What question did this study set out to answer?

The aim is to analyze Beal's Conjecture using a framework based on harmonic coherence and entropy.

March 29, 2026Open Access

Beal's Conjecture: Toward an Entropy-Minimization Resolution via Harmonic Coherence and Hanners Theorem

Key Points

The aim is to analyze Beal's Conjecture using a framework based on harmonic coherence and entropy.
Defined an entropy functional over normalized terms of the Diophantine equation.
Conducted a structural analysis within the harmonic coherence framework.
Assumed a spectral gap (Assumption A3) to derive implications for solutions.
Performed 34 computational tests across large integer domains.
Demonstrated that every solution deviates from entropy equilibrium.
Established a quantifiable lower bound for coprime bases.
Concluded the non-existence of coprime solutions under stated assumptions.

Abstract

A conditional structural analysis of Beal's Conjecture within the Harmonic Coherence (HC) framework. We define an entropy functional over the normalized terms of the exponential Diophantine equation Ax + By = Cz and show that every solution is permanently displaced from the entropy equilibrium, with a quantifiable lower bound for coprime bases. Under Assumption A3 (spectral gap), this displacement implies non-existence of coprime solutions. The result is supported by 34 computational tests (all PASS) over large integer domains. Companion documents: • Contextual Entropy Reduction Theorem • Canonical Reconciliation • HC Bridge Note • Fixed-Point Convergence Theorem • Paper A: Transformer Distillation • Paper B: GW Kerr Ringdown • Paper C: HC Bridge Synthesis

Beal's Conjecture: Toward an Entropy-Minimization Resolution via Harmonic Coherence and Hanners Theorem

Key Points

Abstract

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