Abstract
This essay investigates whether the contour-deformation mechanism established by Alameddine and Hock for handling essential singularities in generalized topological recursion might parallel the explicit formula of analytic number theory.
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Overview
This essay examines the paper "Universal Correlators on Exponentially Ramified Spectral Curves" by Alameddine and Hock (arXiv:2607.17711v1), which develops generalized topological recursion (Gen-TR) for compact spectral curves with essential singularities. The central result is a contour deformation theorem allowing residues at essential singularities to be replaced by residues at meromorphic points.
We explore the speculative analogy between this contour deformation and the Weil explicit formula from analytic number theory, which similarly relocates contour integrals from the critical strip (zeros) to the pole at s=1 and prime singularities. This analogy is rated as failed or negative: while superficially similar in their use of residue calculus, the two theories operate on incompatible topological foundations.
The source paper requires compact Riemann surfaces of finite genus, whereas the Riemann zeta function's analytic theory is inherently non-compact and involves infinite-genus structures. The essay provides an honest assessment of why the analogy breaks down and what kind of source material (non-compact infinite-genus curves) would be required to sustain such a bridge.
This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.