DOI: 10.5281/zenodo.17669807
title: On The Spectral Geometry of Coherence

description: 11 part series, presenting a conditional Hilbert-Pólya operator for the zeta spectrum, a first order quantum gravity pack on a shared cone, and a gauge and gravity unifying superalgebra whose field equations tie primes, light and screen entropy into one testable ledger.
description: The Spectral Geometry of Coherence reconstructs wave, gravity, and prime number structure from a common operator spine. The series develops a two-channel ledger on declared windows, an emergent notion of time read from phase, and a unique first-order, self-adjoint Hamiltonian whose shear is fixed experimentally by group delay. These analytic foundations (Parts 1–4) give rise to a concrete screen entropy scalar, a geometric tension functional (REG), near-horizon timing laws, and chart-covariant relativistic compatibilities.

The work will be of particular interest to curators in theoretical physics and analytic number theory for three reasons:

1. A Conditional Hilbert–Pólya Route (Parts 5 and 6)

Parts 5 and 6 specify a fully explicit Hilbert–Pólya style operator for the Riemann zeta function built directly from the ledger machinery. The operator and test class are fixed via Paley–Wiener constraints, explicit formula machinery, positivity arguments, and a one-hit spectral law for primes.The construction is supplemented by:

a temperature-fixed arithmetic thermostat derived from a Bost–Connes–type system,

a Dirac-style packaging coupling the arithmetic bath to the spectral operator,

and a superconnection that enforces the required symmetry and grading.

All analytic steps are written at theorem–corollary depth. The proof pathway is presented explicitly as conditional: it identifies a concrete operator candidate and the precise analytic checks external referees would need to confirm or refute for a full Hilbert–Pólya resolution.

2. A Superalgebra and Field Equations Unifying Screen Geometry and Arithmetic (Part 10)

Part 10 introduces a graded superalgebra accommodating both gauge fields and gravity, together with an internal arithmetic leg arising from the thermostat above. The associated Ward identities and graded differentials produce a set of “super” field equations that combine Einstein-type curvature forms, gauge slots, and arithmetic curvature into a coherent first-order pack.

The novelty is that the same ledger and capacity scalar controlling screen physics also constrain the arithmetic operators. The resulting equations are not offered as a UV-complete theory, but as a unified low-energy structure with empirically testable metrology consequences.

3. A First-Order Quantum Gravity Pack on the Shared Cone (Part 11)

Part 11 proposes a first-order quantum gravity construction driven by the same operator and ledger spine, including:

a BRST-style differential,

a Ward identity splitting fast and slow channels,

corridor laws and micro-curvature kernels measurable on screens,

and slow-geometry maps acting as potential lensing correctors.

This pack is explicitly first-order, empirically falsifiable, and designed to sit above the fixed analytic results of Parts 1–4.

Summary

The collection presents a unified operator architecture with three major pillars:

A concrete conditional Hilbert–Pólya pathway.

A superalgebra leading to unified field equations.

A first-order quantum gravity pack consistent with the same ledger.

Each pillar contains specific, testable structures suitable for specialist review.