Flat-Minima Scaling Limits and the Riemann Hypothesis: An Exploratory Negative

Abstract

This essay examines whether the non-Gaussian scaling limits of stochastic gradient descent (SGD) at flat minima, as established by Zhang, Mao, and Mukherjee, provide a structural analogy to the value distribution of the Riemann zeta function or critical multiplicative chaos.

The Source Result

The paper Scaling Limits of Constant-Stepsize SGD at Flat Minima (arXiv:2607.16384v1) studies stochastic gradient descent with a constant stepsize α near a minimizer x* where the Hessian vanishes as ‖x − x*‖^{m−2} for m ≥ 2. When m = 2 (quadratic), the invariant law scales as √α and has a Gaussian limit. For m > 2 (flat minima), the authors prove geometric convergence to an invariant law concentrating on the scale α^{1/m}, and the rescaled process converges to the stationary distribution of a nonlinear diffusion dY_t = −h_0(Y_t)dt + Σ^{1/2}dB_t. This yields generally non-Gaussian stationary limits.

The Attempted Bridge

We speculate that this anomalous diffusion might model the critical value statistics of the Riemann zeta function ζ(s), particularly the non-Gaussian fluctuations of log|ζ(1/2 + it)| and the framework of critical multiplicative chaos. The exponent m governing the flatness could correspond to the critical parameter in log-correlated Gaussian fields.

The Assessment

The analogy is rated as a suggestive metaphor at best. While both systems exhibit non-Gaussian limits governed by exponent parameters, the SGD dynamics rely on additive Markovian noise and deterministic flat geometry, whereas zeta-function statistics derive from multiplicative randomness and number-theoretic structures. The correspondence fails to produce a formal analogy because the invariant measures and scaling mechanisms are structurally incompatible.

Key Findings

• The SGD limit distribution for m > 2 is determined by a homogeneous nonlinear drift, distinct from the log-normal/Gaussian behavior of zeta.
• The contraction factor 1 − cα^{m−1} in SGD has no clear counterpart in the spacing statistics of zeta zeros.
• The additive noise structure of SGD cannot replicate the multiplicative independence required for models of the zeta function.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.