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The Riemann $Ξ$-function from primitive Markovian cycles II: Strip rigidity and divisor identification

Douglas F. Watson

TL;DR

The paper develops a strip-based rigidity framework to compare two canonical analytic constructions of a spectral object arising from a primitive dynamical model: a cycle-spectral determinant $P_N$ and the Archimedean Mellin side linked to the Riemann $Ξ$-function. By introducing the seam ratio $R(w)=\frac{Ξ(2w)}{P_N(w)}$ and its normalized form $\widetilde{R}(w)$, the authors prove that, under holomorphy and sector-control hypotheses, the ratio becomes a zero-free strip-unit on admissible overlap strips, forcing $Ξ(2w)$ and $P_N(w)$ to share the same zero divisor on those strips. The bridge is built via a left-strip continuation of the bilateral Laplace transform $\mathcal{B}\Phi^{\star}$, a boundary identity on the left boundary, and a lift to a strip through a holomorphic extension of a real-analytic bridge between the two sides; the key Mellin-Archimedean identification $F_{\mathrm{arch}}(z)=Ξ(2z)$ ties the two constructions together. The work provides a rigorous, conditional pathway to divisor identification and showcases a general mechanism for real-zero phenomena by combining self-duality, modular splitting, and analytic rigidity, with potential applicability beyond the Riemann zeta context.

Abstract

We compare the Riemann $Ξ$--function to a canonical real-entire reference family arising from the cycle Laplacian developed in Paper I. These spectral determinants have only real zeros by self-adjointness. Our main tool is a rigidity lemma for holomorphic functions on horizontal strips. Applied to a normalized seam ratio linking $Ξ(2\cdot)$ to the reference family, this lemma shows that, under explicit holomorphy and boundary nonvanishing hypotheses verified in the forthcoming Paper III, the seam ratio extends to a zero-free holomorphic function of bounded type on each overlap strip. It follows that, on every admissible overlap strip, $Ξ(2\cdot)$ and the reference family have the same zero divisor.

The Riemann $Ξ$-function from primitive Markovian cycles II: Strip rigidity and divisor identification

TL;DR

The paper develops a strip-based rigidity framework to compare two canonical analytic constructions of a spectral object arising from a primitive dynamical model: a cycle-spectral determinant and the Archimedean Mellin side linked to the Riemann -function. By introducing the seam ratio and its normalized form , the authors prove that, under holomorphy and sector-control hypotheses, the ratio becomes a zero-free strip-unit on admissible overlap strips, forcing and to share the same zero divisor on those strips. The bridge is built via a left-strip continuation of the bilateral Laplace transform , a boundary identity on the left boundary, and a lift to a strip through a holomorphic extension of a real-analytic bridge between the two sides; the key Mellin-Archimedean identification ties the two constructions together. The work provides a rigorous, conditional pathway to divisor identification and showcases a general mechanism for real-zero phenomena by combining self-duality, modular splitting, and analytic rigidity, with potential applicability beyond the Riemann zeta context.

Abstract

We compare the Riemann --function to a canonical real-entire reference family arising from the cycle Laplacian developed in Paper I. These spectral determinants have only real zeros by self-adjointness. Our main tool is a rigidity lemma for holomorphic functions on horizontal strips. Applied to a normalized seam ratio linking to the reference family, this lemma shows that, under explicit holomorphy and boundary nonvanishing hypotheses verified in the forthcoming Paper III, the seam ratio extends to a zero-free holomorphic function of bounded type on each overlap strip. It follows that, on every admissible overlap strip, and the reference family have the same zero divisor.
Paper Structure (37 sections, 23 theorems, 114 equations)

This paper contains 37 sections, 23 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Review of the primitive dynamical model
  3. Main results
  4. Organization of the paper
  5. Inputs from Paper I
  6. Self-dual Archimedean completion and forced normalization
  7. Theta-series form and the forced self-dual scale
  8. The Archimedean completion operator
  9. The classical $\Theta$-kernel at the self-dual scale
  10. Mellin transform and the Riemann $\Xi$-function
  11. Cycle spectral determinants and the reference family
  12. Cycle spectrum
  13. Rescaled spectral map and determinant
  14. How Paper II uses $P_N$
  15. Left-strip continuation via modular splitting
  16. ...and 22 more sections

Key Result

Theorem 1.6

Assume the standing assumptions of Remark rem:standing-assumptions, so that the scaling-limit kernel $K_L$ exists and the self-dual normalization may be fixed. Then:

Theorems & Definitions (58)

  • Definition 1.3: Scaling-limit trace and completed trace kernel
  • Definition 1.4: Full centering and half-density normalization
  • Remark 1.5: Standing assumptions and existence of the scaling limit
  • Theorem 1.6: Main theorem (overview)
  • Theorem 1.7: Conditional strip-unit theorem for the normalized seam function
  • Theorem 2.1: Schoenberg--Edrei--Karlin factorization PaperI
  • Remark 2.2
  • Theorem 3.1: Theta-series form of the scaling-limit trace PaperI
  • Lemma 3.2: Forced self-dual normalization
  • proof
  • ...and 48 more