{
  "schema_version": "asymmetry.research-index.v1",
  "generated_at": "2026-07-26T21:00:26.188Z",
  "title": "Asymmetry research index",
  "description": "Audited candidates, published contracts, and a narrow external VVUQ selection in one filterable catalog.",
  "scoring": {
    "scale": "0–5 per dimension",
    "formula": "total = 5*artifact_determinism + 4*claim_exactness + 4*research_value + 3*feasibility + 2*source_strength + 2*nonduplication",
    "labels": {
      "artifact_determinism": "Artifact determinism",
      "claim_exactness": "Claim exactness",
      "research_value": "Research value",
      "feasibility": "Feasibility",
      "source_strength": "Source strength",
      "nonduplication": "Non-duplication"
    },
    "applies_to": "bank-candidate only",
    "warning": "The score ranks contract suitability, not mathematical importance or expected solver ability."
  },
  "counts": {
    "total": 67,
    "bank_candidates": 50,
    "recommended": 20,
    "published_contracts": 14,
    "external_contracts": 3,
    "vvuq_linked": 16
  },
  "snapshots": {
    "bank": "2026-07-26",
    "vvuq": "2026-07-26T21:00:26.188Z",
    "vvuq_open_inventory_count": 24,
    "vvuq_selection_policy": {
      "published_matches": "Attach public market state to an existing Asymmetry contract; never create a second catalog row for the same contract.",
      "external_inclusions": "Include only quantum-research tasks with a concrete output and numerical tolerance that are relevant to the index. External inclusion is not an Asymmetry endorsement or a bank suitability score.",
      "external_exclusions": "Exclude test and arena jobs, synthetic-paper exercises, figure transcription, open-ended extrapolation, other projects' work contracts, and records with contradictory or unfunded settlement metadata."
    }
  },
  "records": [
    {
      "id": "bank:erdos-7-odd-covering-system",
      "native_id": "erdos-7-odd-covering-system",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #7 — a strict odd covering system",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-existential-witness",
      "summary": "There exists a finite covering system of the integers with distinct odd moduli, all greater than one.",
      "claim_boundary": "Failure to find a witness is no evidence for nonexistence; a negative answer requires a separate formal proof contract.",
      "artifact": "Distinct canonical residue/modulus pairs, with every modulus odd and greater than one, plus a complete periodic-coverage certificate.",
      "rail": "proof-carrying-witness",
      "verification": "proof-carrying-witness",
      "profile": "proof-carrying-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "verifiable",
      "status_effect": "An accepted witness proves only the affirmative existential in the pinned statement.",
      "score": 94,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 2,
      "selection_reason": "The witness is finite, the exact Lean statement is pinned, and periodic coverage can be proof-carrying rather than trusted to an opaque search.",
      "tags": [
        "number theory",
        "covering systems",
        "existence"
      ],
      "acceptance": {
        "artifact": "Both exact checkers accept identical normalized congruences and the complete proof object.",
        "claim": "A Lean and number-theory reviewer confirm that the encoded system inhabits the pinned StrictCoveringSystem statement with odd proper ideals.",
        "adjudication": "A dated literature search and the Erdős database maintainers' review find no earlier equivalent system or status update before any resolution label is proposed."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/7"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/7.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-19-efl-finite-remainder",
      "native_id": "erdos-19-efl-finite-remainder",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #19 — the finite remainder of Erdős–Faber–Lovász",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "finite-exhaustion",
      "summary": "If a graph is an edge-disjoint union of n copies of K_n, then its chromatic number is n.",
      "claim_boundary": "The current 'decidable' label does not publish a concrete cutoff or a ready verifier; the draft must remain deferred until those are extracted from the large-n theorem.",
      "artifact": "Explicit bounds theorem, canonical graph generator, and replayable coloring or obstruction certificates for every remaining n.",
      "rail": "finite-exhaustion",
      "verification": "finite-exhaustion",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "decidable",
      "status_effect": "Potential full resolution only after the finite interval is made explicit.",
      "score": 63,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 2
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 3
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "High-value but not contract-ready: the finite range and canonical encoding are not explicit in the authoritative source.",
      "tags": [
        "graph theory",
        "chromatic number",
        "finite exhaustion"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/19"
        }
      ]
    },
    {
      "id": "bank:erdos-23-bipartization-counterexample",
      "native_id": "erdos-23-bipartization-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #23 — triangle-free bipartization counterexample",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "Every triangle-free graph on 5n vertices can be made bipartite by deleting at most n^2 edges.",
      "claim_boundary": "A hard-to-bipartize graph must also be exactly triangle-free and have 5n vertices; a heuristic lower bound is not a certificate.",
      "artifact": "Canonical graph plus an independently checkable MaxCut/odd-cycle-transversal lower-bound certificate.",
      "rail": "proof-carrying-witness",
      "verification": "proof-carrying-witness",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 82,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Excellent reserve candidate, but proving the optimum deletion lower bound requires a substantial certificate rail not yet standardized here.",
      "tags": [
        "graph theory",
        "triangle-free",
        "optimization certificate"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/23"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/23.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-64-power-cycle-counterexample",
      "native_id": "erdos-64-power-cycle-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #64 — minimum-degree graph with no power-of-two cycle",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "Every finite graph of minimum degree at least three contains a cycle of length 2^k for some k at least two.",
