The Spectral Gap Is the Wrong Object: Observable-Projected Gradient Signal-to-Noise and a Reversibility–Mixing Obstruction for Thermodynamic Energy-Based Models

Abstract

Energy-based models trained on thermodynamic samplers estimate their gradient from finite, correlated samples; when the estimate drowns in sampling noise, learning stalls on a plateau. We summarize this with a single computable, differentiable quantity Q — the squared signal-to-noise ratio of the gradient — and report three results. First, the textbook single-spectral-gap diagnostic can be the wrong object by 26–30 orders of magnitude: under Z2 symmetry the slowest mixing mode is exactly orthogonal to every gradient observable, and the fix is an L2(pi) projection, not state-space conditioning. Second, we define an observable-projected, multi-mode predictor Q_struct-perp that tracks the operational gradient-SNR where the naive form fails (45/48 exact-diagonalization cells; 92–99% in RBMs), and prove the factorization Q_op ≈ Q_struct-perp as a conditional. Third, on a real trained MNIST denoising thermodynamic model we measure an antagonism between the reversibility the theorem needs and the decorrelation it needs: the reversible chain shows tau proportional to chain length and never equilibrates, and an optimal equal-acceptance parallel-tempering ladder needs 136 reversible rungs against a 96-rung budget — a thermodynamic-length cost wall.

Key results

• The spectral gap is the wrong object. Under Z2 symmetry the slowest mixing mode is observable-orthogonal, so the single-gap predictor over-predicts the gradient-SNR by 10²⁶–10³⁰×; the correction is an L²(π) projection (proven-here).
• An observable-projected, differentiable predictor. Q_struct^⊥ tracks the operational gradient-SNR (45/48 exact-diag cells; 92–99% in RBMs), with a conditional factorization theorem.
• The reversibility–mixing wall. On a real trained MNIST DTM the reversible chain shows τ ∝ L and never equilibrates; an optimal equal-acceptance ladder needs R* = 136 rungs vs a 96-rung budget.

Cite

@misc{prasanna2026thermodynamic,
  title  = {The Spectral Gap Is the Wrong Object: Observable-Projected Gradient Signal-to-Noise and a Reversibility–Mixing Obstruction for Thermodynamic Energy-Based Models},
  author = {Prasanna},
  year   = {2026},
  note   = {Preprint},
  url    = {https://mlthermo.com/paper}
}

A plain-language tour of these results is in the writings; the complete technical record — every experiment, derivation, and negative result — is in the notebook.