric, correlated placement
  - **observable**
  - No (kNN / ranking)
  - experimental diagnostic
* - **`segregation_approx`** (§5)
  - **observable** (measurement)
  - **Yes** ($\partial/\partial\mathbf{x}$, $\partial/\partial m_{\mathrm{cut}}$)
  - gradient-based / HMC inference *from* positions+masses

The forward-model parameter route is differentiable: the equipartition $\delta$ flows through the coupled multi-mass Poisson solve and the jax.lax.scan sampler (Multi-mass LIMEPY equilibrium (Engine A)). (The retired lambda_seg blend was also differentiable — what it lacked was per-group equilibrium, which is why $\delta$ replaced it.) What was missing on the measurement side was a differentiable observable: a function $f(\text{positions},\text{masses})\to\mathbb{R}$ whose gradient is usable, so segregation can enter a likelihood / HMC without a samplable forward model. That gap is now closed by §5 — mirroring how the differentiable q_approx surrogate closed the same gap for CW04 $\mathcal{Q}$ substructure geometry.

5. Differentiable segregation observables (released core)¶

progenax.diagnostics.segregation_approx provides three differentiable observables that mimic the segregation estimators observers actually report, all sharing one soft mass-cut kernel $w_i=\sigma\!\big((m_i-m_{\mathrm{cut}})/\tau\big)$ (the observer’s “massive bin”; $\tau\to0$ recovers the hard indicator $\mathbb{1}[m_i>m_{\mathrm{cut}}]$):

• soft $\Lambda_{\mathrm{MSR}}$ (lambda_msr_approx): the Allison+2009 MST ratio with a softmin nearest-neighbour distance ($(N{-}1)\langle d_{\mathrm{1NN}}\rangle$ MST proxy, scale-relative temperature $\beta$) and a closed-form random baseline (no Monte-Carlo).

• radial concentration (radial_concentration_approx): $C=\langle r\rangle_{\mathrm{massive}}/\langle r\rangle_{\mathrm{all}}$ about the mass-weighted centroid ($C<1$ segregated).

• soft $\Sigma$–$m$ (sigma_m_approx): $S=\mathrm{corr}_i(w_i,\log\Sigma_i)$ with the Maschberger–Clarke $k$-NN density; the $k$-NN radius uses jnp.sort (the exact order statistic, which has a defined JVP — unlike argsort), so it is differentiable with no radius softening.

All default to 2D-projected positions (observer-faithful), with a 3D flag. Validated in tests/validation/test_segregation_approx_physics.py + tests/unit/diagnostics/ (see the test dashboard for live per-suite counts).

The five figures below are read in order: first that the surrogates are faithful (§5.1), then that they are physically meaningful segregation proxies (§5.2), then which one is best to infer with (§5.3–5.4), and finally that they are usable in gradient-based inference (§5.5). Each figure is followed by its physical interpretation and a correctness verdict.

5.1 Hard-limit convergence — are the surrogates faithful?¶

Interpretation. A differentiable surrogate is only trustworthy if it reduces to the estimator it claims to approximate when the smoothing is removed. That is exactly what this figure tests, and the three curves behave differently for a physically meaningful reason:

• $\Sigma$–$m$ (green, flat at $\sim10^{-11}$). Its only approximation is the soft mass weight — the $k$-NN radius is computed with jnp.sort, the exact order statistic. With the bimodal test masses (0.5 and $10\,M_\odot$) the soft cut at $2\,M_\odot$ is already effectively hard even at $\tau=0.5$ ($\sigma((0.5-2)/0.5)\approx0.05$, $\sigma((10-2)/0.5)\approx1$), so $S$ sits at machine precision throughout. Correct, and exactly the expected behaviour.

• Radial $C$ (orange, converges with $\tau$). Although each weight is near-binary, $C$ aggregates the small residual weight over the $\sim$380 low-mass stars ($380\times0.05\approx19$, comparable to the 20 massive ones), which shifts the mass-weighted centroid and mean radius. As $\tau\to0$ that residual vanishes and $C\to$ the hard ratio at $\sim10^{-11}$. The convergence is real and physical — it measures the aggregate leakage of sub-threshold stars.

• Soft $\Lambda$ (blue, slowest). It carries a second smoothing — the softmin nearest-neighbour distance, controlled by $\beta$. softmin only becomes the true min as $\beta\to0$, and it does so at a finite rate, so $\Lambda$ floors at $5.6\times10^{-3}$ of the hard 1-NN ratio (its intrinsic approximation, not an error). The monotone descent confirms both knobs are mutually consistent.

Verdict: physically correct. The hierarchy (exact $\to$ aggregation-limited $\to$ softmin-limited) is precisely what the construction predicts. In practice one calibrates at a small but finite $\tau,\beta$ (as calibrate_segregation_approx does), trading a known $\lesssim1\%$ bias for clean gradients.

5.2 Response to segregation — are they meaningful proxies?¶

Interpretation. As the massive stars are concentrated into a tighter core, each observable responds in the physically correct direction: the massive population sits at smaller radii ($C\downarrow$, plotted inverted so it rises), in locally denser regions ($S\uparrow$), and with shorter nearest-neighbour spacing ($\Lambda\uparrow$). All three are therefore genuine monotonic segregation proxies, and — critically — they are rank-consistent with the field-standard Allison+2009 $\Lambda_{\mathrm{MSR}}$ ($\rho=0.98$), which is the validated oracle (§1). The radial measure (orange) is the steepest and smoothest; the local NN-based $\Lambda$ (blue) is the noisiest and shallowest — a direct preview of the Fisher ranking in §5.3. (The axis is normalised per-curve, so this panel compares shape and monotonicity, not absolute scale.)