      "claim_boundary": "Checking a cycle basis or a bounded sample of cycles does not prove absence of all forbidden simple cycles.",
      "artifact": "A canonical simple graph, degree evidence, and UNSAT certificates for simple cycles of every length 4,8,... not exceeding its order.",
      "rail": "proof-carrying-witness",
      "verification": "proof-carrying-witness",
      "profile": "proof-carrying-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 91,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 9,
      "selection_reason": "A counterexample has an exact finite meaning; proof-logged SAT can carry the otherwise expensive absence claim.",
      "tags": [
        "graph theory",
        "cycles",
        "counterexample"
      ],
      "acceptance": {
        "artifact": "All degree and no-cycle checks pass in both implementations and every proof hash is recorded.",
        "claim": "A graph theorist confirms that the cycle encoding is for simple cycles and matches the pinned k>=2 formulation.",
        "adjudication": "A dated search confirms the graph is not a rediscovery and independent reviewers reproduce all UNSAT proofs."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/64"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/64.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-97-four-equidistant-counterexample",
      "native_id": "erdos-97-four-equidistant-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #97 — convex polygon with four equidistant neighbors at every vertex",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-geometric-counterexample",
      "summary": "Every convex polygon has a vertex for which no four other vertices are equidistant from it.",
      "claim_boundary": "Floating-point coordinates or numerical near-equalities cannot establish the metric predicate.",
      "artifact": "Algebraic coordinates with isolating intervals, convex-order certificate, and exact squared-distance equalities.",
      "rail": "formal-algebraic-witness",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 70,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 3
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "The algebraic-number and convexity certificate stack is not mature enough for a low-trust contract.",
      "tags": [
        "geometry",
        "convexity",
        "algebraic coordinates"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/97"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/97.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-107-happy-ending-disproof",
      "native_id": "erdos-107-happy-ending-disproof",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #107 — disproof track for the Happy Ending formula",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-geometric-counterexample",
      "summary": "The least f(n) forcing n points in convex position is 2^(n-2)+1 for every n at least three.",
      "claim_boundary": "A configuration with no convex n-gon at one size is merely a lower bound unless it crosses the conjectured threshold in the correct direction.",
      "artifact": "Exact point configurations alone prove only lower bounds; a disproof needs enough certified information to establish f(n) differs from the formula.",
      "rail": "two-sided-finite-certificate",
      "verification": "finite-exhaustion",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Potential disproof.",
      "score": 65,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 2
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 3
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "The obvious witness contract would overclaim; exact f(n) correspondence requires a much stronger two-sided certificate.",
      "tags": [
        "geometry",
        "convex position",
        "two-sided bound"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/107"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/107.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-128-dense-half-subgraph-counterexample",
      "native_id": "erdos-128-dense-half-subgraph-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #128 — triangle-free dense-half counterexample",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "A graph in which every at-least-half-sized induced subgraph has more than n^2/50 edges must contain a triangle.",
      "claim_boundary": "Sampling dense subsets is insufficient; the certificate must cover every subset satisfying the pinned cardinality inequality.",
      "artifact": "Canonical graph plus a minimum-induced-edge optimization certificate over all qualifying vertex subsets.",
      "rail": "proof-carrying-witness",
      "verification": "proof-carrying-witness",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 75,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Precise but the universal subset optimization certificate is substantially harder than the selected graph rails.",
      "tags": [
        "graph theory",
        "triangle-free",
        "subset optimization"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/128"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/128.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-242-erdos-straus-counterexample",
      "native_id": "erdos-242-erdos-straus-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #242 — a certified Erdős–Straus exception",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-number-theory-counterexample",
      "summary": "For every n greater than two, 4/n is a sum of three distinct positive unit fractions.",
      "claim_boundary": "A failed bounded search is not a counterexample unless the denominator bounds are proved complete.",
      "artifact": "n plus a formally justified finite search bound and completeness proof excluding every x<y<z.",
      "rail": "proof-carrying-counterexample",
      "verification": "proof-carrying-witness",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 74,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 3
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "The completeness proof, not integer arithmetic, is the hard artifact and is not yet pinned.",
      "tags": [
        "unit fractions",
        "counterexample",
        "complete search"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/242"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/242.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-287-egyptian-gap-counterexample",
      "native_id": "erdos-287-egyptian-gap-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #287 — Egyptian fraction with all gaps at most two",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "Every strictly increasing representation of one as a sum of reciprocals of distinct integers greater than one has a consecutive denominator gap at least three.",
      "claim_boundary": "The known (2,3,6) example has maximum gap three and shows sharpness; it does not disprove the conjecture.",
