{
  "name": "State of Proof",
  "url": "https://stateofproof.org/",
  "description": "A public record of source-locked mathematical claims, scoped checks, and precise open questions.",
  "notice": "Public case record — no headline claim has been validated by State of Proof",
  "lastMaterialChange": "2026-08-26",
  "method": "https://stateofproof.org/method",
  "policies": "https://stateofproof.org/policies",
  "dockets": [
    {
      "id": "2026-08-23-alpoge-s6",
      "slug": "alpoge-s6",
      "url": "https://stateofproof.org/dockets/alpoge-s6",
      "field": "Complex geometry",
      "claim": "A completed (3,4,∞) modular family gives a compact complex threefold diffeomorphic to S⁶",
      "statuses": [
        "audit-in-progress",
        "human-review-required"
      ],
      "notice": "Audit in progress · limited local results only · headline claim remains human-review-required",
      "summary": "The 108-page manuscript claims that a completed (3,4,∞) family of complex two-tori produces a compact complex threefold diffeomorphic to S⁶. Exact finite identities and a stipulated local torsion mechanism have been reproduced locally. Analytic quotient and gluing, relative meridian signs, integral homology, and the headline theorem remain unresolved.",
      "scope": "This first pass covers the page-2 monodromy identities, the conditional arithmetic of Theorem 7.17, a candidate local Kähler-differential torsion channel, and the smallest unresolved geometric and topological gates. It is not a whole-paper validation.",
      "source": {
        "label": "108-page source manuscript",
        "url": "https://alpo.ge/s6.pdf",
        "version": "retrieved 2026-08-24",
        "sha256": "283bba102dd1d5dc346af81b28145bdaaea6654398d5032e76e97bafb9a858f2",
        "locked": "2026-08-24",
        "retrieved": "2026-08-24T14:30:22Z",
        "note": "Self-hosted PDF with no embedded byline, version, or date. Attribution is based on the public announcement that linked this exact file."
      },
      "lastMaterialChange": "2026-08-24",
      "dependencies": [
        {
          "node": "S6-C1",
          "label": "Page-2 monodromy identities and exact finite skeleton",
          "deps": "locked matrices",
          "status": "reproduced-in-scope"
        },
        {
          "node": "S6-C2",
          "label": "Candidate local torsion mechanism in the stipulated double-crossing model",
          "deps": "source-to-model applicability",
          "status": "reproduced-in-scope"
        },
        {
          "node": "S6-C3",
          "label": "Relative meridian signs and zero residual toric translation",
          "deps": "local quotient maps",
          "status": "open-dependency"
        },
        {
          "node": "S6-C4",
          "label": "Analytic quotient, freeness, gluing, and collar compatibility",
          "deps": "Theorems 4.5 and 6.2",
          "status": "human-review-required"
        },
        {
          "node": "S6-C5",
          "label": "Integral homology, simple connectivity, and six-manifold recognition",
          "deps": "S6-C3 · S6-C4",
          "status": "human-review-required"
        }
      ],
      "evidenceRoutes": [
        {
          "route": "computational",
          "note": "Exact rational and integer replay reproduced the encoded page-2 identities and the conditional Smith arithmetic. The geometry that supplies those inputs was not established.",
          "log": false
        },
        {
          "route": "formal",
          "note": "A custom exact local-module certificate reproduced a candidate torsion differential inside the stipulated model. Model applicability and the global implication remain open.",
          "log": false
        },
        {
          "route": "literature",
          "note": "The printed CDP20 reduction was compared with the manuscript's proposed local mechanism; specialist source-to-model review is still required.",
          "log": false
        },
        {
          "route": "expert",
          "note": "Required next: independent complex-geometric and topological review of the sign, gluing, and integral-topology gates.",
          "log": false
        }
      ],
      "findings": [
        "All encoded page-2 matrix identities and the conditional Theorem 7.17 integer/Smith arithmetic were reproduced locally.",
        "A nonzero local torsion differential was reproduced inside the stipulated double-crossing model; this does not show that the model follows from the manuscript's global hypotheses.",
        "Changing one relative sign changes the candidate fundamental-group parameter from |p|=1 to |p|=7, so the meridian-sign gate is load-bearing.",
        "The headline S⁶ complex-structure claim remains human-review-required. Nothing in this docket is a validation, proof, certification, or referee report."
