Birefringent Splitting of Critical Curves and the Statistics of Zeta Zeros: A Speculative Computational Bridge

Abstract

We speculate that the bifurcation of photon-sphere critical curves observed in the rotating García–Díaz black hole under nonlinear electrodynamic coupling provides a formal analogy for the level-repulsion phenomenon in the distribution of Riemann zeta zeros.

Overview

This essay mines a 2026 preprint by Guzmán, Fathi, and Villanueva (arXiv:2606.20284v1) that studies high-frequency light propagation in a rotating black hole coupled to nonlinear electrodynamics (NLED). The authors show that the electromagnetic field acts as a birefringent medium, causing the Fresnel quartic characteristic equation to factorize into two distinct quadratic optical metrics. Each metric possesses its own family of unstable photon orbits (critical curves) that project onto the observer’s celestial sphere as two separate contours Γ₊ and Γ₋. In the Maxwell limit (zero nonlinear coupling), these curves collapse to a single degenerate contour; when the coupling is switched on, they split linearly in the perturbation parameter.

We propose that this bifurcation structure provides a formal analogy—though not an isomorphism—for the level repulsion observed in the spacing statistics of Riemann zeta zeros. The analogy is rated formal: both systems exhibit a linear splitting of a spectral invariant when a degeneracy is lifted by a small parameter. On the zeta side, the “splitting” corresponds to the deviation from Poisson statistics (zero correlation) toward the GUE statistics (avoided crossings) conjectured by Montgomery and observed numerically by Odlyzko.

Key Structural Parallels

• Source (NLED black hole): A one-parameter family of critical constants (ξ, k) describing photon impact parameters. At β = 0 (Maxwell), there is one curve; for β ≠ 0, it splits into two branches with separation Δξ ∝ β.
• Target (Zeta zeros): The nearest-neighbor spacing δₙ of zeros on the critical line. In a hypothetical “integrable” limit (β = 0), spacings would be Poissonian (zero probability of zero spacing); in the “chaotic” limit (β > 0), the GUE repulsion creates a linear vanishing of the small-spacing probability, effectively splitting the sequence into rigidly separated levels.

Proposed Tests

The essay proposes computational experiments—explicitly not yet executed—that would partition the zeta zeros into two subsequences (e.g., by parity of index or by sign of the Hardy Z-function derivative) and compute a “birefringent width” statistic analogous to the angular separation of Γ₊ and Γ₋ in the black-hole shadow. If this statistic scales linearly with a tunable deformation parameter in a family of L-functions, the analogy would be supported.

Obstructions

The primary failure mode is the absence of a local constitutive relation on the number-theoretic side. The black-hole splitting arises from a spatially varying response matrix mapping (D, B) ↦ (E, H); no analogous local field theory governs the Riemann zeros. Consequently, the correspondence cannot be promoted to a structural isomorphism.

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