{
  "id": "2026-07-29-maxwell-conjecture",
  "slug": "maxwell-conjecture",
  "recordState": "active",
  "canonicalUrl": "https://stateofproof.org/dockets/maxwell-conjecture",
  "url": "https://stateofproof.org/dockets/maxwell-conjecture",
  "field": "Potential theory",
  "claim": "Five positive point charges in R³ have at least 24 nondegenerate critical points, contradicting the proposed (5−1)² bound",
  "notice": "Examination underway · one conditional exact polynomial check reproduced · counterexample unresolved",
  "summary": "The manuscript claims five positive charges yield at least 24 nondegenerate critical points. A local exact-algebra check reproduces the displayed limiting polynomial's classification of 21 real, nondegenerate critical points, conditional on manual transcription. The polynomial's derivation from the charge potential, 24-point persistence, Morse perturbation, global finiteness, and the counterexample remain unresolved.",
  "scope": "This completed route covers only Lemma 2's exact critical-point and Hessian classification for the displayed limiting polynomial. Gates G1 and G3–G5 remain unexamined; no headline theorem disposition follows.",
  "evidencePacketUrl": "https://stateofproof.org/artifacts/maxwell-conjecture/lemma2/README.md",
  "evidenceRelease": null,
  "publicMeaning": {
    "plainEnglish": "The manuscript proposes more stationary points around five idealized positive charges than a proposed bound allows. One exact check confirms the 21-point classification of an intermediate polynomial written in the paper. It does not yet show that the five-charge potential has 24 such points.",
    "ifHolds": "If the argument holds, it would give a mathematical counterexample to the proposed numerical bound. It would concern the stated idealized point-charge model, not by itself a claim about a practical physical system.",
    "potentialConcerns": "The intermediate polynomial must still be derived from the charge model. The proposed points must persist and remain distinct, a nearby positive-charge configuration must have nondegenerate critical points, and the total critical set must be finite.",
    "ifNot": "If the claimed configuration or count does not hold up, that would not by itself prove the proposed bound; it would only mean this proposed counterexample has not established an exception."
  },
  "sourcePreservation": {
    "state": "version-pinned-source-with-public-archive",
    "summary": "The source is pinned to arXiv v1. A public Wayback snapshot was found, but it has not been hash-compared with the locked docket file.",
    "url": "https://web.archive.org/web/20260731150356/https://arxiv.org/pdf/2607.27197v1",
    "label": "Wayback snapshot from 2026-07-31 (not hash-compared)",
    "sha256": null,
    "checkedAt": "2026-08-27",
    "lockedSource": {
      "url": "https://arxiv.org/pdf/2607.27197v1",
      "label": "arXiv 2607.27197v1",
      "version": "arXiv v1",
      "sha256": "4178ddc5a4efcdc11726bea6d6d20785575c55564095dbf53bfaa6d26055a958",
      "locked": "2026-08-26",
      "retrieved": "2026-08-26T20:09:20Z"
    },
    "formalArtifact": null
  },
  "authorContact": {
    "state": "no-contact-recorded",
    "summary": "No author outreach, correspondence, response, or author approval is recorded for this docket.",
    "lastUpdated": "2026-08-27",
    "responsePublished": false,
    "response": null,
    "contactUrl": "mailto:team@materialshift.com?subject=State%20of%20Proof",
    "note": "No author response is currently published; the record steward can receive a response or correction request."
  },
  "recordProvenance": {
    "maintainedBy": "Material Shift",
    "recordType": "scoped local replay record",
    "summary": "Material Shift transcribed the displayed polynomial and ran its own exact checker. This is not author-supplied code, independent specialist verification, peer review, or a referee report.",
    "lastUpdated": "2026-10-06",
    "docketId": "2026-07-29-maxwell-conjecture",
    "canonicalUrl": "https://stateofproof.org/dockets/maxwell-conjecture",
    "steward": "Material Shift",
    "lastMaterialChange": "2026-10-06",
    "sourceLock": {
      "sha256": "4178ddc5a4efcdc11726bea6d6d20785575c55564095dbf53bfaa6d26055a958",
      "locked": "2026-08-26",
      "retrieved": "2026-08-26T20:09:20Z"
    }
  },
  "routeProvenance": [
    {
      "route": "computational",
      "state": "queued",
      "lastUpdated": "2026-08-26",
      "methodAndProvenance": "Planned Material Shift route derived from the locked source and claim map; no run is recorded for the charge-to-polynomial derivation.",
      "inputs": [
        "locked source",
        "published claim map"
      ],
      "performedBy": "Not yet run",
      "machineAssistance": "AI-assisted claim mapping and route preparation",
      "independentSpecialistReview": "None recorded",
      "artifact": null,
      "logPublished": false,
      "publicArtifactUrl": null,
      "note": "Planned: derive the displayed polynomial from the five-charge potential with exact Taylor coefficients and analytic remainder, then examine persistence."