      "artifact": "A canonical sorted list of natural denominators; exact rational arithmetic checks the sum and gaps.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 93,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 4
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 3,
      "selection_reason": "The entire mathematical claim collapses to transparent exact rational checks on one finite list.",
      "tags": [
        "number theory",
        "unit fractions",
        "counterexample"
      ],
      "acceptance": {
        "artifact": "Both arithmetic implementations and the concrete Lean replay accept the identical list hash.",
        "claim": "A number theorist and Lean reviewer confirm the list negates the exact strict-mono, sum, and max-gap theorem.",
        "adjudication": "A dated search confirms the example and resolution are not already known."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/287"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/287.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-307-prime-reciprocal-product",
      "native_id": "erdos-307-prime-reciprocal-product",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #307 — prime reciprocal sums whose product is one",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-existential-witness",
      "summary": "There exist finite sets P and Q of primes such that the product of their reciprocal sums is exactly one.",
      "claim_boundary": "The known coprime-set relaxation is not the prime-set problem; every member of both submitted sets must be certified prime.",
      "artifact": "Two sorted prime sets with primality certificates; exact rational arithmetic checks the identity.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "verifiable",
      "status_effect": "Affirmative resolution only.",
      "score": 93,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 4
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 4,
      "selection_reason": "A positive solution has a small, exact, auditable semantic core even if the integers themselves are enormous.",
      "tags": [
        "number theory",
        "primes",
        "unit fractions"
      ],
      "acceptance": {
        "artifact": "Both certificate/arithmetic implementations and the concrete Lean replay accept identical P and Q hashes.",
        "claim": "A number theorist confirms exact correspondence with prime sets, not the relaxed coprime variants.",
        "adjudication": "A dated literature and computational-record search confirms the identity is not previously known."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/307"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/307.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-364-powerful-triple",
      "native_id": "erdos-364-powerful-triple",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #364 — three consecutive powerful numbers",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "There is no n for which n, n+1, and n+2 are all powerful numbers.",
      "claim_boundary": "A probable factorization or a triple with a squareful-looking partial factorization is not sufficient.",
      "artifact": "n and complete certified prime factorizations of n, n+1, and n+2, with all exponents at least two.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "verifiable",
      "status_effect": "Negative resolution only.",
      "score": 91,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 8,
      "selection_reason": "Any discovery has an exceptionally crisp proof artifact: three exact complete factorizations.",
      "tags": [
        "number theory",
        "factorization",
        "counterexample"
      ],
      "acceptance": {
        "artifact": "Both exact factor checkers and the Lean concrete replay accept the same n and factorizations.",
        "claim": "A number theorist verifies correspondence with powerful numbers, including the treatment of zero and one.",
        "adjudication": "A dated search confirms the triple is new and truly resolves the recorded conjecture."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/364"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/364.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-366-full-neighbor",
      "native_id": "erdos-366-full-neighbor",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #366 — a 2-full n followed by a 3-full n+1",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-existential-witness",
      "summary": "There exists n>0 such that n is 2-full and n+1 is 3-full.",
      "claim_boundary": "The known reverse-direction pair 8,9 does not satisfy this ordering.",
      "artifact": "n with complete certified prime factorizations of n and n+1.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "verifiable",
      "status_effect": "Affirmative resolution only.",
      "score": 93,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 4
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 5,
      "selection_reason": "The exact direction and exponent conditions are easy to encode and independently replay.",
      "tags": [
        "number theory",
        "factorization",
        "existence"
      ],
      "acceptance": {
        "artifact": "Both exact checkers and the Lean replay accept the same n hash and complete factors.",
        "claim": "A number theorist confirms the 2-full/3-full direction and the n>0 condition.",
        "adjudication": "A dated search confirms the pair is not already known."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/366"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/366.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-375-distinct-divisor-counterexample",
      "native_id": "erdos-375-distinct-divisor-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #375 — consecutive composites without distinct prime representatives",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "Every block of k consecutive composite integers admits k distinct primes, one dividing each integer.",
      "claim_boundary": "Merely failing a matching heuristic is not enough; the Hall obstruction must be exact and factors complete.",
      "artifact": "n,k, complete factorizations, and a subset of positions whose union of prime divisors is smaller than the subset.",
      "rail": "proof-carrying-witness",
      "verification": "proof-carrying-witness",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 78,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Strong reserve candidate, but complete factorizations for an entire block make the artifact materially heavier than higher-ranked arithmetic witnesses.",
      "tags": [
        "number theory",
        "matching",
        "factorization"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/375"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/375.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-398-brocard-extra-solution",
