O1: Projection in L²(π), Not State-Space Conditioning

The object behind $Q_{struct}^{\perp}$ is an orthogonal projection in $L^2(\pi_\theta)$ — never a conditioning on a state-space event — and the stationary signal-to-noise it feeds does not depend on whether you project in function space or pass to a fundamental domain.

The question

Obligation O1 is the foundational leg of the claim $Q_{op} \approx Q_{struct}^{\perp}$, recorded against the proof-sketch's assumption package A1–A8. Two doubts had to be closed first. Is the cluster projection $\Pi_C$ — which keeps only the observable-relevant eigenmodes — secretly a conditioning of the chain on some region of state space, making the symmetry-zero story an artifact of a state-space cut? And does the predictor depend on whether we project in the full space or quotient by the symmetry group? This is the complete technical record beneath it.

The setup

Work in $H := L^2(\pi_\theta)$ with $\langle u,v\rangle_\pi := E_\pi[\bar u v]$. Under A2 ($\pi_\theta$-reversibility) the kernel $P_\theta$ is self-adjoint; under A3 (irreducibility + aperiodicity) it has a real $\langle\cdot,\cdot\rangle_\pi$-orthonormal eigenbasis $\{\phi_j\}$ with $\phi_1 = 1$ and $P_\theta\phi_j = \sigma_j\phi_j$ (Levin–Peres Lemma 12.2). The cluster projection is $\Pi_C u := \sum_{j\in C^*(O)}\langle u,\phi_j\rangle_\pi\,\phi_j$. This page adds A9 (symmetry-compatibility) as the conjunction Sym-$\pi$ (invariant $\pi_\theta$), Sym-P ($G$-equivariant $P_\theta$), Sym-f ($G$-invariant observables). Critically, A9 is independent of A2 ($A2 \perp A9$): a reversible kernel can be non-equivariant, so A9 is genuine extra content about the energy + scan, not a consequence of reversibility.

The result

O1.a [solid] — $\Pi_C$ is idempotent and self-adjoint in $\langle\cdot,\cdot\rangle_\pi$, hence an orthogonal projection. By Parseval, $\sum_a\lVert\Pi_C f_a\rVert^2_\pi = \sum_a\sum_{j\in C^*(O)}\hat f_{a,j}^2$. Pure linear algebra on the reversible eigenbasis.

O1.b [solid] (core) — projection $\neq$ conditioning. Conditioning on an event $S$ is, in $L^2$, multiplication by $\mathbb{1}_S$: projection onto the support-subspace $H_S$. If $\Pi_C$'s range $V_C$ equaled some $H_S$, then $H_S$ would be $P_\theta$-invariant, forcing $P_\theta(x,y)=0$ across the $S$/$S^c$ boundary; reversibility makes the flow vanish both ways, so $S$ and $S^c$ are both absorbing — the chain is reducible, contradicting A3 unless $S\in\{\varnothing,\Omega\}$. The precise obstruction is dynamical isolation, which A3 excludes. Under the canonical free $Z_2$ flip $s\cdot x = -x$, the "even sector" $H^+$ is a function-parity sector, not a support-subspace of any event; $s$ has no fixed point, so "even configurations" is not even a well-defined event. Consequently the odd slowest mode is annihilated by $L^2(\pi)$-orthogonality, not conditioning: for even $f_a$, $\langle f_a,\phi_2\rangle_\pi = 0$ exactly. This is the spine's Corrected-candidate (1), now derived rather than asserted.

O1.c proven-here (conferred 2026-06-01) — the first terminal proven-here block in the wiki. The restriction map $\iota: H^G \to L^2(\bar\pi)$, $(\iota f)([x]) := f(x)$, is a unitary intertwiner: an isometry (using orbit-sum weights $\bar\pi([x]) := \sum_{z\in[x]}\pi(z)$), surjective, satisfying $\iota\,P_\theta|_{H^G} = \bar P\,\iota$. Hence $\bar P$ is $\bar\pi$-reversible with the same invariant eigenvalues $\{\sigma_j\}$. Term by term — overlaps, mean, $\operatorname{Var}_\pi$, autocovariance $\sum_{j\ge 2}\sigma_j^k\hat f_{a,j}^2$, $\tau_{int}$, $T_O$ — the stationary objects are identical in the $H$-projection and fundamental-domain pictures. The data phase matches for any data law $p$ (Sym-f alone drives the orbit regrouping). Therefore $Q_{struct}^{\perp} = (K/2)\lVert g\rVert^2 / T_O$ is invariant outright — no $|G|$ rescaling survives, since neither $\lVert g\rVert^2$ nor $T_O$ is an $L^2$ function norm. The $H$-form is canonical: the isotypic decomposition exists for any symmetry, and it is what the $B$+$K$ estimator on the full $\Omega$ delivers.

Numerically: exp1's single-site Gibbs confirmed detailed balance to residual $\le 5\times 10^{-18}$, the free-$Z_2$ quotient DB residual was $7\times 10^{-18}$, and the odd-mode / even-observable overlap measured $\le 3.5\times 10^{-17}$ (RBM; exp1's Ising values are smaller, $\sim 10^{-24}$ to $10^{-29}$) — exactly the $\hat f_{a,2}=0$ of O1.b.

Scope and caveats

The proof holds only in regime A1–A4 + A9, and the flip is narrow. Only O1.c is proven-here; O1.a/b are textbook one-liners [solid]. The factorization $Q_{op}\approx Q_{struct}^{\perp}$ stays [conjectured] — O1 is foundational, not sufficient; O2–O6 remain its own obligations. Risk 1 stays [open]: O1.c proves the projection mechanism, not that the predictor tracks $Q_{op}$ at scale (exp3 leaves it sharpened, untested at adequate equilibration — no tag flip). The empirical corroboration is post-hoc mode filtering, not a quotient chain — falsifier F2 was never run, so O1.c is analytical content with no experimental confirmation, which is why it rests on proven-here, never validated.

Two independent exits from the proven regime must not be conflated (since $A2\perp A9$): (3a) non-reversible kernels (A2 fails $\to$ F3) — this is exp2's alternating block-Gibbs, the main at-scale support, sitting outside the regime via the A2 leg, not an A9 counterexample; (3b) non-equivariant kernels (Sym-P fails $\to$ the intertwining $\iota$ breaks). The full finite-$B$ $Q_{op}$ carries a bias leg that is picture-invariant only under a pushed-forward start law — but that is O3's domain, kept deliberately separate.

What this feeds: O1 is the backbone under the symmetry-zero result and the entry point for O2–O6; the next obligations turn this projection content into the full $Q_{op}\approx Q_{struct}^{\perp}$ factorization.

Sources

• Levin & Peres, Markov Chains and Mixing Times (2017), Lemma 12.2 — real $\langle\cdot,\cdot\rangle_\pi$-orthonormal eigenbasis; cited for the spectral decomposition only.