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

The Erdős–Straus Project Archive preserves complete mathematical readers, proof sources and reproducible finite computations. This 8 September 2026 edition adds Exact bounded congruence gluing and arithmetic successors for Erdős–Straus shells: a 67-page contribution with 12 proof files and 58 named statements. The new contribution is the default PDF preview. The five files of the preceding archive edition remain available unchanged. The contribution develops bounded congruence transport, valuation coordinates, shared-variable CRT, integral lattice and cyclic-character maps, divisor-shell incidence, and complete ordered-witness reconstructions. It includes corrected all-shell witness fibres and exact finite-character transport for the auxiliary torus matrix with entries (3,1;1,5). Complete definitions, hypotheses, domains, codomains, coordinate calculations and proofs appear in the PDF and TeX sources. The supplementary source archive supplies portable exact checkers, their finite certificates, the statement-to-proof crosswalk, repair findings and source attribution. The original ordered-witness comparison through p ≤ 5000 checks 186,882 shells and 16,815,504 residual divisors. These computations have the stated finite scope. The contribution does not establish universal Erdős–Straus positivity, an empty prime shell system, or a Navier–Stokes existence or blowup theorem. Ongoing extensions are recorded in later editions when their proofs and artifacts are ready. Cite this edition using 10.5281/zenodo.22666493. The continuing project reference is 10.5281/zenodo.20401937. The supplementary reader carries both references on its first page. Preserved archive: 31 August 2026 edition. The following description documents the inherited 619-page reader and its original companion archive. This is the corrected 31 August 2026 successor in the existing Erdős–Straus Project Archive concept-DOI chain. It replaces no earlier record: the immutable predecessor remains available as version 2026.08.30.1. The reader and deterministic source/provenance archive carry the exact definitions, maps, calculations, standard proofs, Python certificates, Lean sources and compiled objects, literature crosswalks, and machine-readable state for this checkpoint. The edition corrects the finite post-167 candidate census. On the exact domain of fresh primes q<291, q=3 (mod 4), and gcd(q,L_167)=1, there are 21 candidates and 210 strict pairwise product comparisons. Q=59 ranks first and Q=239 second. The retained R=239 theorem is therefore explicitly a historically chosen non-greedy continuation, not an optimality theorem. At its proved finite scope, the R=239 carrier computation has 192 exact gcd-profile fibres, 384 tagged channels, 34,445 constructive CRT/positive-denominator witnesses, 20,524,419,041,579,843 forced one-step lifts, and 67,689,004,732,642,841 residue-unforced lifts. The exact post-R=239 refresh of the inherited R=167 coefficient newly forces 21,618,387,592,200 branch-1 rows and 17,407,298,355,642 branch-25 rows, or 39,025,685,947,842 total, leaving exactly 67,649,979,046,694,999 survivors. The refreshed cumulative forced count is 2,757,919,929,517,785,001 of 2,825,569,908,564,480,000. The corrected candidate census is sealed by a 45-probe Lean module. A separate 28-probe Lean module seals the post-R=239/R=167 coefficient refresh. The archive contains exactly those two current compiled .olean objects, their generated sources, and their complete raw, sealed, and independent validation provenance, alongside separate standard proofs and Python certificates. The R=107 arithmetic-progression lead remains credited to Reddit user u/CommonCareful3149. The project reconstructs, checks, and extends the supplied observation without a priority claim and without silently treating the contributor's unreceived two-page note as proof. The principal reader has 619 pages. The Erdős–Straus conjecture remains open. Finite carrier survivors are not counterexamples, shell-emptiness claims, density theorems, or endpoint claims; the finite candidate ranking is not an unbounded optimality theorem.