      ],
      "openSteps": [
        {
          "field": "Complex geometry and topology",
          "question": "Do the two local quotient maps induce the same meridian exponent sign, and does the toric zero-section give μ=0?",
          "why": "The already-passing Smith calculation decides simple connectivity only after these geometric inputs are fixed.",
          "evidence": [
            {
              "route": "computational",
              "note": "both sign branches and their group arithmetic are already encoded"
            }
          ],
          "scope": "specialist geometric trace",
          "docket": "2026-08-23-alpoge-s6"
        },
        {
          "field": "Complex geometry",
          "question": "Do the analytic quotient and gluing steps establish a compact complex threefold with the claimed compatibility?",
          "why": "Finite identities cannot supply freeness, proper discontinuity, convergence, or global gluing.",
          "evidence": [
            {
              "route": "expert",
              "note": "Theorems 4.5 and 6.2 identified as the review surface"
            }
          ],
          "scope": "specialist proof review",
          "docket": "2026-08-23-alpoge-s6"
        }
      ],
      "history": [
        {
          "date": "2026-08-24",
          "type": "check-reproduced",
          "docket": "Finite matrix identities and conditional Smith arithmetic reproduced locally"
        },
        {
          "date": "2026-08-24",
          "type": "open-step",
          "docket": "Relative meridian signs recorded as the highest-risk next gate"
        },
        {
          "date": "2026-08-24",
          "type": "new-docket",
          "docket": "108-page source locked and five load-bearing gates mapped"
        }
      ],
      "stewardship": "Selected and maintained by Material Shift as protocol research. No external sponsor or author involvement is recorded for this case."
    },
    {
      "id": "2026-07-29-maxwell-conjecture",
      "slug": "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",
      "statuses": [
        "claim-mapped"
      ],
      "notice": "Intake record · five gates mapped · no audit gate has run",
      "summary": "The manuscript claims a configuration of five positive charges with at least 24 nondegenerate critical points of its Coulomb potential. Since (5−1)²=16, that would contradict the proposed Maxwell bound. The source is locked and five gates are mapped; no computation or mathematical audit has run in this lab.",
      "scope": "The planned audit covers exact rescaling and Taylor coefficients, an exhaustive exact classification of 21 limiting critical points, persistence of 21 plus 3 branches, the Morse perturbation, and global finiteness.",
      "source": {
        "label": "arXiv 2607.27197v1",
        "url": "https://arxiv.org/pdf/2607.27197v1",
        "version": "arXiv v1",
        "sha256": "4178ddc5a4efcdc11726bea6d6d20785575c55564095dbf53bfaa6d26055a958",
        "locked": "2026-08-26",
        "retrieved": "2026-08-26T20:09:20Z",
        "note": "Nine-page arXiv v1 source. The locked record shows no journal reference or referee disposition."
      },
      "lastMaterialChange": "2026-08-26",
      "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": "G1",
          "status": "claim-mapped"
        },
        {
          "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"
        }
      ],
      "evidenceRoutes": [
        {
          "route": "computational",
          "note": "Planned: independent symbolic coefficient comparison, exact real-root isolation, and a separately implemented Hessian replay.",
          "log": false
        },
        {
          "route": "expert",
          "note": "Planned: specialist review of quantitative persistence, transversality, and global analytic bounds.",
          "log": false
        }
      ],
      "findings": [
        "No audit gate has run and no mathematical finding has been issued.",
        "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 real algebra",
          "question": "Can the displayed limiting polynomial and all 21 real critical points be derived and certified exactly from the locked source?",
          "why": "This is the first finite, falsifiable gate underneath the claimed 24-point configuration.",
          "evidence": [
            {
              "route": "computational",
              "note": "planned exact algebra and certified root isolation"
            }
          ],
          "scope": "audit not started",
          "docket": "2026-07-29-maxwell-conjecture"
        }
      ],
      "history": [
        {
          "date": "2026-08-26",
          "type": "new-docket",
          "docket": "arXiv v1 source locked and five audit 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."
