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A Chiral Adelic Dirac Operator and the Spectral Realization of the Riemann Zeros

James C. Hateley

TL;DR

The paper advances an operator-theoretic approach to the Riemann zeros by constructing a self-adjoint, chiral adelic Dirac system on the idèle class space, with a real-place Floquet background that creates spectral gaps. Arithmetic data enter through a prime-indexed mass deformation built from spherical Hecke operators, yielding isolated gap eigenvalues that are encoded in a spectral-shift function and organized by a separated trace formula. A Dirac–Hilbert–Pólya perspective emerges: zeros of global $L$-functions are realized as spectral-shift discontinuities induced by adelic deformations, not as raw eigenvalues of a single operator, and finite-prime truncations provide computable models for exploring arithmetic spectral flow. The framework unifies functional-equation symmetry, Floquet band-gap structure, and Euler-product-type arithmetic factors, offering a rigorous path toward an adelic Dirac interpretation of the Riemann hypothesis and a practical numerical program for investigating the spectral realization of arithmetic data.

Abstract

This paper develops a chiral adelic operator framework in which the functional--equation symmetry of global $L$--functions is realized directly in the spectrum of a Dirac--type Hamiltonian. Working on the idèle class space, we place a real--place Floquet Hamiltonian into an off--diagonal chiral form to obtain a global adelic Dirac operator with an exact involutive symmetry implemented by real reflection and idelic inversion. Arithmetic information is incorporated through a prime--indexed mass deformation built from spherical Hecke operators; when the coefficient functions are even, the perturbed operator preserves the chiral symmetry and produces isolated $\pm$--paired eigenvalues inside the spectral gaps of the Floquet background. These eigenvalues appear as jump discontinuities of the Dirac spectral shift function, while a separated adelic trace formula expresses the trace as a product of a Floquet orbital factor and a prime--indexed Euler--factor--type term whose logarithmic derivatives yield a prime--orbit expansion reminiscent of the explicit formula. This structure motivates a Dirac reinterpretation of the Hilbert--Pólya idea, identifying the nontrivial zeros of $ζ(s)$ not with the raw spectrum of a single operator but with the spectral--shift discontinuities of a chiral adelic Dirac system under controlled prime--indexed deformations, with finite--prime truncations providing computable models that converge distributionally and enable numerical exploration of arithmetic spectral flow.

A Chiral Adelic Dirac Operator and the Spectral Realization of the Riemann Zeros

TL;DR

The paper advances an operator-theoretic approach to the Riemann zeros by constructing a self-adjoint, chiral adelic Dirac system on the idèle class space, with a real-place Floquet background that creates spectral gaps. Arithmetic data enter through a prime-indexed mass deformation built from spherical Hecke operators, yielding isolated gap eigenvalues that are encoded in a spectral-shift function and organized by a separated trace formula. A Dirac–Hilbert–Pólya perspective emerges: zeros of global -functions are realized as spectral-shift discontinuities induced by adelic deformations, not as raw eigenvalues of a single operator, and finite-prime truncations provide computable models for exploring arithmetic spectral flow. The framework unifies functional-equation symmetry, Floquet band-gap structure, and Euler-product-type arithmetic factors, offering a rigorous path toward an adelic Dirac interpretation of the Riemann hypothesis and a practical numerical program for investigating the spectral realization of arithmetic data.

Abstract

This paper develops a chiral adelic operator framework in which the functional--equation symmetry of global --functions is realized directly in the spectrum of a Dirac--type Hamiltonian. Working on the idèle class space, we place a real--place Floquet Hamiltonian into an off--diagonal chiral form to obtain a global adelic Dirac operator with an exact involutive symmetry implemented by real reflection and idelic inversion. Arithmetic information is incorporated through a prime--indexed mass deformation built from spherical Hecke operators; when the coefficient functions are even, the perturbed operator preserves the chiral symmetry and produces isolated --paired eigenvalues inside the spectral gaps of the Floquet background. These eigenvalues appear as jump discontinuities of the Dirac spectral shift function, while a separated adelic trace formula expresses the trace as a product of a Floquet orbital factor and a prime--indexed Euler--factor--type term whose logarithmic derivatives yield a prime--orbit expansion reminiscent of the explicit formula. This structure motivates a Dirac reinterpretation of the Hilbert--Pólya idea, identifying the nontrivial zeros of not with the raw spectrum of a single operator but with the spectral--shift discontinuities of a chiral adelic Dirac system under controlled prime--indexed deformations, with finite--prime truncations providing computable models that converge distributionally and enable numerical exploration of arithmetic spectral flow.

Paper Structure

This paper contains 23 sections, 16 theorems, 249 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. The Global Adelic Hamiltonian
  3. Automorphic seed and physical Hilbert space
  4. The real--place Dirac operator
  5. The global chiral adelic operator
  6. Standing Assumptions
  7. A Chiral Adelic Trace Formula
  8. Spectral Shift, Gap Eigenvalues, and a Dirac--Hilbert--Pólya Principle
  9. Finite--Prime Dirac Truncations
  10. Numerical Implementation and Algorithmic Framework
  11. Step 1: Floquet discretization.
  12. Step 2: Hecke sampling.
  13. Step 3: Mass shifts.
  14. Step 4: Fiber eigenvalues.
  15. Step 5: Spectral shift function.
  16. ...and 8 more sections

Key Result

Lemma 1

There is a canonical unitary isomorphism where $d^{\times}x_{\infty}=dx_{\infty}/|x_{\infty}|$, and the restricted tensor product is taken with respect to the spherical vectors $1_{\mathbb{Z}_{p}^{\times}}$ at all but finitely many primes.

Figures (1)

  • Figure 1: A numerical approximation to the truncated spectral shift $\xi_{\mathrm{Dirac}}^{(S)}(\lambda)$ for the prime set $S=\{2,3,5,7,11,13,17,19,23,29\}$ using the Mathieu potential $U(y)=2\cos(2\pi y)$ and a reference energy $E_{\ast}$ placed in the first spectral gap. The plotted staircase exhibits the expected odd symmetry, with unit jumps located at the finite--prime gap eigenvalues $\pm\lambda_{k}^{(S)}$ arising from the real Floquet bands and the even arithmetic mass deformation. After applying a chiral affine normalization to the lowest gap eigenvalues, the initial segment of the staircase aligns horizontally with the first several imaginary parts of the nontrivial zeros of $\zeta(s)$, illustrating how finite--prime truncations of the arithmetic Dirac operator encode a structured, symmetric gap spectrum compatible with the qualitative Dirac--Hilbert--ólya heuristic.

Theorems & Definitions (36)

  • Lemma 1: Restricted tensor product structure
  • proof
  • Definition 1
  • Proposition 2: Structure of $\mathcal{H}_{\mathrm{phys}}$
  • proof
  • Lemma 3: Logarithmic unitary
  • proof
  • Lemma 4: Floquet band structure
  • proof
  • Proposition 5: Chiral symmetry at the real place
  • ...and 26 more