The UAP_Ω Theorem, Slow Decrease, Cyclotomic Discount Rates, and the Two Final Gaps for H′
ZFCρ Paper XLIX

Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.

Abstract

Paper 48 reduced H′ to two core open problems (prime-layer cancellation, UBPD) and three weak auxiliary inputs. This paper advances all five directions.

First, UAP_Ω^{SW} (Ω-shell-in-AP equidistribution in the Siegel-Walfisz range) is derived from Gafni-Robles 2025 AP Selberg-Delange expansion. The three auxiliary inputs become theorems. Coprime-shell numerical verification: ratio ≈ 1.00 ± 0.01.

Second, the Slow Decrease Theorem is proved: A(X) = Σ_{p≤e^X} η(p)/p satisfies one-sided bounded decrease (A(X+h)-A(X) ≥ -Ch). This eliminates one of two Tauberian conditions.

Third, the prime-layer gap is precisely identified: not slow decrease (proved), but the boundary pseudofunction behavior of (D(1+s)-C)/s. A natural and sufficient intermediate target is Hölder continuity of D(1+it) at t=0. The ℓ²/H² route is fully assessed and shown insufficient.

Fourth, the Cyclotomic Discount Rate CDR(p,d) = ρ_E(Φ_d(p))/(λ·log Φ_d(p)) ≈ 0.97 (Zsigmondy explanation) establishes the complete physical mechanism for UBPD.

Fifth, the Restricted Quasi-Super-Additivity (RQSA) hypothesis is proposed. H′'s two core open problems now have precise mathematical targets: the direct Tauberian input for prime-layer is (D-C)/s boundary pseudofunction behavior; UBPD reduces to RQSA + quotient rigidity + benchmark-split rigidity.

Keywords: integer complexity, UAP_Ω, slow decrease, Hölder continuity, cyclotomic discount rate, Zsigmondy, RQSA, UBPD, H-prime closure, ZFCrho