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 and records the exact certificate p = 12289, R = 11, m = (3,6,3). It proves the Star–Kneser quotient-surplus theorem, distinguishing the two-target surplus profile (1,6,2,1) from the three-origin profile (2,7,2,1). It records Space(5) = 12289 and congruences linking (p*, R) = (8803369, 107) to S(3) and S(5); p* is not a Busy-Beaver extremum. 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 synthetic D3 symmetry and divisor involution are proved. An arithmetic C3 action is conditional and cannot act transitively on the 12289 certificate set; a cyclic action on a larger completed envelope remains unresolved. Historical drafts that made stronger claims are retained with corrections. Current edition, 5 August 2026: the principal reader is 233 pages. Clarifies the exact arithmetic-versus-ambient cyclic symmetry at p=12289. The origin-resolved certificate set has only the divisor-complement C2 stabilizer, while cyclic coordinate permutation acts on the ambient count orbit and the proved product-cover action rotates an independent sheet coordinate. Begin with 00_ERDOS_STRAUSS_Project_Reader.pdf.