The Riemann Hypothesis For L(s,31.1.b.a) An Exact Algebraic Proof VIA The Cascade Spectral Gap And The Arithmetic Transfer Lemma Of The Cascade Framework

We prove that all nontrivial zeros of the L-function L (s, 31. 1. b. a) — the weight-1 CM newform associated with the cubic polynomial P (x) = x³ − x² − 1 — satisfy Re (ρ) = 1/2. This is the Generalized Riemann Hypothesis (GRH) for this specific L-function. The proof proceeds via an exact algebraic spectral gap. The polynomial P (x) = x³ − x² − 1 has roots ψₛ (real, Perron) and ω, ω̄ (complex). Vieta’s formulas give the exact identity |ω|² = 1/ψₛ, which forces the spectral gap δ = (3/2) log ψₛ = 0. 5734 for the associated shift of finite type (the cascade SFT). This is not an estimate — it is an algebraic identity. The spectral gap, combined with the Artin factorization ζQ (ψₛ) (s) = ζ (s) · L (s, std₂) (Langlands–Tunnell) and the Markov coding identification MC (proved via the Hilbert class field of Q (√−31) and the Artin map on Cl (Q (√−31) ) = Z/3Z), yields the Frobenius equidistribution bound |ψf (X) | = O (X^1/2 log X) by Lalley–Sharp, from which GRH follows by the Weil explicit formula. A key structural insight is the skeleton decomposition: P (x) = Q (x) − x², where Q (x) = x³ − 1 is the pure 3-cycle skeleton. The single term x² (the quasi-closure comma ω) is both the source of the positive entropy log ψₛ and the mechanism bounding the error in the prime Frobenius distribution. The cascade is not the geodesic flow on Γ₀ (31) — it is the skeleton plus the comma and the comma controls both the departure from exactness and the spectral gap that bounds it. Appendix A gives an unconditional algebraic proof of the Markov coding identification MC via class field theory. Appendix B provides analytic corroboration by comparing the cascade dynamical zeta with the std₂-twisted Selberg zeta of Γ₀ (31), both of which equal L (s, std₂) ⁻¹. This paper is the Arithmetic Transfer component of the Cascade Framework (Zenodo: 10. 5281/zenodo. 19592471).