The Conductor-21 L-Function: Certified Zeros, Exact Laws, and a Density–Prime Operator

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 records the abelian reduction RH(χ₈) ⟺ GRH(ψ), for an explicit cubic Hecke character ψ, by Artin induction (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 establishing completeness below T = 6.55, 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, identified through the Weil/Li positivity dictionary with GRH(ψ) and 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 — the open seam GRH(ψ) — is instrumented, not crossed.