Conformal Prediction and Zero Spacing Uncertainty: A Speculative Bridge from Medical AI to the Riemann Hypothesis

Abstract

This essay proposes a speculative transfer of the conformal prediction methodology—deployed in foundation model evaluation for cancer diagnosis—to the statistical analysis of Riemann zeta zero spacings.

Overview

The source paper investigates trustworthiness in medical AI via conformal prediction, a statistical technique that provides finite-sample, distribution-free coverage guarantees for prediction sets. We mine this structure for a potential bridge to the Riemann Hypothesis (RH), specifically regarding the uncertainty quantification of zeta zero spacings.

The Core Analogy

In the medical domain, conformal prediction uses a calibration set of past diagnoses to construct sets that capture future diagnoses with guaranteed probability. We speculate that an analogous procedure could be applied to the sequence of normalized zero spacings δ_n = (γ_n+1 − γ_n) log(γ_n/2π)/2π. By treating an initial segment of zeros as a "calibration set," one could algorithmically generate prediction intervals for future spacings that are valid without assuming the GUE hypothesis.

Strength of the Correspondence

We rate this a suggestive metaphor. While both domains involve sequential data and coverage guarantees, the analogy faces a fundamental tension: medical data is stochastic and exchangeable (conditionally), whereas zeta zeros are deterministic and strictly ordered. The "coverage" in the RH context would describe algorithmic success rates over random splits of the zero sequence, not a probabilistic truth about the critical line itself.

Proposed Experiments

The essay outlines unexecuted Wolfram Language experiments that would apply split conformal prediction to Odlyzko's canonical zero data. The goal is to test whether distribution-free intervals for spacings achieve nominal coverage and whether they exclude zero (which would correspond to a double zero, violating RH) with high confidence.

Assessment

This is an exploratory speculation. The analogy breaks down on the deterministic nature of the zeros and the long-range correlations (Montgomery–Odlyzko) that violate the exchangeability assumed by conformal prediction theory. The value lies in precisely identifying this boundary: where statistical learning theory meets the rigid structure of the zeta function.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.