Character Stripping, Deterministic Square-Function, and the Precise Localization of H′
ZFCρ Paper L

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

Abstract

We precisely localize the remaining difficulty for Hypothesis H′ in the ZFCρ framework. The 94% positive bias of η(p) is identified as an arithmetic masquerade from p mod 12 structure, with an a priori derivation from the {2,3}-cost basis of integer complexity, and removable by character stripping. Four candidate proof routes are rigorously excluded. The stripped dyadic block sums B_j decay at approximately the white-noise rate |B_j| ≈ C · 2^−j/2 — exponentially faster than the provable bound |B_j| ≤ C/√j — and this polynomial-to-exponential gap is formalized as the deterministic square-function hypothesis DSF_α. The Tauberian condition (slowly decreasing) is proved unconditionally from the Guy upper bound and Chebyshev's prime count. The prime-power layer converges absolutely in a neighborhood of Re(s) = 1, making it analytic past the boundary and non-critical for the prime-layer singularity structure. The full closure chain reduces to two conditions: DSF_α (block regularity) and BL (block localization summability). Twenty-four unconditional results at scales N = 10⁷, 10⁹, 10¹⁰ support the localization.

Paper 49 identified two open problems for H′; the present paper transforms them from vague targets into precisely stated, numerically verified, formally testable mathematical objects.

Keywords: integer complexity, character stripping, deterministic square-function, DSF, dyadic block sums, Tauberian, ZFCrho, H-prime, prime deficiency, p mod 12