Generalised Eigenvalue Geometry and the Hilbert-Pólya Conjecture: A Speculative Correspondence

Abstract

This essay examines the structural analogy between the generalized eigenvalue problem governing semantic adversarial attacks and the spectral interpretation of the Riemann Hypothesis.

Overview

This essay explores whether the mathematical structure of generalized eigenvalue problems arising in adversarial machine learning can inform the search for a spectral proof of the Riemann Hypothesis (RH). The source paper, "Generalised Eigenvalue Geometry of Semantic Adversarial Attacks," introduces the attackability index λ*(x), defined as the largest generalized eigenvalue of a matrix pencil (A,B) constructed from the Jacobians of two neural network embeddings. This quantity governs the worst-case displacement of a target model's representation under semantic perturbations.

The Proposed Analogy

We investigate whether this "two-metric" spectral problem—optimizing a Rayleigh quotient between two pullback metrics—serves as a finite-dimensional toy model for the Hilbert-Pólya conjecture, which posits that the non-trivial zeros of the Riemann zeta function ζ(s) correspond to the eigenvalues of a self-adjoint operator H. The dictionary maps the matrix pencil (A,B) to the operator H (equipped with a suitable metric), and the attackability index λ* to the squared imaginary part of a zero (t_n²).

Assessment of Strength

The correspondence is rated as a suggestive metaphor only. While both frameworks invoke spectral invariants to characterize critical behavior (decision-boundary crossing vs. zero location), the source lacks the essential analytic number-theoretic structures: the functional equation, the Euler product, and the global phase space symmetries found in Connes' approach or quantum chaos models. The source is local and finite-dimensional, whereas the RH spectral problem is global and infinite-dimensional.

Key Findings

• The generalized eigenvalue λ*(x) measures the maximal distortion between two embedding geometries; no analogous "disagreement" between two metrics is known to govern zeta zeros.
• The linearized flip condition in the source resembles an instability condition, but zeta zeros are fixed global constants, not locally variable.
• We propose (unexecuted) numerical experiments discretizing model Hamiltonians to test for spectral coincidence, predicting that the generalized eigenvalues will not align with zeta zero ordinates without imposing additional symmetry constraints absent from the source.

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