We present an arithmetic support framework detailing the structural coherence of Diophantine equations and motives over hyper-elliptic spaces. Operating within the Universal Relational-Geometric Coherence Law (URCL) framework, we map the stability fields of generalized Frey curves onto a parameterized family of non-local arithmetic operators modulated by the synchopeshing trace-map recurrence. We analyze the algebraic tracking behavior of Hecke eigenvalue sequences under the dominant eigenvalue condition λ_dom = φ > 1, where φ represents the golden ratio fixed point. By applying uniform irreducibility parameters and Ribet's level-lowering theorem, we demonstrate that hypothetical non-coprime configurations or unconfined rational cycles introduce non-subexponential growth trajectories that break the modularity of the associated Galois representations. This mathematical behavior establishes a strict, non-perturbative containment boundary across the conductor space, mapping out the structural bounds of the Beal, ABC, and Hodge conjectures within scale space. This integration connects number-theoretic stability bounds directly to the geometric protection mechanisms of the URCL. Pipeline Disclosure: The core conceptual translation—substituting direct multi-conjecture unified proof assertions with the classical frameworks of arithmetic geometry, Frey curves, Ribet's level-lowering, and motivic cohomology stability bounds—was fully designed and authorized by the author. Initial layout organized via Grok (xAI); rigorous mathematical validation, motivic filtration alignment, and production-ready LaTeX typesetting finalized via Gemini (Google).