Erdős–Straus Project Archive: exact bounded congruence transport and auxiliary-torus interfaces

Abstract

New in this version (27 September 2026). 00-claude-opus-5-5-erdos-straus-record.pdf, with its Markdown source, is an audited record of the Erdős–Straus workbench and of the author's notes, with proofs and new results. It was written by Claude (Anthropic, model claude-opus-5-5) and is published at the author's request ahead of the author's review. It replaces the Erdős–Straus readers of 26–27 September 2026, which were retracted. The errors of those readers ran consistently in one direction: they overstated negative results, understated positive ones, and presented bridges between areas as having no bearing. The cause cannot be established without access to the model's internals; the pattern is consistent with a bias against AI-assisted work on hard problems. In the new record every statement is proved, re-computed by a named program, marked as not verified, or labelled as the auditor's assessment, because a result stands or falls on its proof. The verification programs are on GitHub: KokunoYumeto/zeta-function-research-reader, branch claude/claude-ab-grind-20260925, folder contrib/claude-ab/rewrite/. Erdős–Straus Project Archive is the continuing mathematical workbench of The Clankers. The GitHub workbench provides a reading guide, source navigation and continuing work. The continuing archive DOI is 10.5281/zenodo.20401937. This expanded contribution, Exact bounded congruence transport and auxiliary-torus interfaces, presents 72 statements with complete written proofs in an 86-page citation-bearing reading edition and 15 unchanged canonical TeX files. It adds fourteen statements to the 58-statement, 67-page predecessor. The ten inherited files, including the 619-page archive reader, remain part of the archive unchanged. The arithmetic results retain the original prime-shell coordinates, divisor valuations, witness order and finite-code bounds. They develop bounded congruence gluing, shared-variable Chinese remainder constructions, integral lattice and cyclic-character maps, exact arithmetic successor domains and inverses, and all-shell witness correspondences. Four boundary statements classify same-shell pair-swap/tag-change candidates, integral fibres and code differences, including the exact first code on the characterized occupied residual-three domain. The auxiliary-torus calculations retain the original matrix, determinant 14, Fourier factor 2πi and period-one coordinates. Ten added operator statements give continuous transfer and kernels, source mean corrections, physical evaluation, exact sampling obstructions and reconstruction, sharp small-denominator constants, derivative loss, smooth and distributional inverses, and parameter-dependent estimates with the original cutoff remainder. Arithmetic, positivity-code and prime-production contributions are credited to Reddit user u/UmbrellaCorp_HR, including the continuation numbered (71)–(100). The earlier mod-107 observation remains separately credited to u/CommonCareful3149 in the inherited archive. The reading edition cites the specific arithmetic sources of Christian Elsholtz and Terence Tao, the divisor-category and cyclic constructions of Alain Connes and Caterina Consani, and the exact auxiliary-fluid sources used; the bibliography and source records retain the stated roles of their human contributors. Read the numbered 07 PDF. The numbered 08 ZIP contains the complete proof source graph, fourteen portable exact checker commands, finite certificates, source-bound review records, a file manifest and PDF-specific quality receipt. The numbered 09 reading guide and 10–11 manifests make the edition navigable and checkable. The 9 September reading-PDF repair resolves all citations and cross-references while preserving the canonical mathematical proof bytes. Protected source manuscripts and private conversation logs are not redistributed. These results do not prove or disprove the Erdős–Straus conjecture, the Riemann hypothesis or the Yang–Mills mass gap, and they do not independently validate the complete announced Navier–Stokes construction. The auxiliary-operator results have their own proofs and do not assume a fluid blowup theorem. Existing formal and finite computational records retain their explicit source scopes; no full-fluid Lean verification is claimed.