Exp 8 — Hypernetwork Retrofit: Building the Missing Substrate

We built the substrate the audit said did not exist, and checked that a network can hand an energy model couplings that are differentiable all the way through — without ever touching the operational hurdle that still blocks the program.

The question

The substrate audit returned a clean zero: of eight published thermodynamic-ML substrates, none lets a network produce differentiable EBM couplings. So the scan's §3 retrofit row recommended building one — a hypernetwork $W = g_\phi(u)$ that emits RBM coupling matrices from a teacher-identifying code $u$. This run asks three preregistered questions, no more: can the G1b gate be CONSTRUCTED (couplings NN-produced, the active target differentiable through $J=W$); is the R3 "crossable-multimodality" sweet spot non-empty; and is R4 (the verify-or-HALT diagnostic battery) verifiable. It explicitly does not touch G2 — the operational factorization tier stays [conjectured].

The setup

Pre-commitment was frozen (bb342b0, 2026-06-09) before any implementation — see experiments/exp8-hypernet-retrofit/ for the freeze, runner, and per-cell results.json. The run was 447 s wall on a laptop CPU (cap 3 h; midpoint trigger projected 419 s → PROCEED), JAX 0.9.1 in x64. The grid is $m \in \{4,5,6\}$ hidden units $\times\ s \in \{1,2,3,4\}$ split-axis positions $\times$ 4 teacher codes $u$ — 192 cells, with a verdict basis of 72 ($m\in\{4,5,6\}\times s\in\{2,3\}\times 4u\times$ verdict checkpoints).

The load-bearing design choice was the derivative-agreement test for P1(c). Rather than autodiff through an eigendecomposition, the registered Change-4 route builds a JAX mirror of the selection-free $Q_{struct}^{\perp}$ via a deflated resolvent — smooth wherever $\gamma_u > 0$, no eigh. We then compare autodiff against central finite differences along 5 frozen directions.

The result

P1 — G1b CONSTRUCTED (construction/formula-level, never "empirically validated"):

• (a) $J_u$ finite at all 48 $(m,s,\text{ckpt})\times 4u$ entries; $\min \|J_u\|_F = 1.58 > 0$.
• (b) nontrivial $u$-dependence: $\min_{u\neq u'}\|W(u)-W(u')\|_F \ge 2.66$ at every final checkpoint (threshold $10^{-6}$).
• (c) at $(4,2)$, anchor = first $\le 0.5$ crossing (step 66): the value-agreement gate $|\tilde Q_{JAX}-Q_{numpy}|/|Q_{numpy}| = 7.06\times10^{-15}$ (gate $10^{-10}$); autodiff-vs-FD best-$h$ relative errors $\{6.5\text{e-}10, 2.1\text{e-}10, 9.1\text{e-}10, 1.0\text{e-}9, 9.9\text{e-}7\}$, all $\le 10^{-4}$ → PASS.

The no-eigh route was load-bearing, observed not assumed: the four quotient kernels at the anchor had min adjacent eigenvalue gaps $\sim 1.1\text{e-}16$ to $3.3\text{e-}16$ — numerically degenerate spectra that would have made eigh-autodiff ill-conditioned exactly here. The pre-freeze skeptic's MAJOR was necessary, not precautionary.

P2 — R3 non-empty (empirically measured): 11/72 verdict-basis cells have $N_{qual}\ge 1$. Both stability riders are clean — identical $N_{qual}$ across the $\epsilon_{overlap}$ sweep $\{0.003, 0.01, 0.03, 0.1\}$ (zero verdict flips), and the multiway-ordering rider is vacuously stable (zero $q\ge 3$ saddles in all 192 cells; the machinery armed but never fired). Every qualifying event sits $\ge 0.0529$ from its nearest filter threshold. Qualifying-cell $\gamma_{eff}$ spans $9.0\text{e-}4$ to $8.1\text{e-}2$; $U_{thermo}$ from 0.51 to 7.08. Descriptive prevalence over all 192 cells: 19 cells with $N_{qual}\ge 1$, per-$s = \{8,5,6,0\}$, all at $m\in\{5,6\}$.

P3 — R4 verifiable (verify-or-HALT, no HALT fired): max A2 detailed-balance residual $5.2\text{e-}17$ ($TOL_{SYM}=10^{-10}$); the global Cheeger bound $\gamma \le 2\hat\Phi(A) + 10^{-10}$ holds on every merge-tree cut (min margin $+3.93\text{e-}4$); zero Sym fallbacks — the $\mathbb{Z}_2$ quotient was legitimate in all cells.

P4 — split-axis response (descriptive): the capacity-allocation tension is real. Measured $\mathrm{rank}(J_u)=r_{EBM}(2m-r_{EBM})$ everywhere (the rank-$r$ manifold tangent dimension), so the observable span contracts monotonically along $s$. At $s=4$ ($r_{EBM}=1$) the $M_t\ge 2$ teachers are unmatchable — $(5,4)$ stalls at a genuine stationary point (per-$u$ KL up to 2.63), $N_{qual}=0$ everywhere at $s=4$.

Scope and caveats

This decides G1b/R3/R4 only, scoped to this family/architecture/teacher set ($N\le 12$, one seed table, 4 teachers at $\beta_t = 3$). No tag moves. The code $u$ is a teacher-identifying frozen code — a construction witness, not strong encoder evidence. P1(c) certifies the selection-free $Q$ form; the hard-$C^*$-selection non-smoothness (Risk 2) stays a theorem-side boundary, unresolved here. $N_{qual}=0$ at $m=4$ everywhere. And critically: nothing here touches A6/A7 at scale — G2 remains the binding empirical hurdle, the row's G2/A6/A7 sub-question stays NOT-DETERMINED-WITHOUT-RUN, and $\to$ [validated] requires the full A1–A8 + plateau + F4 regime.

What this feeds: G1b is now CONSTRUCTED and R3 is shown non-empty, so the differentiable substrate exists and contains crossable multimodal structure — clearing the way to attack G2 (operational factorization at scale), the one tier this run deliberately left [conjectured].