Spectral Rigidity and Sobolev Norm Bounds for the Riemann Zeta Function via Fano Graphon Refinements
Year: 2026
Type: preprint
Abstract: We propose a geometric confinement mechanism for the non-trivial zeros of the Riemann zeta function $\mathbf{\zeta(s)}$ by embedding its local arithmetic symmetries into the Szegedy-Lovasz graphon limit of the projective plane $PG(2,q)$ over finite fields $\mathbb{F}_q$. By explicitly calculating the exact singular spectrum of the iterated tensor refinements of the incidence matrix over $PG(2,q)$, we demonstrate a strict separation between the trivial Perron-Frobenius eigenvalue and the universal non-trivial spectral radius $\lambda = \sqrt{q}$. Formulating a Strengthened Trace Equivalence Conjecture between the continuous logarithmic phase space of $\mathbf{\zeta(s)}$ and the cylindrical state space of the graphon limit, we construct a proto-unitary isomorphism $U_0$ on a dense subspace and characterize the scale-absorption functional map $\Phi$ via Sobolev trace bounds. We show that within this framework, the amplitude of continuous spectral modes is rigidly pinned to the radial geometry $R(x) = \sqrt{x}$, yielding the stationary spectral condition $\mathrm{Re}(s) = 1/2$ along the critical line. A comprehensive cylinder stress test validates the analytical predictions for all perturbations, and a uniqueness theorem based on the characteristic polynomial of the Fano plane demonstrates that the projective structure is the unique geometric realization of the TEC-Weil framework. 🌐 Live SPARQL Semantic Queries & Knowledge Graph Architecture This research deposit is integrated with the Blandino Research Knowledge Graph (BR-KG). Direct SPARQL query interfaces and machine-readable RDF endpoints are available below for automated data extraction and semantic interoperability: Live SPARQL Query: Full Micro-Node Graph Exploration (Q1–Q72) — Dynamic table query extracting all 72 indexed nodes, properties (P1–P12), and verification suite relationships. Interactive 2D Knowledge Graph Visualizer — Real-time 2D network rendering of cluster dependencies and topological factorizations. Live SPARQL Query: Linked Open Data & External DOI Cross-Mappings — Cross-reference mappings between internal nodes (Q3–Q36), Zenodo DOIs, FAIRsharing Record 8936, and Wikidata URIs. Public REST API Endpoint (RDF/JSON-LD) — Standardized endpoint for programmatic query execution.
Author: Blandino, Massimiliano
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203781657
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