A Conditional Height-Dominance Architecture for Beal's Conjecture: A Zookeeper-Style Reduction with a Power-Residue Sieve Companion (DRAFT)

Abstract

This paper formulates a conditional reduction architecture for Beal's conjecture alongside an effective power-residue sieve companion. Positioned within the Height-Dominance and Zookeeper reduction series, the manuscript approaches Beal's conjecture ($A^x + B^y = C^z$ with $\gcd(A,B,C)=1$ and exponents $\ge 3$) by separating the deterministic structural reduction from an unconditional effective sieve bound for primitive solutions. Part A introduces a direct Height-Dominance architecture modelled on the structural pattern of Zookeeper-style reductions for the Riemann hypothesis: an intrinsic cluster definition, a cancellation coordinate system, and a three-lemma endgame centred on a coercive verifier-complement HD3'. Part B combines a power-residue verifier $V_q$ with the per-row form of the Bombieri–Montgomery large sieve to establish an effective, uniform upper bound $T_K(H) \ll_{K,\varepsilon} H^{2/3}/\Sigma(Q;K)$ on primitive solutions. (i) External preprint inputs: Standard Jacobi-sum bounds (Weil 1948, Ireland & Rosen) and the classical Bombieri–Montgomery large sieve; (ii) Structural component: Power compression gives the elementary bound $\text{rad}(ABC) < H^{1/x+1/y+1/z}$, while the coercive complement HD3' acts as a local-global bridge forcing a common prime factor; (iii) Diagnostic / numerical evidence: 1000-row empirical near-miss dataset and negative-control root-coherence calibrations; (iv) Open internal bridge: Establishing the coercive local-global complement HD3' by arithmetic means remains an open problem. This version is an advanced research draft and conditional architecture. The local-global passage HD3' is stated as a conjecture; no claim of an unconditional proof of Beal's conjecture is made. Changes in Version 1.0 (July 2026) Initial preprint release. Deterministic Architecture (Part A): Formulates the Zookeeper-style Height-Dominance reduction and the coercive complement HD3' for Beal's conjecture. Sieve Companion (Part B): Derives an unconditional, effective uniform bound $T_K(H) \ll H^{2/3}/\Sigma(Q;K)$ for primitive solutions using power-residue verifiers. Diagnostic Evidence: Includes a 1000-row empirical near-miss dataset and diagnostic calibrations. DE/EN: English and German PDFs published synchronously; combined PDF generated from EN plus DE.