    },
    {
      "id": "2026-02-14-openai-first-proof",
      "slug": "openai-first-proof",
      "url": "https://stateofproof.org/dockets/openai-first-proof",
      "field": "Proof evaluation",
      "claim": "Ten independent First Proof attempts form a calibration corpus; attempt 2 is the first blinded audit target",
      "statuses": [
        "claim-mapped"
      ],
      "notice": "Calibration intake · five gates mapped · no audit gate has run",
      "summary": "This case is not one theorem and will not receive a corpus-wide pass or fail. The locked packet contains ten independent proposed solutions. Attempt 2 is the first blinded negative-control target; OpenAI's public assessment is held out until the lab's status and evidence are frozen.",
      "scope": "The protocol locks and compares the original and revised packets, segments all ten attempts, audits attempt 2 under label withholding, applies a common evidence rubric, and compares frozen dispositions with external labels only after unblinding.",
      "source": {
        "label": "OpenAI First Proof attempt packet",
        "url": "https://cdn.openai.com/pdf/26177a73-3b75-4828-8c91-e8f1cf27aaa0/oai_first_proof.pdf",
        "version": "revised packet",
        "sha256": "27e97e4f61610cb31120c68bc8845b7a7b65bb2bfd044ffc4bb6f3b4d17f8102",
        "locked": "2026-08-26",
        "retrieved": "2026-08-26T20:09:53Z",
        "note": "The current and original packets are preserved separately. OpenAI's public label for attempt 2 is a held-out comparator, not a State of Proof finding."
      },
      "lastMaterialChange": "2026-08-26",
      "dependencies": [
        {
          "node": "G1",
          "label": "Source-version lineage for the original and revised packets",
          "deps": "two locked packet hashes",
          "status": "claim-mapped"
        },
        {
          "node": "G2",
          "label": "Ten-way claim segmentation",
          "deps": "G1",
          "status": "claim-mapped"
        },
        {
          "node": "G3",
          "label": "Blinded mathematical audit of attempt 2",
          "deps": "sealed comparator sources",
          "status": "claim-mapped"
        },
        {
          "node": "G4",
          "label": "Common scoped rubric across reviewed attempts",
          "deps": "frozen attempt maps",
          "status": "claim-mapped"
        },
        {
          "node": "G5",
          "label": "Comparator unblinding and calibration",
          "deps": "frozen lab outputs",
          "status": "claim-mapped"
        }
      ],
      "evidenceRoutes": [
        {
          "route": "computational",
          "note": "Planned: deterministic extraction and structured diff of the 67-page original and 90-page revised packets.",
          "log": false
        },
        {
          "route": "literature",
          "note": "Planned: pre-approved source set for the blinded attempt-2 dependency audit.",
          "log": false
        },
        {
          "route": "expert",
          "note": "Planned: independent specialist review with prior-exposure disclosure and no access to held-out labels.",
          "log": false
        }
      ],
      "findings": [
        "No audit gate has run and no mathematical finding has been issued.",
        "The public external label for attempt 2 is comparator data only. It cannot be copied into the lab's result."
      ],
      "openSteps": [
        {
          "field": "Automorphic forms and audit methodology",
          "question": "Does attempt 2's compact/Howe-vector construction satisfy its equivariance, support, central-character, and nonvanishing requirements?",
          "why": "It tests whether the protocol can reach an attempt-level result without contaminating the audit with a known external label.",
          "evidence": [
            {
              "route": "expert",
              "note": "blinded source-grounded review planned"
            }
          ],
          "scope": "audit not started",
          "docket": "2026-02-14-openai-first-proof"
        }
      ],
      "history": [
        {
          "date": "2026-08-26",
          "type": "new-docket",
          "docket": "Original and revised packets locked; five calibration 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."
