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

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 (ap^bq=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.