Multi-component Eddington equilibria (Engine B)

Engine B (MultiComponentCluster.from_density_profiles) builds Plummer, EFF, and King density components into ONE shared self-consistent potential (a single cumulative-trapezoid Poisson pass — no ODE), inverts each component’s DF by the generic eddington_invert in that shared $\Psi$ (optionally Osipkov-Merritt per component, $r_{a,j}$), and samples a true joint equilibrium with no external virial rescale. It complements Engine A (DF-defined lowered-isothermal/LIMEPY): A starts from a DF family, B from prescribed densities — and the two engines must agree where they overlap. Test file: tests/validation/test_engine_b_physics.py (6 tests; plus the TestEngineB unit suite in tests/unit/cluster/test_multicomponent.py); figures + PASS/FAIL gate: scripts/validate_multicomponent_eddington.py.

What is verified¶

Each row maps to assertions in test_engine_b_physics.py; the Measured column is the standalone validation-script run (2026-06-10, ALL PASS, 11/11; run twice at close-out with identical tables). The headline mix is a Plummer halo ($r_h=2$ pc, 60% of the mass) + EFF $\gamma=5$ core ($a=0.8$ pc, $r_t=9$ pc, 40%).

The King A-vs-B anchor is the trust anchor: the same physical model ($W_0=5$, $r_c=1$ pc) built by two independent engines — A’s lowered-isothermal DF + coupled Poisson ODE vs B’s prescribed King density + Poisson quadrature + Eddington inversion — agrees in the sampled radial CDF (KS 0.0002) and dispersion profile (max dev 0.0003). The exact-quadrature $Q_j$ oracle alone is necessary, not sufficient ($2T_j + W_j = 0$ holds for any positive $f$ in a consistent $(\Psi, d\Psi/dr)$ pair); the cross-engine and closed-form-DF anchors carry the inversion-correctness burden.

Figure¶

Three physics findings of the arc¶

1. The $E^{7/2}$ oracle needs the untruncated zero point. With the truncated zero point $\Psi = \Phi(r_t) - \Phi$ the energies shift by $c = M/\sqrt{r_t^2+a^2}$ and the power law picks up an $O(3.5c/E) \approx 30\%$ deviation at $E = 0.1\Psi_0$ even at $r_t = 100a$. The law is tested with $\Psi = -\Phi$ (untruncated), and the truncated case is covered by the exact closed form including the boundary term $f(E) = \big[20k(2b^3\sqrt{E} - 2b^2E^{3/2} + \tfrac{6}{5}bE^{5/2}

   • \tfrac{2}{7}E^{7/2}) + 5kc^4/\sqrt{E}\big]/(\sqrt{8},\pi^2)$,$b = E+c$-- a stronger amplitude+shape oracle ($8.86\times10^{-6}$). Zero-point physics is part of the oracle definition.

2. Not every density mix has an equilibrium — and the gate proves it. The plan’s drafted EFF core $a=0.4$ in the Plummer-halo shared potential has a genuinely negative Eddington DF ($\min f/\max|f| = -0.20$, resolution-independent; verified with the closed-form two-Plummer oracle, since $\gamma=5$ EFF $\equiv$ Plummer). The $f_j \ge 0$ realizability gate refuses such builds, naming the component, its $f_{\min}$, and a remedy; a close-out sweep located the gate flip between $a = 0.65$ (refused) and $a = 0.68$ (realizable). The headline therefore uses $a = 0.8$. Realizability is physics, not a numerical nuisance.

3. The hybrid-sampling $Q_j$ plateau is predicted physics, not bias. Engine B samples a hybrid: positions from the prescribed $\rho_j$, speeds from $f_j$. A hard-truncated component has $\rho(r_t) > 0$ — an edge offset no ergodic $f(E)$ can carry (the Eddington pair represents $\rho(\Psi) - \rho(0)$) — so the truncated halo’s sampled $Q_j$ plateaus below 0.5. The exact-quadrature hybrid expectation predicts 0.4953; an 18-seed $\times$ 16k campaign measured $0.4947 \pm 0.0014$ ($0.4\sigma$ from the prediction — this is the headline figure the theory pages cite). The in-script PASS/FAIL table above runs a faster 3-seed $\times$ 16k check ($0.4917 \pm 0.0062$ for the halo, consistent with the 18-seed campaign within its larger SEM); both sit on the prediction, not on idealized 0.5. The gate is against the prediction, never a tuned offset.

Numerics bug found and fixed: the King $dW/dr$ staircase¶

jnp.gradient of the piecewise-linearly interpolated $\psi$ grid is a staircase whose ringing the Eddington $d^2\rho/d\Psi^2$ + Abel $1/\sqrt{E-\Psi}$ weight focuses into $f(E\to\Psi_0)$: a single King component read $\min f/\max|f| = -0.679$ (the true King ergodic DF is strictly positive). Fix: integrate King’s own Poisson identity $d\psi/d\xi = -(9/\hat\rho_0)\,\xi^{-2}\int_0^\xi \hat\rho(\psi(s))\,s^2\,ds$ by cumulative trapezoid of the closed-form density — never differentiate interpolated data in an Abel-type inversion. Re-measured at close-out: $\min f/\max|f| = +5.1\times10^{-7}$.

Differentiability¶

The full Engine B build (build_engine_b_state) is differentiable: AD matches central FD through the shared-potential pass + Eddington inversion in the halo scale $r_h$ ($5.57\times10^{-9}$), the mass fraction $t$ ($7.83\times10^{-7}$), and the per-component anisotropy radius $r_{a,j}$ ($2.00\times10^{-8}$) — all far inside the 10^-3 gate. One designed exception: the King component’s internal subgraph (its 1-D profile solve and the derive_r_t domain choice) is constant w.r.t. the differentiated parameters — King outputs enter via the scale and the shared $\Psi$; the domain is a construction-time decision.

How to run¶

# physics tests (6 validation tests; the unit suite covers realizability + extraction pins)
pytest tests/validation/test_engine_b_physics.py -q
pytest tests/unit/cluster/test_multicomponent.py -q

# regenerate the 5-panel figure with the 11-row PASS/FAIL table
env -u VIRTUAL_ENV uv run --no-sync python scripts/validate_multicomponent_eddington.py

What this suite does not test¶

• No N-body integration of the ICs — equilibrium is established by virial oracles, cross-engine agreement, and DF-moment checks on the initial conditions, not by evolving them and measuring stationarity.

• The hard-truncation edge approximation is quantified, not eliminated — a hard-truncated prescribed density is only approximately stationary at the edge ($\rho(r_t)>0$ cannot be carried by any ergodic $f(E)$); the sampled global-$Q$ gate is 0.02 (not 0.002) for this reason, and the per-component sampled $Q_j$ is gated on the hybrid prediction.

• Speed-table acceleration (Phase 1.5-style) is deferred — Engine B’s quadrature sampling path is the oracle; tables get added only if profiling shows Engine B sampling matters at $N \gtrsim 10^5$.

References¶

Eddington inversion and the $f \propto E^{7/2}$ Plummer DF: Binney & Tremaine (2008); the King model is King (1966), the EFF profile Elson et al. (1987). Engine A (the DF-defined family it cross-validates against) is documented at Multi-mass LIMEPY equilibrium (Engine A); the lowered-model-family roadmap at The differentiable lowered-model family (Engine A). This validation backs the Phase-2 Engine-B close-out.