The Conductor-21 L-Function: Certified Zeros, Exact Laws, and a Density–Prime Operator
Year: 2026
Type: preprint
Abstract: We study one explicit L-function — the degree-8 Artin/Hecke L-function L(s, χ₈) of PSL(2,7) at conductor 21¹⁰ — and the rigid six-member family it belongs to. The paper proves the abelian reduction RH(χ₈) ⟺ GRH(ψ) for an explicit cubic Hecke character ψ (no automorphy hypothesis), exhibits χ₈ by an explicit 8×8 monomial realization, and certifies its low zeros: 153 ball-certified bracket locations on [0, 27.92] with certified one-sided Turing counts, plus a derivative-level interval theorem. An arithmetic-free finite-section model is solved exactly — value laws, a Γ/Mosco scaling limit with certified rate, and an Airy caustic envelope theorem with arb-certified constants, sharpened at the band edge to a derived edge law. An operator H = D + V is then built from density and primes with the zeros as output only; it reconstructs the spectrum, passes object-identifying controls, is provably positive at finite range (on the certified spectrum inputs), and localizes the one open seam — equivalent to GRH(ψ), the reduction displayed as a conditional Hilbert–Pólya bridge (Theorem W). Every load-bearing statement carries an explicit grade; certificate-grade claims ship with machine-checkable certificates, and refutations are reported with the same prominence as theorems. The wall is instrumented, not crossed.
Authors: Stenberg, Selina, Balashov, Ilya
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203615407
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