Why Partially Coherent Airy Beams Resist a Phase-Space Bridge to the Riemann Hypothesis: An Honest Assessment

Abstract

We examine the phase-space theory of partially coherent accelerating beams developed by Ponomarenko and Hajati (arXiv:2601.11416v1) for potential structural correspondences with the Riemann Hypothesis.

The Source System

The paper advances a phase-space description of partially coherent, accelerating, non-diffracting beams using the Wigner distribution function (WDF). The underlying dynamics are governed by the paraxial Schrödinger equation with the Hamiltonian Ĥ = k̂²/2 + x̂, whose eigenfunctions are Airy functions Ai[2^(1/3)(x − ε)] with continuous eigenvalues ε ∈ ℝ. Partial coherence is introduced via a statistical mixture of these eigenstates weighted by a Gaussian noise spectrum p(ε). The resulting Wigner function W_G(X, k, z) is given in closed form as a smoothed Airy function in phase space, evolving by a shear flow (X, k) → (X − kz, k).

Superficial Resonances with the Riemann Hypothesis

Three features suggest a possible bridge to the Riemann Hypothesis (RH). First, the Wigner function is the central object of the quantum chaos approach to RH, which seeks a Hamiltonian whose spectrum encodes the zeta zeros. Second, Airy functions govern the edge statistics of Gaussian Unitary Ensembles (GUE), which conjecturally model the local statistics of Riemann zeros. Third, the coherence parameter ξ_c controls a smooth deformation of the ideal beam, reminiscent of deformation parameters in zeta function theory.

Why the Analogy Fails

The correspondence fails at the level of formal analogy. The Airy Hamiltonian has a continuous spectrum (the real line), whereas the Riemann zeros form a discrete set. The “noise” in the source is additive Gaussian convolution in the spectral domain, not the multiplicative chaos or log-correlated Gaussian fields relevant to random multiplicative functions and zeta moments. The phase-space flow is parabolic (constant acceleration), not the hyperbolic expansion/contraction (xp Hamiltonian) conjectured to underlie the zeta zeros. Consequently, the smoothed Wigner function retains a fundamentally different structure from the deformed zeta function.

Diagnostic Conclusion

This essay rates the analogy as a failed structural correspondence. A viable bridge would require a source system with discrete spectrum approximating the zeta zeros, hyperbolic phase-space dynamics, and multiplicative rather than additive statistical structure.

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