December 5, 2025Open Access

A Mean Min--Max Lemma for the Equation a+b=c with Absolute Constants and Applications

Key Points

Analysis reveals the importance of energy control in Diophantine equations, enhancing theoretical foundations further.
Key evidence includes operational details for content–height mean principle and proofs of the Beal corollary.
Technical supplement provides insights into dyadic stratification and global gain with relevant equations for reproducibility.
Findings support the utility of spectral methods in advancing the comprehension of Diophantine equations and their applications.

Abstract

Technical supplement to ‘A Mean Min–Max Lemma for a+b=c with Absolute Constants and Applications’. Consolidates full proofs and operational details for the content–height mean principle and the Beal corollary (apᵇq=cr, p, q, r≥3). Appendix A standardizes absolute values, weights, and the product formula; B establishes the exact Poincaré identity on K3 and energy/gap control; C builds the dyadic stratification, block weights, and global gain; D bridges heights and radical with comparison constant αcmp (K) and slack δcmp; E closes the coprime Beal case via finite window and local elimination (congruences, LTE). Includes parameter table, dependency graph, seeds, logs, and reproducibility notes. Keywords: Diophantine equations; content–height; Szpiro/ABC; Beal; Arakelov geometry; spectral methods; LTE; Zsigmondy; reproducible research.

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