    },
    {
      "id": "2025-10-27-nesterov-point-convergence",
      "slug": "nesterov-point-convergence",
      "url": "https://stateofproof.org/dockets/nesterov-point-convergence",
      "field": "Optimization",
      "claim": "The stated Nesterov accelerated-gradient iterates converge to a minimizer for every L-smooth convex objective with a minimizer",
      "statuses": [
        "claim-mapped"
      ],
      "notice": "Intake record · four gates mapped · no audit gate has run",
      "summary": "The locked v2 manuscript claims point convergence of the Nesterov iterates, not merely convergence of objective values. Four load-bearing gates are mapped from the recurrence and energy inequality through the unique-cluster-point argument. No gate has run.",
      "scope": "The audit target is Theorem 3.5 in arXiv v2 under its stated finite-dimensional assumptions. The withdrawn v1 side claim, infinite-dimensional extensions, FISTA, OGM, and other damping regimes are outside this docket.",
      "source": {
        "label": "arXiv 2510.23513v2",
        "url": "https://arxiv.org/pdf/2510.23513v2",
        "version": "arXiv v2",
        "sha256": "31dafeda57a8b0f933fbadefe20d32543ca36ea176292dfaf240d179f3b273f0",
        "locked": "2026-08-26",
        "retrieved": "2026-08-26T20:09:53Z",
        "note": "Seventeen-page v2 source. The withdrawn v1 argument for r∈[1,3) is preserved only as source lineage and is not the audit target."
      },
      "lastMaterialChange": "2026-08-26",
      "dependencies": [
        {
          "node": "G1",
          "label": "Recurrence equivalence and discrete energy inequality",
          "deps": "locked v2 assumptions",
          "status": "claim-mapped"
        },
        {
          "node": "G2",
          "label": "Boundedness of the auxiliary and NAG sequences",
          "deps": "G1",
          "status": "claim-mapped"
        },
        {
          "node": "G3",
          "label": "Scalar convergence bridge and all tₖ hypotheses",
          "deps": "G2",
          "status": "claim-mapped"
        },
        {
          "node": "G4",
          "label": "Unique cluster point and common xₖ,yₖ limit",
          "deps": "G3",
          "status": "claim-mapped"
        }
      ],
      "evidenceRoutes": [
        {
          "route": "computational",
          "note": "Planned: exact recurrence replay and independent algebraic derivation of the energy inequality.",
          "log": false
        },
        {
          "route": "literature",
          "note": "Planned: source comparison for the cited scalar convergence lemma.",
          "log": false
        },
        {
          "route": "expert",
          "note": "Planned: finite-dimensional convex-analysis review of the cluster-point and common-limit steps.",
          "log": false
        }
      ],
      "findings": [
        "No audit gate has run and no mathematical finding has been issued.",
        "Version 2, not version 1 or an invented version 3, is the locked audit target."
      ],
      "openSteps": [
        {
          "field": "Convex optimization",
          "question": "Does the stated recurrence yield the one-step energy inequality under exactly the manuscript's tₖ condition?",
          "why": "Every later boundedness and convergence step depends on this first algebraic gate.",
          "evidence": [
            {
              "route": "computational",
              "note": "planned independent algebraic replay"
            }
          ],
          "scope": "audit not started",
          "docket": "2025-10-27-nesterov-point-convergence"
        }
      ],
      "history": [
        {
          "date": "2026-08-26",
          "type": "new-docket",
          "docket": "arXiv v2 locked and four audit gates mapped; no gate run"
        },
        {
          "date": "2026-01-19",
          "type": "source-updated",
          "docket": "v2 published; the withdrawn v1 r∈[1,3) route retained as lineage only"
        }
      ],
      "stewardship": "Selected and maintained by Material Shift as protocol research. No external sponsor or author involvement is recorded for this case."
