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, the complete pinned Lean 4 development with an extracted-copy build receipt, a searchable locator for every public local-file reference, earlier 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. Among permutations of the three origin labels, the stabilizer of (3,6,3) is the divisor-complement C2. An order-three arithmetic permutation cycling the origin fibres requires equal fibre cardinalities and therefore does not exist for the 12289 certificate. The proved C3 action cycles the ambient count-vector orbit and rotates the added coordinate of the product cover while preserving every arithmetic origin label. Whether a natural non-product arithmetic enlargement carries a genuine origin-cycling C3 action is not resolved here. Current edition, 5 August 2026: the principal reader is 237 pages. Adds the exact six-element coupling theorem comparing the commuting C6 and reflected D3 laws, the eighteen-state two-sheet phase group with diagonal C6 and antidiagonal D3 subgroups, the singular 12289 count-orbit D3 action, and the sixty-four-cell completed sign-sedenion algebra; it also repairs the verification archive with its complete three-module Lean tree, a clean extracted build, and a locator for every public artifact reference. Begin with 00_ERDOS_STRAUSS_Project_Reader.pdf.