{
 "what": "Polynomial maps two 2026 papers print, decided in exact rational arithmetic: Gao's counterexamples to the Jacobian conjecture (constant Jacobian determinant expanded symbolically, non-injectivity by two rational points with one image) and the weak Markus–Yamabe fields ((JX + I)^N = 0 as a polynomial identity, distinct zeros by evaluation).",
 "generated": "2026-10-05",
 "seconds": 90.1,
 "sources": {
  "gao": {
   "arxiv": "2608.00222v1",
   "title": "Counterexamples to the Jacobian conjecture in dimensions greater than two",
   "author": "Shuhong Gao",
   "license": "CC BY 4.0",
   "pdfSha256": "483208235e32ae69aaf83832916451707c78aa5a4410486c860950bd3eb6860f",
   "tex": "gao-2608.00222.tex",
   "aiDisclosure": "\"AI disclosure: The main idea and framework are due to the author, and Claude Fable 5 assisted in the proofs and in the writing up of the paper.\" (title footnote)",
   "checkingStatement": "\"All polynomial identities asserted below ... were verified by exact rational arithmetic; the verification scripts are available from the author.\" (p. 2); pp. 24 and 28 promise ancillary files with F6 and F7, which arXiv does not carry"
  },
  "chv": {
   "arxiv": "2608.05392v1",
   "title": "The weak Markus–Yamabe conjecture fails in dimension 14",
   "authors": "Álvaro Castañeda, Gerardo Honorato, Francisco Valenzuela-Henríquez",
   "license": "CC BY 4.0",
   "pdfSha256": "5a7b6ecf19a0c3e01cb231de12b4936c50158eb813823f510dcd9828f62117ab",
   "tex": "chv-2608.05392.tex",
   "provenance": "Phi (Prop 6.2) equals, all 11 components, the map of gist.github.com/Spacerat/08b4a43f6b6ca57178efabc220170ce8 (revision 2224dace of 2026-07-20, its only revision; the file 11varcubicjacobiancounter.py is git blob 1f974e34), which says: \"The explicit 11-variable construction, simplification, and verification code in this file were generated by ChatGPT (OpenAI).\" The appendix checks the 18-dimensional unipotency \"at a sample rational point\".",
   "gist": {
    "url": "https://gist.github.com/Spacerat/08b4a43f6b6ca57178efabc220170ce8",
    "owner": "Spacerat",
    "revision": "2224dace71e8763a8621a7f557bbc545a53aa820",
    "revisionDate": "2026-07-20T06:58:41Z",
    "revisions": 1,
    "file": "11varcubicjacobiancounter.py",
    "fileBlob": "1f974e34b1d42f09efd1df83db11375b68177f6c",
    "fileSha256": "06ed27d398dc7017e679cfcd1d10840cb3495829dc2ccfb9bb2fe441a4ec2d36",
    "fileBytes": 13916,
    "stored": false,
    "checked": "2026-10-05: the revision from api.github.com/gists/08b4a43f6b6ca57178efabc220170ce8 (history holds one entry); the blob is `git hash-object` of the file bytes and is the id in the gist raw URL. Until 2026-10-05 this record called the blob the revision."
   }
  }
 },
 "code": {
  "decide.py": "c9f9cfac98bc4a4c00182ec3adc68bdf94748bfc7ed7c42eb9c17f7a0f4f9136",
  "poly.py": "3a9ec0288bf8cdd6a4c54abd148f81303d895e783b2589dd49285d63b800db47"
 },
 "byVerdict": {
  "CERTIFIED": 7,
  "PARTIAL": 1
 },
 "rows": [
  {
   "id": "gao-g",
   "claim": "G: a polynomial map of C^3 with det J = 2 that is not injective (a counterexample to the Jacobian conjecture), printed in full",
   "source": "arXiv 2608.00222, Thm 3.5",
   "verdict": "CERTIFIED",
   "checks": [
    {
     "name": "det JF is identically the printed constant 2",
     "ok": true,
     "detail": "constant 2"
    },
    {
     "name": "component degrees are the printed [4, 11, 12]",
     "ok": true,
     "detail": "[4, 11, 12]"
    },
    {
     "name": "2 distinct rational points with one image: F is not injective",
     "ok": true,
     "detail": "image (1, 5/4, 0)"
    }
   ],
   "terms": [
    3,
    14,
    16
   ]
  },
  {
   "id": "gao-f4",
   "claim": "F4: a polynomial map of C^4 with det J = -44/9 that is not injective (a counterexample to the Jacobian conjecture), printed in full",
   "source": "arXiv 2608.00222, Thm 4.3",
   "verdict": "CERTIFIED",
   "checks": [
    {
     "name": "det JF is identically the printed constant -44/9",
     "ok": true,
     "detail": "constant -44/9"
    },
    {
     "name": "component degrees are the printed [4, 11, 12, 21]",
     "ok": true,
     "detail": "[4, 11, 12, 21]"
    },
    {
     "name": "3 distinct rational points with one image: F is not injective",
     "ok": true,
     "detail": "image (1, -1289/270, -113/90, 1453/90)"