    },
    {
      "id": "2026-07-10-openai-cycle-double-cover",
      "slug": "openai-cycle-double-cover",
      "url": "https://stateofproof.org/dockets/openai-cycle-double-cover",
      "field": "Graph theory",
      "claim": "Every finite bridgeless undirected graph has a cycle double cover",
      "statuses": [
        "claim-mapped"
      ],
      "notice": "Intake record · four gates mapped · no audit gate has run",
      "summary": "The three-page manuscript claims the cycle double cover conjecture and ships with a public Lean artifact. Both sources are locked. The first gate is prose-to-Lean statement and definition correspondence; the repository has not been built or mathematically audited in this lab.",
      "scope": "The planned audit covers prose-to-Lean correspondence, a fresh pinned Lean replay, independent reconstruction and finite stress testing of Lemma 2.2, and the exact hypotheses of the cited graph reductions and flow inputs.",
      "source": {
        "label": "Three-page OpenAI manuscript",
        "url": "https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf",
        "version": "PDF + Lean 577e9d9",
        "sha256": "b4797f5053d9067329b3dcfcbf913f8bb40d13467453b1300f6d78d08460fc13",
        "locked": "2026-08-26",
        "retrieved": "2026-08-26T20:09:53Z",
        "note": "Public manuscript with no asserted arXiv, journal, or referee disposition in the locked source."
      },
      "lastMaterialChange": "2026-08-26",
      "dependencies": [
        {
          "node": "G1",
          "label": "Paper-to-Lean statement and definition correspondence",
          "deps": "locked PDF · pinned commit",
          "status": "claim-mapped"
        },
        {
          "node": "G2",
          "label": "Fresh pinned Lean build and kernel replay",
          "deps": "G1",
          "status": "claim-mapped"
        },
        {
          "node": "G3",
          "label": "Independent reconstruction and finite stress tests for Lemma 2.2",
          "deps": "G1",
          "status": "claim-mapped"
        },
        {
          "node": "G4",
          "label": "Cited reductions and nowhere-zero-flow inputs",
          "deps": "primary source hypotheses",
          "status": "claim-mapped"
        }
      ],
      "evidenceRoutes": [
        {
          "route": "formal",
          "note": "Planned: fresh Lean 4.31.0 replay, compiled-source scan, axiom closure, and hashed outputs.",
          "log": false
        },
        {
          "route": "computational",
          "note": "Planned: independent finite property tests for cubic bridgeless multigraphs and admissible flows.",
          "log": false
        },
        {
          "route": "literature",
          "note": "Planned: lock the primary graph-reduction and flow theorems and verify their exact hypotheses.",
          "log": false
        },
        {
          "route": "expert",
          "note": "Planned: graph-theorist review of the cycle-decomposition inference and scope correspondence.",
          "log": false
        }
      ],
      "findings": [
        "No Lean build, replay, or mathematical audit gate has run in this lab.",
        "A successful formal build would support only the encoded endpoint; it would not by itself establish correspondence with the prose theorem."
      ],
      "openSteps": [
        {
          "field": "Graph theory and formal verification",
          "question": "Do the Lean definitions and endpoint match the manuscript's advertised theorem across loops, parallel edges, connectedness, bridgelessness, and cycle conventions?",
          "why": "Formal replay is meaningful only after the encoded statement is shown to match the public claim.",
          "evidence": [
            {
              "route": "formal",
              "note": "pinned source available; replay not started"
            }
          ],
          "scope": "audit not started",
          "docket": "2026-07-10-openai-cycle-double-cover"
        }
      ],
      "history": [
        {
          "date": "2026-08-26",
          "type": "new-docket",
          "docket": "Manuscript and Lean commit locked; four audit 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."
