Why Temperature Chaos in Directed Polymers Resists a Gibbs-Measure Bridge to the Riemann Hypothesis: An Exploratory Negative Assessment

Abstract

This essay examines the structural correspondence between the temperature-chaos phenomenon in the continuum directed random polymer (CDRP) and the statistical behavior of Riemann zeta zeros at separated heights.

Overview

This essay mines the paper Temperature chaos in directed polymers (arXiv:2607.18194v1) for patterns that might transfer to the Riemann Hypothesis. The source paper studies the Continuum Directed Random Polymer (CDRP), a model in the KPZ universality class driven by white noise. Its main result is that the free energies of the polymer at two different inverse temperatures β₁ and β₂ become asymptotically independent when β₂ ≫ β₁ ≫ 1, a phenomenon termed "temperature chaos."

The Proposed Analogy

The apparent structural similarity lies in the notion of asymptotic independence across a parameter gap. In the polymer model, the parameter is inverse temperature β; in the Riemann zeta function, one might imagine the imaginary part T (the height on the critical line). The speculation asks whether the statistics of zeta zeros in an interval [T, 2T] become "independent" of those in a distant interval [T², 2T²] as T → ∞, mimicking the decoupling of free energies.

Assessment of Strength

The analogy is rated a suggestive metaphor. While both systems exhibit decorrelation across scales, the mechanisms are fundamentally incompatible. The polymer result depends on the Gibbs variational principle (reweighting by exp(-βH)), the resampling of white-noise disorder, and the geometry of last-passage percolation. The Riemann zeros are deterministic, possess no tunable temperature parameter, and their statistical properties arise from the prime number explicit formula rather than a disordered energy landscape. The essay provides a precise analysis of these failure modes.

Proposed Tests

Despite the negative assessment, we outline a proposed computational experiment to test the (weak) analogy: computing empirical correlations between normalized zero spacings in widely separated intervals. The expected outcome—vanishing correlation—is consistent with known GUE universality but does not validate the specific Gibbs-mechanism analogy.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.