Erdős–Straus Project Archive: Residual Divisor Shells, Quotient Surpluses, and Finite Verification

Abstract

This record gathers the Erdős–Straus project under its existing concept DOI. The principal paper develops the residual divisor-shell equivalence and its origin counts, proves the Star–Kneser quotient-surplus theorem, and records the exact certificate at p = 12289 with R = 11, m = (3,6,3), and surplus profile (1,6,2,1). It also treats the divisor-complement involution, Wick-completed support coordinates, the mod-107 congruence package, coefficient-space circle inversion, and the Fable–Circle–Sarnak identities. The accompanying archives contain the complete LaTeX source, a research logbook, replayed symbolic and finite checks, a pinned Lean 4 development for the formalized core, earlier versions and drafts with a correction guide, and a reviewed working corpus. DOI_LINEAGE.csv records the earlier Erdős–Straus records and related project lineages. The full Erdős–Straus conjecture remains open. The divisor involution is proved; a cyclic action on a larger completed arithmetic envelope remains unresolved. Historical drafts that made stronger claims are retained with corrections. Current edition, 4 August 2026: the principal reader is 57 pages. This edition extends the quotient-to-blade bridge through the arithmetic carrier, completed tessarine and sedenion coordinates, the half-rapidity divisor coordinate, shellwise involution reduction, and rigidified denominator towers with exact Frobenius transitions. Begin with 00_ERDOS_STRAUSS_Project_Reader.pdf.