    },
    {
      "route": "symbolic",
      "state": "completed-local",
      "lastUpdated": "2026-10-06",
      "methodAndProvenance": "Material Shift exact checker of the manually transcribed Lemma 2 polynomial. No manuscript code or third-party mathematical package ran; no independent specialist review is recorded.",
      "inputs": [
        "arXiv:2607.27197v1 PDF SHA-256 4178ddc5a4efcdc11726bea6d6d20785575c55564095dbf53bfaa6d26055a958",
        "displayed limiting polynomial, manually transcribed from printed pp. 4–5"
      ],
      "performedBy": "Material Shift research workflow",
      "machineAssistance": "AI-assisted analysis and public-record preparation",
      "independentSpecialistReview": "None recorded",
      "artifact": {
        "url": "/artifacts/maxwell-conjecture/lemma2/replay-manifest.json",
        "label": "Public replay manifest · exact Lemma 2 route",
        "sha256": "ce89beeed3f6d88aef98cbbc3d29d48e9d91a680cc8e55cd4121bcc4df262e53"
      },
      "logPublished": true,
      "publicArtifactUrl": "/artifacts/maxwell-conjecture/lemma2/replay-manifest.json",
      "note": "Exact case splitting and Cartesian Hessian evaluation give 21 critical points of the displayed polynomial, all nondegenerate, with 10 (++−) and 11 (+−−). The charge-potential derivation and headline remain open."
    },
    {
      "route": "expert",
      "state": "queued",
      "lastUpdated": "2026-08-26",
      "methodAndProvenance": "No specialist report, sign-off, or external review is recorded.",
      "inputs": [
        "locked source",
        "published claim map"
      ],
      "performedBy": "Not yet run",
      "machineAssistance": "AI-assisted claim mapping and route preparation",
      "independentSpecialistReview": "None recorded",
      "artifact": null,
      "logPublished": false,
      "publicArtifactUrl": null,
      "note": "Planned: specialist review of quantitative persistence, transversality, and global analytic bounds."
    }
  ],
  "dependencies": [
    {
      "node": "G1",
      "label": "Exact rescaling and Taylor skeleton",
      "deps": "locked v1 source",
      "status": "claim-mapped"
    },
    {
      "node": "G2",
      "label": "Exhaustive exact classification of 21 limiting critical points",
      "deps": "displayed limiting polynomial; G1 derivation remains open",
      "status": "reproduced-in-scope"
    },
    {
      "node": "G3",
      "label": "Persistence and separation of 21 plus 3 critical-point branches",
      "deps": "G2",
      "status": "claim-mapped"
    },
    {
      "node": "G4",
      "label": "Morse perturbation via transversality with positive charges",
      "deps": "G3",
      "status": "claim-mapped"
    },
    {
      "node": "G5",
      "label": "Global finiteness of the complete critical set",
      "deps": "G4",
      "status": "claim-mapped"
    }
  ],
  "findings": [
    "An exact local check of the displayed limiting polynomial reproduces Lemma 2's 21 real critical points and nondegenerate Hessian signatures: 10 (++−), 11 (+−−).",
    "This result is conditional on manual transcription of the displayed polynomial. Its derivation from the five-charge potential (G1) has not been examined.",
    "Persistence, Morse perturbation, global finiteness, and the claimed Maxwell counterexample remain unresolved; no specialist review is recorded.",
    "The intake target is five charges and at least 24 nondegenerate critical points, not a three-charge or five-equilibrium construction."
  ],
  "openSteps": [
    {
      "field": "Potential theory and exact asymptotics",
      "question": "Does the five-charge potential yield the displayed limiting polynomial with the stated exact coefficients and analytic remainder?",
      "why": "The completed 21-point calculation applies to the displayed polynomial; its connection to the charge configuration depends on this derivation.",
      "evidence": [
        {
          "route": "symbolic",
          "note": "conditional exact 21-point polynomial classification completed"
        },
        {
          "route": "computational",
          "note": "charge-to-polynomial expansion still planned"
        }
      ],
      "scope": "G1 exact derivation; G2 local result remains conditional",
      "docket": "2026-07-29-maxwell-conjecture"
    }
  ],
  "materialHistory": [
    {
      "date": "2026-10-06",
      "type": "check-reproduced",
      "docket": "Exact 21-point and Hessian classification of the displayed Lemma 2 polynomial reproduced locally; charge-model derivation and counterexample remain open"
    },
    {
      "date": "2026-08-27",
      "type": "record-provenance",
      "docket": "Record classified as queued; source preservation, route provenance, and author-contact state published"
    },
    {
      "date": "2026-08-26",
      "type": "new-docket",
      "docket": "arXiv v1 source locked and five examination gates mapped; no gate run"
    }
  ],
  "stewardship": "Selected and maintained by Material Shift as protocol research. No external sponsor or author involvement is recorded for this case."
}