The Proof Program: O1–O6 and the Conditional Factorization

This is the diagnostic ledger that enumerates exactly what must be true for the corrected predictor to factorize — and is honest that all but one piece is textbook reversible-MCMC math.

The question

The spine conjectures that the operational gradient-SNR equals the structural predictor: $Q_{op} = \lVert g\rVert^2 / \mathbb{E}\lVert \hat{g}-g\rVert^2 \asymp Q_{struct}^{\perp}(\theta,K)$, with $Q_{struct}^{\perp} = (K/2)\lVert g\rVert^2 / T_O$. The question this entry answers is bookkeeping, not physics: which sub-claims would carry that factorization from [conjectured] toward [proven-here], and which of them are actually discharged? The source synthesis/proof-sketch-q-struct-perp.md is a diagnostic page — it moves no tag itself.

The setup: the assumption package A1–A9

The factorization is asserted only in a fixed-$\theta$, finite-dimensional, reversible regime. The package is nine tagged assumptions:

• A1 fixed $\theta$; A2 $\pi_\theta$-reversibility (self-adjoint $P_\theta$, real spectrum — single-site random-scan Gibbs and random-scan 2-block; not alternating-scan); A3 irreducibility + aperiodicity ($\sigma_1=1>\sigma_2\geq\dots\geq\sigma_d\geq 0$); A4 observable regularity ($f_a\in L^2(\pi_\theta)$).
• A5 burn-in adequacy $B\gtrsim c/\gamma_{eff}$; A6 window adequacy, sharpened to $K \gg \tau_{int}[f_a]$ for the slowest $a$ (finite-window relative error $\approx 1/(\gamma K)=1/(2\cdot\mathrm{ESS})$).
• A7 overlapping-bulk relaxation — a rate condition $\gamma_{bulk}:=1-\max_{j\notin C^*(O),\,\mu_\sigma>0}\sigma_j\geq\Omega(1)$, not an eigenvalue-ordering gap (the cluster and bulk $\sigma$-ranges may interleave; that gap is $<0$ on exp1). The $\mu_\sigma>0$ restriction is essential or A7 is vacuous on its own motivating instances.
• A8 threshold robustness ($C^*(O)$ insensitive to $\varepsilon$); A9 symmetry-compatibility (a finite group $G$ with invariant $\pi_\theta$, equivariant $P_\theta$, invariant $f_a$ — canonically the free global $\mathbb{Z}_2$ flip; independent of A2).

experiments/exp1-exact-diag/ satisfies A9 only for the even pairwise observables under random-scan single-site Gibbs — not the odd bias gradients nor the $b\neq 0$ follow-up.

The result: O1 proven-here, O2–O6 solid

The six obligations carry explicit dependencies. O1 is foundational; the rest reference its projection-vs-conditioning resolution.

• O1 — projection in $L^2(\pi_\theta)$, not state-space conditioning. The observable-projection $\Pi_C u = \sum_{j\in C^*(O)}\langle u,\varphi_j\rangle_\pi\,\varphi_j$ is an orthogonal function-subspace projection under the single unmodified measure $\pi_\theta$ — categorically different from conditioning on a sector of configurations. For the global $\mathbb{Z}_2$ flip there is no fixed-point set in $\Omega=\{-1,1\}^N$, so "$\pi|_{even}$" is a category error; even gradients live in $H^+$, the odd slow mode $\varphi_2$ in $H^-$, with $\langle f_a,\varphi_-\rangle_\pi=0$. The dedicated O1 entry holds the rigorous argument; the load-bearing lemma O1.c (fundamental-domain invariance of the stationary SNR objects, via an intertwining isometry making $T_O$, $\lVert g\rVert^2$, $Q_{struct}^{\perp}$ identical term-by-term) is the proven-here lemma — the wiki's first terminal tag. O1.a/O1.b are [solid]. Empirically exp1+exp2 measured $\langle f_a,\varphi_2\rangle_\pi \leq 3.5\times10^{-17}$ — confirmation the $H$-projection form works, not proof that conditioning is unnecessary.

O2–O6 are a compact summary — each is textbook reversible-MCMC math, graded [solid], NOT proven-here, each derived on its own page:

• O2 — the fixed-$\theta$ MCMC-CLT: $S_a(\theta)=2\tau_{int}[f_a]\,\mathrm{Var}_\pi[f_a]$, $\sum_a S_a = 2T_O$, so $T_O$ is half the aggregate asymptotic variance. The finite-window correction $\approx 1/(\gamma K)$ sharpens A6.
• O3 — bias subdominance: $\mathrm{Bias}_a^2/(S_a/K)\approx \hat{m}_{j^*}^2\sigma_*^{2(B+1)}/(4\cdot\mathrm{ESS})=o(1)$; the window-averaging (A6) carries subdominance, so A5 is sufficient but not the binding gate.
• O4 — $[\Pi_C, P_\theta]=0$ on the reversible eigenbasis (membership criterion is reversibility, not sampler name — MALA/full-refresh HMC are in, ULA/alternating-scan out); degeneracy is repaired by the basis-invariant spectral-subspace definition.
• O5 — the harmonic-mean identity $T_O = (\mu_C/\gamma_{eff})(1-\gamma_{eff}/2)(1+o(1))$, off-cluster tail subdominant as $O(\gamma_{eff}/\varepsilon)$ under A7.
• O6 — the aggregation closure: for the Euclidean $Q_{op}$, $\mathbb{E}\lVert\hat{g}-g\rVert^2=\sum_a \mathrm{MSE}_a$ is exact and regime-free (the trace drops the real-but-invisible off-diagonal $\mathrm{Cov}[\hat{g}_a,\hat{g}_b]$), assembling $Q_{op}=Q_{struct}^{\perp}(1+o(1))$ in regime A1–A8 + plateau + F4.

Scope and caveats

After all six closed, the spine tag was split by researcher conferral, not by self-flip: the conditional factorization (regime $\Rightarrow Q_{op}\approx Q_{struct}^{\perp}$) is [solid]; the operational/unconditional claim stays [conjectured]. The gates are open at scale — exp3 fired F1 (median ratio $\approx 2.13$, under-equilibrated), $\lvert C^*(O)\rvert \sim 7\text{k}-12\text{k}$, and its alternating-block kernel is F3 (so A2 also fails). The factorization is a conditional assembly over the single proven-here lemma; this page is diagnostic and no tag flip happens here.

The six falsifiers are the operational handles for exp3: F1 finite-$K$ bias dominates $T_O/K$ (ratio $>1.5$); F2 $H$-projection vs fundamental-domain SNR diverge ($\gg 1$, O1 fails); F3 non-reversibility breaks the spectral-sector argument (tracking degrades $> O(20\%)$); F4 positive-phase MSE comparable to $2T_O/K$; F5 moving-$\theta$ scope-falsifier ($\lVert\Delta\theta\rVert\tau_{int}/K>0.1$); F6 threshold-$\varepsilon$ instability for $C^*(O)$.

What this feeds: the operational claim now rests entirely on closing A7 and $K\gg\tau_{int}$ at scale — the exp3 redesign targets F1/F3 directly, since those are the gates exp3 already tripped.

Sources

• Younes (1999), §7 Thm 3 — the fixed-$\theta$ MCMC-CLT variance object $S(\theta)$ grounding $T_O$ (object provenance only, no SNR factorization).
• Yuille (2004) — CD-from-data endpoint bias eigen-expansion (Risk 6 row; outside A5).
• Levin & Peres (2017), Markov Chains and Mixing Times — reversible-chain eigenbasis, geometric bias, time-average variance (the [solid] backbone for O2–O5).