    }
   ],
   "terms": [
    3,
    32,
    39,
    114
   ]
  },
  {
   "id": "gao-f5",
   "claim": "F5: a polynomial map of C^4 with det J = 160/29 that is not injective (a counterexample to the Jacobian conjecture), printed in full",
   "source": "arXiv 2608.00222, Thm 4.4",
   "verdict": "CERTIFIED",
   "checks": [
    {
     "name": "det JF is identically the printed constant 160/29",
     "ok": true,
     "detail": "constant 160/29"
    },
    {
     "name": "component degrees are the printed [3, 12, 14, 16]",
     "ok": true,
     "detail": "[3, 12, 14, 16]"
    },
    {
     "name": "2 distinct rational points with one image: F is not injective",
     "ok": true,
     "detail": "image (1, -89/116, -605/116, -109/58)"
    }
   ],
   "terms": [
    2,
    24,
    34,
    54
   ]
  },
  {
   "id": "gao-f6",
   "claim": "F6: the map the printed construction defines (n = 5), det J = -290, not injective",
   "source": "arXiv 2608.00222, Thm 4.5 (the full map is not printed; the promised ancillary files are not on arXiv)",
   "verdict": "CERTIFIED",
   "checks": [
    {
     "name": "the rebuild reproduces the printed X_1..X_4 and F6,1",
     "ok": true
    },
    {
     "name": "det J(S) = gamma and the stage determinant is -290",
     "ok": true
    },
    {
     "name": "rebuilt sizes",
     "ok": true,
     "detail": "terms [6, 342, 421, 507, 904], degrees [7, 38, 40, 42, 44]"
    },
    {
     "name": "det JF is identically the printed constant -290",
     "ok": true,
     "detail": "constant -290"
    },
    {
     "name": "component degrees are the printed [7, 38, 40, 42, 44]",
     "ok": true,
     "detail": "[7, 38, 40, 42, 44]"
    },
    {
     "name": "2 distinct rational points with one image: F is not injective",
     "ok": true,
     "detail": "image (1, 1, 2, -508, -660)"
    }
   ],
   "terms": [
    6,
    342,
    421,
    507,
    904
   ]
  },
  {
   "id": "gao-f7",
   "claim": "F7: the map the printed construction defines (n = 5), det J = 119377, not injective",
   "source": "arXiv 2608.00222, Thm 4.13",
   "verdict": "PARTIAL",
   "checks": [
    {
     "name": "the rebuild reproduces the printed F7,1",
     "ok": true
    },
    {
     "name": "rebuilt sizes",
     "ok": true,
     "detail": "terms [7, 14445, 17727, 21410, 25518], degrees [7, 86, 89, 92, 95]"
    },
    {
     "name": "the factors' determinants: det J(S) = gamma, the stage 119377, det J_(w2,w3)(H3, H4) = 1",
     "ok": true
    },
    {
     "name": "2 distinct rational points with one image: F7 is not injective",
     "ok": true
    },
    {
     "name": "det J(F7) constant: rests on the paper's factorization lemma (direct expansion of 14,000-26,000-term rows is out of reach here)",
     "ok": null
    }
   ],
   "terms": [
    7,
    14445,
    17727,
    21410,
    25518
   ]
  },
  {
   "id": "chv-x14",
   "claim": "X on R^14 (degree 7): JX has spectrum {-1} at every point and X has three zeros, so the weak Markus–Yamabe conjecture fails in dimension 14",
   "source": "arXiv 2608.05392, Thm 5.1",
   "verdict": "CERTIFIED",
   "checks": [
    {
     "name": "(JX + I)^k = 0 identically (k = 14 <= 14): every eigenvalue of JX is -1 at every point",
     "ok": true
    },
    {
     "name": "3 distinct zeros of X",
     "ok": true
    }
   ],
   "nilpotencyIndex": 14
  },
  {
   "id": "chv-phi",
   "claim": "Phi, a cubic map of C^11 with det J = -2 that is not injective (the gist that first published it says ChatGPT generated it)",
   "source": "arXiv 2608.05392, Prop 6.2",
   "verdict": "CERTIFIED",
   "checks": [
    {
     "name": "det J Phi is identically -2 (11 x 11, expanded directly)",
     "ok": true
    },
    {
     "name": "three distinct points with one image",
     "ok": true
    }
   ]
  },
  {
   "id": "chv-xhat18",
   "claim": "X-hat on R^18 (degree 3): spectrum {-1} everywhere, three zeros — decided as a symbolic identity where the paper's appendix checks one sample point",
   "source": "arXiv 2608.05392, Thm 8.1",
   "verdict": "CERTIFIED",
   "checks": [
    {
     "name": "(JX + I)^k = 0 identically (k = 17 <= 18): every eigenvalue of JX is -1 at every point",
     "ok": true
    },
    {
     "name": "3 distinct zeros of X",
     "ok": true
    }
   ],
   "nilpotencyIndex": 17
  }
 ]
}