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. This work supersedes and corrects Kokuno Yumeto, Miroku Akagi, and Maya Sakuyah, The Z3 Center Symmetry: Arithmetic Lattices, Automorphic Spectral Triples and QCD, 10.5281/zenodo.16936041. Its usable three-sector and modular constructions are retained as source material; the present paper replaces the earlier unsupported global identifications with the exact support-indexed statements and stated hypotheses proved here. 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, 5 August 2026: the principal reader is 206 pages. This edition adds the research-logbook provenance for the finite regular-simplex Tomita sector and its explanatory support diagram. It records the exact synthesis-map kernel, support corner, faithful trace state, trivial modular flow, central density generator, visual audit, structural boundary, and checker agreement for dimensions two through nine. Begin with 00_ERDOS_STRAUSS_Project_Reader.pdf.