Verdict: physically correct and meaningful. Every curve moves the right way and tracks the published diagnostic; the differences in steepness are real information content, not artefacts.

5.3 Fisher information — which observable should you infer with?¶

Interpretation. Fisher information is the inverse of the smallest achievable posterior variance: $\mathrm{Var}(\hat\theta)\gtrsim1/\mathcal{I}$ (Cramér–Rao). A higher bar means a tighter, more confident inference of the segregation strength from that one number. The ranking has a clean physical reading:

• Radial concentration wins ($\sim5\times$) because it is a global statistic — every star’s radius shifts when the core tightens, so the signal $\mathrm{d}\mu/\mathrm{d}\theta$ is large while the realisation-to-realisation variance stays small.

• $\Lambda_{\mathrm{MSR}}$ and $\Sigma$–$m$ are local measures (nearest-neighbour spacing, $k$-NN density). They respond only through the immediate neighbourhood of the massive stars, which is both a weaker average signal and a noisier one (small-$N$ local estimates) — hence $\sim5\times$ less Fisher information.

This is the concrete, quantitative payoff of differentiability: the gradient $\mathrm{d}\mu/\mathrm{d}\theta$ is computed exactly by autodiff rather than by noisy finite differencing, so the identifiability comparison is itself trustworthy.

Verdict: meaningful, with stated scope. The ranking is evaluated at a single operating point ($\theta=0.3$, $N=400$, bimodal masses); the ordering (global $\gg$ local) is robust and physically expected, but the absolute factors are configuration-dependent and should be re-measured for a specific survey/cluster regime before being quoted as a forecast.

5.4 Projection: 2D vs 3D — how much segregation signal survives?¶

Interpretation. Observers never see 3D positions — they see the cluster projected on the sky. This figure quantifies, per observable, how much segregation information that projection destroys, using the 2D/3D Fisher ratio as “fraction of signal surviving”:

• Radial concentration: $\sim50\%$ survives. Projecting a 3D radius $r$ to a sky radius $R\le r$ partially scrambles the radial ordering (foreground/background stars land at small projected radius), so the centrally-concentrated-massive-star signal is diluted by about half. Physically sensible.

• $\Sigma$–$m$: only $\sim32\%$ survives — the largest loss. A 3D local density becomes a 2D surface density, and line-of-sight superposition adds uncorrelated neighbours to each star’s $k$-NN ball, injecting noise into the very quantity the correlation depends on. Reasonable that it degrades most.

• Soft $\Lambda_{\mathrm{MSR}}$: ratio $\approx1.2$ — essentially projection-insensitive. The nearest-neighbour ratio is dimensionless and local; 2D NN distances still separate a dense core from a diffuse halo about as well as 3D ones.

Verdict: physically correct. Global, line-of-sight-sensitive measures lose the most ($\Sigma$–$m$ $>$ radial); the dimensionless local ratio loses least. This is the expected ordering, and it directly motivates the research question below.

5.5 Differentiability and end-to-end recovery¶

Interpretation. Panel (a) is the differentiability certificate: autodiff and finite-difference gradients coincide to machine precision, so the observables can be dropped into a gradient-based optimiser or HMC sampler with confidence. The shapes are themselves informative — $\partial C/\partial m_{\mathrm{cut}}$ and $\partial\Lambda/\partial m_{\mathrm{cut}}$ peak near 3–$4\,M_\odot$, because that is where the mass cut sweeps through the bimodal mass gap and the “massive bin” membership changes fastest, while $\partial S/\partial m_{\mathrm{cut}}\approx0$ (flat green) because the $\Sigma$–$m$ correlation is insensitive to exactly where the cut falls inside the gap. Panel (b) closes the loop: descending on a single differentiable observable recovers the generating segregation strength exactly — the minimal end-to-end demonstration that gradient-based inference of segregation now works (the observable-side complement of the differentiable forward-model knob $\delta$ in MultiComponentCluster.from_mass_segregation).

Verdict: correct. Gradients are exact; inference through the observable succeeds.

5.6 Reading the results — recommendations¶

Run it yourself¶

# (1) Λ_MSR diagnostic — analytic validation tests (released core)
pytest tests/validation/test_mass_segregation_physics.py -v

# (2) differentiable observables — validation + unit tests, and the 5 figures
pytest tests/validation/test_segregation_approx_physics.py tests/unit/diagnostics/ -v
python scripts/validate_segregation_approx.py     # -> seg_*.png (5 figures, PASS/FAIL)

# (3) primordial generator — physics suite, Λ_MSR diagnostic figures, end-to-end
pytest tests/validation/test_segregation_equilibrium_physics.py -v
python scripts/validate_mass_segregation.py     # -> lambda_msr_*.png
python scripts/validate_cluster_ic.py           # -> cluster_ic_*.png
#   figures land in validation/plots/; the curated copies embedded here live in
#   docs/website/50-validation/figures/

# (4) the EQUILIBRIUM mass-segregation route (Engine A, differentiable delta)
#     -> see multimass-equilibrium.md
python scripts/validate_multimass_equilibrium.py

# (5) experimental mass-weighted Σ metric + density-correlated placement
PYTHONPATH=src:src/experimental pytest tests/experimental/unit/test_mass_density.py tests/experimental/unit/test_masses.py -v

References¶

Theory at Mass segregation. Diagnostic Allison et al. (2009); energy-ordered generator Baumgardt et al. (2008); mass-weighted local-$\Sigma$ Maschberger & Clarke (2011). The differentiable-observables design is recorded in the repository’s planning notes.