Enumeration-Listing Reductions and the Incremental Complexity of the Riemann Hypothesis

Abstract

This essay proposes a speculative analogy between the enumeration-to-listing reduction for projected subgraphs established by Bringmann, Fischer, and Wang and the incremental verification of Riemann zeta zeros.

Overview

This paper explores a speculative bridge between algorithmic graph theory and analytic number theory. The source work by Bringmann, Fischer, and Wang establishes a generic reduction from listing algorithms (which output all solutions at once) to enumeration algorithms (which output solutions incrementally with bounded delay) for projected tree and hypergraph problems. Their main tool, the "Cheater's Lemma," shows that if the j-th solution can be retrieved in time roughly proportional to j, then one can achieve enumeration with preprocessing time independent of the number of solutions.

We propose mapping this complexity-theoretic structure onto the problem of verifying the Riemann Hypothesis. In this analogy, "listing" corresponds to computing the total count of zeros up to height T (the function N(T)), while "enumeration" corresponds to the sequential verification of individual zeros with constant or polylogarithmic delay between consecutive zeros. The Cheater's Lemma would suggest that if finding the j-th zero scales linearly in j, then sublinear-delay enumeration should be possible.

The analogy is rated as a suggestive metaphor only. While the complexity tradeoffs appear structurally similar, the Riemann zeta function lacks the finite splitting properties and self-reducibility that are essential to the graph-theoretic reduction. The essay identifies precisely where the correspondence breaks down and proposes computational experiments that would test whether the "delay" statistic of zeta zero verification exhibits the independence properties predicted by the analogy.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.