Abstract
This essay examines the entropy certificate method of Williamson for proving lower bounds in secret sharing schemes and assesses its potential structural analogy to the Weil explicit formula for the Riemann zeta function.
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Overview
This essay mines the secret-sharing construction of Williamson (arXiv:2608.17047v1) for patterns that might transfer to the Riemann Hypothesis. The source paper uses an entropy certificate—a carefully constructed linear combination of conditional entropies and mutual informations—which, when averaged over a finite symmetry group, telescopes to yield a quadratic lower bound on share size. This technique of enforcing global bounds via local positivity and symmetry-driven cancellation bears a superficial resemblance to the Weil explicit formula, where positivity of a functional constrains the location of zeta zeros.
We assess the strength of this analogy and find it to be weak—at best a suggestive metaphor. The source structure relies on finite submodular entropy functions and combinatorial access structures, while the Riemann zeta function requires complex analysis, Mellin transforms, and an Euler product. The group averaging in the source is a finite sum over a discrete automorphism group; the functional equation of zeta is not a finite averaging process. Consequently, we cannot construct a dictionary mapping the entropy certificate to the explicit formula without violating the axioms of one domain or the other.
The essay provides a detailed honest exit analysis: we state the source structure faithfully, explain what made it superficially resonant with RH, identify precisely where the analogy fails (the inability to align the telescoping mechanism with the explicit formula), and describe what kind of source paper (e.g., one involving continuous entropy on infinite graphs) might repair the bridge. We also propose a hypothetical computational test on the Ihara zeta function of a regular graph, which serves as a discrete toy model for the Riemann zeta function.
Rating: Suggestive Metaphor (no formal analogy established).
This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.