    },
    {
      "id": "2026-08-01-openai-nonsofic-group",
      "slug": "openai-nonsofic-group",
      "url": "https://stateofproof.org/dockets/openai-nonsofic-group",
      "field": "Group theory",
      "claim": "The unit group of the binary Leavitt algebra L_F₂(1,2) is not sofic",
      "statuses": [
        "claim-mapped"
      ],
      "notice": "Intake record · four gates mapped · no audit gate has run",
      "summary": "Chapter 3 of OpenAI's updated Ten Advances manuscript claims that the binary Leavitt algebra's unit group is not sofic. The pinned Lean commit predates a substantive prose revision by four days. Paper-to-Lean and version correspondence is therefore the first gate; no build or mathematical audit has run.",
      "scope": "The planned audit covers original-to-revised source lineage, paper-to-Lean correspondence, a fresh pinned formal replay, the exact hypotheses of cited theorems, and an independent specialist reconstruction of the revised proof.",
      "source": {
        "label": "Updated Ten Advances manuscript",
        "url": "https://cdn.openai.com/pdf/ten-proofs-oai.pdf",
        "version": "updated 2026-08-06",
        "sha256": "ebc561ab5c53dbd240e17a8fdb6fffeb648591eca85dbfc7466f563638f8c566",
        "locked": "2026-08-26",
        "retrieved": "2026-08-26T20:09:54Z",
        "note": "The updated Chapter 3 is the audit target. The original PDF is preserved separately. No journal or referee disposition is asserted in the locked sources."
      },
      "lastMaterialChange": "2026-08-26",
      "dependencies": [
        {
          "node": "G1",
          "label": "Paper-to-Lean and August 2/August 6 version correspondence",
          "deps": "original PDF · revised PDF · pinned commit",
          "status": "claim-mapped"
        },
        {
          "node": "G2",
          "label": "Fresh pinned formal replay and axiom closure",
          "deps": "G1",
          "status": "claim-mapped"
        },
        {
          "node": "G3",
          "label": "Cited-theorem hypothesis table",
          "deps": "primary literature",
          "status": "claim-mapped"
        },
        {
          "node": "G4",
          "label": "Independent audit of the revised expander and Leavitt-algebra route",
          "deps": "G1 · G3",
          "status": "claim-mapped"
        }
      ],
      "evidenceRoutes": [
        {
          "route": "formal",
          "note": "Planned: build and kernel-check the pinned Lean target, then keep its result separate from prose correspondence.",
          "log": false
        },
        {
          "route": "literature",
          "note": "Planned: exact-hypothesis table for every load-bearing cited theorem.",
          "log": false
        },
        {
          "route": "expert",
          "note": "Planned: independent group-theory reconstruction of the August 6 proof route.",
          "log": false
        }
      ],
      "findings": [
        "No Lean build, replay, or mathematical audit gate has run in this lab.",
        "The August 2 Lean commit predates the substantive August 6 manuscript revision, so repository proximity cannot establish correspondence."
      ],
      "openSteps": [
        {
          "field": "Group theory and formal verification",
          "question": "Which exact Lean endpoints, witnesses, and definitions correspond to the revised paper's unit-group claim?",
          "why": "The formal artifact predates the revised proof and may encode a materially different route or endpoint.",
          "evidence": [
            {
              "route": "formal",
              "note": "versioned sources are locked; correspondence audit not started"
            }
          ],
          "scope": "audit not started",
          "docket": "2026-08-01-openai-nonsofic-group"
        }
      ],
      "history": [
        {
          "date": "2026-08-26",
          "type": "new-docket",
          "docket": "Updated and original manuscripts plus Lean commit locked; four gates mapped; no gate run"
        },
        {
          "date": "2026-08-06",
          "type": "source-updated",
          "docket": "OpenAI published a substantive Chapter 3 revision; the August 2 Lean commit predates it"
        }
      ],
      "stewardship": "Selected and maintained by Material Shift as protocol research. No external sponsor or author involvement is recorded for this case."
    }
  ]
}