Exp 7 — Barrier/Conductance: A Scoped Green Light

A direct, zero-extra-compute test of whether a slow mode is escapable rather than fundamentally trapped — answered on a tiny controlled RBM, where the answer is a scoped yes.

The question

Our chain probes (experiments/exp4-*, experiments/exp6-*) show $\tau \propto L$ at scale, but $\tau \propto L$ cannot separate a genuine $\exp(cN)$ barrier from a slow-but-escapable one. The A2↔A6 obstruction in the spine needs non-chain evidence: a direct crossability test. The pre-registered question (frozen §3 confirmatory pre-commitment, commit fba065a): does a non-empty, crossable, observable-relevant sweet spot exist on the controlled RBM family?

The setup

The seed-0 confirmatory grid — the pre-registered pilot_seed=0 / EXP7_NS=20260607 tuple, untouched by the pilot/re-pilot work on seeds $\{1,2,3\}$ — is $m \in \{4,5,6\}$ dense plus a held-out $m=7$, across $M \in \{1,2,3,4\}$ and $\beta \in \{0.5,1,2,3\}$: $48 + 16 = 64$ cells. Run 2026-06-08 on laptop CPU, float64, exact-diag (not GPU), runtime $2239$ s ($\sim 37$ min).

Each cell builds basins and measures (not OLS-predicts) masses, barriers, MFPTs, and overlaps, then applies four frozen filters: (i) mass $\pi(C) \ge 0.004$; (ii) separation $B^* \ge \log 2$; (iii) measured-MFPT crossability via full-chain targeted-hitting $(I - P_{S \setminus C})t = 1$ in both directions, M-matrix certified, against a deep-escape budget $B_{budget} = \log(2m \cdot 250 / C_{deep})$; (iv) observable relevance $r_e \ge \varepsilon_{overlap}$. A cell qualifies when $N_{qual} \ge 1$; strength is $U_{thermo} = \sum_e r_e \cdot B^*_e$ (existence-only — magnitude is descriptive).

The result

TT2 green-light candidate. $25$ of $64$ cells have $N_{qual} \ge 1$ at the primary $\varepsilon_{overlap} = 0.01$, firing decision-rule branch 2. The qualifying region is concentrated at $M=2$ ($13/16$ cells) with selected $M=3,4$ cells ($6/16$ each); at $m=7$ only $M=2$ qualifies. $U_{thermo} \in [0.52, 5.87]$.

The F3 candidate-cut $\to \gamma_{eff}$ proxy passes on held-out seed 0. The single-seed specialisation log γ_eff,step = α + s·log Φ̂, checked against the frozen lock (slope band $[0.527, 1.673]$, $\rho_{min} = 0.70$), lands all three dense strata in band:

stratum   n    slope s_m   in band   Spearman ρ_m   ≥0.70   OK
m=4       16   1.0330      yes       0.750          yes     yes
m=5       16   1.1178      yes       0.979          yes     yes
m=6       16   1.3444      yes       0.956          yes     yes

The $m=6$ stratum was never seen by the seeds-$\{1,2,3\}$ calibration: a genuine out-of-sample confirmation, not a re-tune. $f_{valid} = 48/48 = 1.0$.

Two registered steps were omitted by the runner and supplied post-hoc (audit cheeger_slope_audit.py, stage_a_results.json left untouched). The acceptance-blocking global Cheeger check $\gamma_{global} = 1 - \sigma_2 \le 2\hat\Phi$ holds for all 64 cells ($\max \gamma_{global}/(2\hat\Phi) = 0.4139$, 0 violations). The stronger chain $\gamma_{global} \le \gamma_{\perp} \le 2\hat\Phi$ holds on the $63$ cells with a defined $\gamma_{\perp}$ — not everywhere: the one UNRESOLVED cell ($m=7, M=1, \beta=3$) has no $\gamma_{\perp}$. The verdict stands because this check passes. The $\varepsilon_{overlap}$ sweep $\{0.003, 0.01, 0.03, 0.1\}$ shows existence at all four thresholds — no flip.

Scope and caveats

This is a scoped existence demonstration on a tiny RBM ($N \le 14$, seed 0, $M/\beta$ analogy axis) — never a fundamentality verdict. Honest qualifiers, all registered:

• The tie-ordering screen (decision [3]) was vacuous: $\max n_{multiway} = 0$ over all 64 cells, so "no flip" is vacuously true — the screen was never exercised. (The $\varepsilon_{overlap}$ sweep was a genuine no-flip.)
• There is no literal net↔EBM expressivity-split knob in this RBM; the $M/\beta$ axis is an analogy. The result is existence over that analogy axis, not proof an intermediate-split sweet spot drives it.
• The pooled OLS association of $U_{thermo}$ with $\beta$ is positive (slope $0.506$, 95% CI $[0.192, 0.820]$, post-hoc spec) but not monotone: in strata $(5,3)$, $(6,3)$, $(6,4)$, $U_{thermo}$ peaks at $\beta=2$ and falls at $\beta=3$.
• Diagnostics J1/J2 are retired, recorded-not-gated. J1 fails its old $0.25$ gate on held-out seed 0 too (median rel-err $0.558$), vindicating its retirement; J2 deferred $\varphi_2/\varphi_3$ and missed the literal model-lock-before-any-held-out discipline (narrower discipline held). Neither touches the verdict.
• Prefactors are measured, kernel-specific, and not universal. No spine tag flips: conditional factorization stays [solid], the operational claim stays [conjectured]. A $U_{thermo}$-predicts-the-sweet-spot $\to$ [validated] is researcher-conferred, never self-applied.

What it feeds next

Exp 7 supplies the non-chain, direct-crossability evidence the A2↔A6 antagonism needs: on this small family the obstruction is operationally escapable within the 250-sweep budget — pushing toward the non-fundamental reading, on this family only. The at-scale / fundamentality question stays open.

Feeds: Stage B — a deferred GPU DTM-anchor + ACP-exchange gate, blocked behind both this passing Stage A and a favorable substrate-reachability answer; the path to → validated remains an at-scale run with A7 and $K \gg \tau_{int}$ met.