{"id":"doi:10.5281/zenodo.18675951","kind":"article","title":"Spectral Density Depletion in Arithmetic Cayley Graphs: From the Primorial Ladder to the Davis Law of Ramanujan","authors":["Bee Rosa Davis"],"author_count":1,"year":2026,"published":"2026-02-18","venue":"Zenodo (CERN European Organization for Nuclear Research)","cited_by":0,"doi":"10.5281/zenodo.18675951","arxiv_id":null,"nct_id":null,"source_url":"https://doi.org/10.5281/zenodo.18675951","archive_url":"https://space-kitty.com/papers/doi:10.5281/zenodo.18675951","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W7130433131","pdf_url":null,"abstract":"A nine-link chain from finite graph eigenvalues to the zeros of automorphic L-functions, realizing a concrete instance of the Hilbert–Pólya program through the Langlands correspondence. The Problem The Hilbert–Pólya conjecture proposes that the nontrivial zeros of the Riemann zeta function are eigenvalues of a self-adjoint operator on some Hilbert space. Three pieces are missing: the operator, the space, and the geometry that generates it. The Discovery We construct a primorial ladder — a sequence of Cayley graphs on (ℤ/Nkℤ)* where Nk is the product of the first k primes — and subject it to a five-branch diagnostic framework spanning spectral geometry, statistical mechanics, Fisher information geometry, Hodge theory, and renormalization group flow. Seven cross-branch identities are verified to machine precision at Levels 0–2. At Level 3 (N = 2310, 480 vertices), the classical Ramanujan bound |µ| ≤ 2√(d−1) fails. Twenty-four of 480 eigenvalues exceed the bound. The mechanism is exact: generators projecting entirely onto quadratic non-residues mod 11 produce coherent constructive interference — a phenomenon we identify as spectral density depletion, the precise spectral analog of superfluid density depletion at an obstacle rim in Landau's critical velocity theory. The Corrected Bound The Davis Law of Ramanujan replaces the classical bound with a character-dependent, generator-aware threshold: |µχ| ≤ ρactive(S, χ) The corrected bound is exact: the ratio of actual spectral maximum to the Davis bound equals 1.000 across all 480 eigenvalues. The classical Ramanujan bound is recovered as the special case where spectral density is uniform — the \"Landau criterion\" of spectral graph theory. The Chain to Number Theory The paper then traces a nine-link chain from graph eigenvalues to the completed L-function of an elliptic curve: Link 1: Eigenvalues are partial L-functions [computationally verified] Link 2: The graph trace formula is the explicit formula [computationally verified] Link 3: The abelian obstruction and its diagnosis [computationally verified] Link 4: The LPS construction provides both abelian and non-abelian settings [established] Link 5: The LPS adjacency operator is a Hecke operator [established] Link 6: Jacquet–Langlands lifts to GL(2) [established] Link 7: The adelic limit, governed by arithmetic coherence length Carith = q²/p [addressed] Link 8: The archimedean completion [computationally verified] Link 9: Functional equation verification [computationally verified] The quaternion algebra B ramified at {11, ∞} forces every archimedean parameter with no free choices: conductor N = 11, weight k = 2, gamma factors Γℝ(s) · Γℝ(s+1), root number ε = +1. Since dim S2(Γ0(11))new = 1, the newform is uniquely f = η(τ)²η(11τ)² — the elliptic curve 11a1. The completed L-function Λ(s, f) matches the LMFDB to eight digits. A 158-factor partial Euler product from primes p ≤ 4973 converges to six decimal places. Computational Verification Exact character tables via Burnside class algebra for PSL(2, Fq) at q = 3, 7, 11, 13, 19 Cuspidal Hecke eigenvalue extraction through isotypic decomposition (match error < 10−13) GUE statistics confirmed in the non-abelian setting (Casimir collapse) Seven cross-branch identities verified to machine precision All code available in the companion repository What This Is — and What It Is Not This is not a proof of the Riemann Hypothesis. It is a research program with verified computational foundations, a precise failure mechanism, a new diagnostic law, and a complete chain connecting finite spectral data to the zeros of automorphic L-functions. The L-function is complete at every place of ℚ: finite places from graph eigenvalues, the ramified place from the Steinberg representation, and the infinite place from the quaternion algebra. 34 pages · 16 sections · 10 figures · 20 tablesPreprint, February 2026 This paper extends the Davis Field Equations framework developed in The Geometry of Sameness (Davis, 2024) and is part of the Geometric Intelligence Series.","abstract_license":"CC-BY (publisher license, via OpenAlex)","license":"cc-by","is_open_access":true,"retracted":false,"peer_reviewed":null,"categories":["Analytic Number Theory Research"],"details":null,"provenance":"openalex","attribution":"Abstract of \"Spectral Density Depletion in Arithmetic Cayley Graphs: From the Primorial Ladder to the Davis Law of Ramanujan\" by Bee Rosa Davis, https://doi.org/10.5281/zenodo.18675951, used under its Creative Commons Attribution license (CC-BY); markup removed, otherwise unmodified.","topics":[{"slug":"landmark-math-results","name":"Landmark Results in Pure Mathematics","domain_slug":"math-cs"}],"cited_in_dossiers":[]}