GitHub - elliotglazer/diophantine-z-q: A proof that Z is Diophantine in Q.

Version 9

This repository contains Z is Diophantine in Q, Version 9, dated November 7, 2026. It adds quartics in 523 variables with exact symbolic coefficients and 540 variables with every coefficient written out in decimal. The original large construction and the earlier quartic proofs remain in the manuscript. The new reductions were contributed by Gabriel Conant with ChatGPT.

The project uses the collective agentic notes from Stefan Baranczuk, Jose Brox, Gabe Conant, Sohail Farhangi, James Freitag, Elliot Glazer, Jennifer Park, and Caroline Terry.

Version 9 integrates the odd-product argument, smaller anchor, direct targets and asymmetric formal-depth refinement. The new inventory binds every addition and preserves the existing literature-version qualifications. Version 8 remains in the manuscript archive.

Version 7 incorporates four further audits of the Gaussian appendix and their repairs. See the audit summary for the changes and review limits.

The Version 7 full handoff includes the Version 7 public snapshot, selected research notes, and the unchanged Version 6 full handoff, which preserves the Version 5 and earlier November 4 archives. The current-only handoff omits the embedded historical archive. Neither includes a conversational transcript.

The three polynomials

For each polynomial, the input and all witnesses range over the rationals. The asserted defining property is

$$ q\in\mathbb Z \quad\Longleftrightarrow\quad \exists,\mathbf w\in\mathbb Q^{m-1},\quad P(q,\mathbf w)=0. $$

The 540-variable table contains all coefficients in decimal and occupies 708,848 bytes. The 523-variable table uses the fixed integer $S=3^{2^{26}}$, not an extra variable. Reproduction and convention checks cover both.

The original construction is specified by an expression tree. The Version 4 monomial table writes every coefficient as an ordinary integer. Its SHA-256 is 094f6a7ed17e7cc8d5d17a2f809127bbaffd0fe343cb2a32d18ac97e61279876.

The Version 6 table lists every monomial and specifies its integer coefficient using the fixed constant $S=3^{2^{27}}$. It has 55,401 coefficients independent of $S$, seven linear in $S$, and one equal to $64S^2$. Its SHA-256 is 6b798de7d00ac3c7d725ca0a69d39f1eac1ddb02fee2756cb6323770b952af32. The large coefficients are not written out digit by digit. Version 9 provides a smaller complete decimal table, while retaining this older proof baseline.

The earlier 962-variable polynomial builds on an independent reconstruction of the public agentic improvements notes, followed by changes of coordinates, smaller arithmetic circuits, and 32 certified eliminations. It is not identified with the unavailable original 1,315-variable table. Its 919 quadratic residuals and their complete sum of squares are included. The manuscript also gives the notes' degree-four lower-bound argument over the rationals. Thus degree four is necessary for a single defining polynomial; the variable and monomial counts are not claimed minimal.

Verification scope

The finite checks verify the source program, algebraic coordinate changes, projection-preserving eliminations, complete sum-of-squares expansion and coefficient statistics. Independent checks and rejection tests accompany the package. These checks do not prove the source system's arithmetic characterization of the integers.

The manuscript supplies that arithmetic argument. The Version 9 extension connects the new arithmetic arguments to the inherited proofs. The Version 7 inventory records exact theorem applications, source versions, hypotheses, conventions and translations. Version 6 proves the needed Gaussian ray-field estimate within the manuscript, following the method of Frei–Tinková without assuming their theorem. The replacement uses Mitsui's published prime-ideal estimate, Green–Tao's one-parameter results, and ordinary-prime transference.

The inventory retains source-version qualifications. In particular, the applications of Dodos–Kanellopoulos and Kollár were checked in the stated author versions of published works; their final typeset statements were not compared. This is not a claim that every cited source's final published text has been inspected.

The Version 4 inventory and the earlier 82-record inventory retain their own frozen scopes. Structural inventory checks detect stale or mismatched records; they are not proofs of mathematical theorems.

The Lean formalization is incomplete. The accepted snapshot contains 48 helper modules; the main bounded-polynomial declaration still contains sorry. AI review and successful finite checks do not constitute independent human peer review or a completed formal proof.

Earlier work and provenance

The October 11, 2026 manuscript, A conditional Diophantine definition of Z in Q from four arithmetic inputs, was produced by Sohail Farhangi with ChatGPT 6. Its PDF is preserved without modification. Its provenance and conditional scope are distinct from Version 9.

The November manuscript was produced by Elliot Glazer with Astra 6, Sol 6.1, and Opus 5.5, with auditing from Fable 5.1. The Version 5 manuscript, Version 4 manuscript, Version 3 manuscript and Version 2 manuscript remain historical references, with their separate polynomial packages and inventories. Earlier research handoffs identify their own snapshots.

The project builds on OpenAI's October 6, 2026 preprint, Hilbert's tenth problem over the rational numbers. The rank-one elliptic-curve and local-divisibility strategy goes back to Thanases Pheidas, as credited in the manuscript. The current construction also credits the collective agentic notes for its formulas and degree obstruction, and Zhi-Wei Sun for the prime-pattern strategy.

The manuscript author field is blank, and CITATION.bib intentionally omits author fields. See archival notes for citation and DOI status, and third-party notices for source attribution and redistribution scope.