      "native_id": "erdos-398-brocard-extra-solution",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #398 — a new Brocard–Ramanujan solution",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "The only natural-number solutions to n!+1=m^2 occur at n=4,5,7.",
      "claim_boundary": "A near-square, modular candidate, or probabilistic computation is not a solution.",
      "artifact": "Canonical arbitrary-precision n and m; exact factorial and square equality.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 94,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 1,
      "selection_reason": "A discovery would have an almost ideal contract: two integers and one exact identity, with no interpretive gap.",
      "tags": [
        "number theory",
        "factorials",
        "Diophantine equation"
      ],
      "acceptance": {
        "artifact": "Both arithmetic implementations and the concrete proof replay accept identical n,m bytes.",
        "claim": "A number theorist confirms the natural-number indexing and exact set inequality.",
        "adjudication": "Independent reviewers search the Brocard–Ramanujan literature and computational records before proposing a status change."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/398"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/398.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-458-prime-lcm-counterexample",
      "native_id": "erdos-458-prime-lcm-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #458 — a prime-index LCM inequality failure",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "For every zero-based prime index k, lcm(1,...,p_(k+1)-1) is less than p_k times lcm(1,...,p_k).",
      "claim_boundary": "The Formal Conjectures source uses zero-based Nat.nth indexing; the public description must display both k and the actual primes to prevent an off-by-one claim.",
      "artifact": "k with certified consecutive primes and exact prime-power descriptions of both LCMs.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 90,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 11,
      "selection_reason": "The claim is a single exact inequality, provided the prime indexing and LCM certificates are made explicit.",
      "tags": [
        "number theory",
        "primes",
        "lcm"
      ],
      "acceptance": {
        "artifact": "Both prime/LCM implementations and the Lean replay agree on k, primes, LCM hashes, and the failed inequality.",
        "claim": "A number theorist confirms zero-based indexing and strict-inequality negation.",
        "adjudication": "A dated search confirms the failing index is not already reported."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/458"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/458.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-475-valid-ordering-finite-remainder",
      "native_id": "erdos-475-valid-ordering-finite-remainder",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #475 — valid orderings in finite fields",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "finite-exhaustion",
      "summary": "Every finite nonzero subset of F_p has an ordering with all nonempty partial sums distinct.",
      "claim_boundary": "Four asymptotic range theorems do not by themselves expose a numeric cutoff; no finite-search contract should be bonded until the union of ranges is explicit.",
      "artifact": "An explicit large-prime cutoff plus proof-logged verification for every remaining prime and subset orbit.",
      "rail": "finite-exhaustion",
      "verification": "finite-exhaustion",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "decidable",
      "status_effect": "Potential full affirmative resolution.",
      "score": 68,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 3
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 3
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Promising after theorem-bound extraction, but the current source only establishes a qualitative finite remainder.",
      "tags": [
        "additive combinatorics",
        "finite fields",
        "finite exhaustion"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/475"
        }
      ]
    },
    {
      "id": "bank:erdos-488-multiples-density-counterexample",
      "native_id": "erdos-488-multiples-density-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #488 — a finite multiples-density counterexample",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "For every finite A excluding 0 and 1 and every m>n>=max(A), the density through m of integers divisible by some a in A is less than twice the density through n.",
      "claim_boundary": "The strict inequality and the formal source's exclusions 0,1 are claim-bearing; equality is already a failure.",
      "artifact": "Sorted A and integers n,m; exact inclusion-exclusion and direct counting implementations verify the failed rational inequality.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 93,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 4
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 6,
      "selection_reason": "A compact finite witness has two naturally independent exact counting algorithms.",
      "tags": [
        "number theory",
        "density",
        "inclusion-exclusion"
      ],
      "acceptance": {
        "artifact": "Both counting algorithms and Lean replay agree on both counts and the exact failed inequality.",
        "claim": "A number theorist checks the B, interval, and strict-inequality correspondence.",
        "adjudication": "A dated search confirms the counterexample is not already known."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/488"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/488.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-506-circle-count-finite-remainder",
      "native_id": "erdos-506-circle-count-finite-remainder",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #506 — minimum circle counts for small point sets",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "finite-exhaustion",
      "summary": "Determine the minimum number of circles determined by n planar points not all on one circle.",
      "claim_boundary": "The source explicitly says the intended nondegeneracy condition is unclear; contract code cannot resolve that editorial ambiguity.",
      "artifact": "Would require a formally chosen nondegeneracy condition and exact realizability certificates for oriented matroids.",
      "rail": "not_ready",
      "verification": "deferred-or-rejected",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "decidable_but_statement_ambiguous",
      "status_effect": "Undefined until statement review.",
      "score": 40,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 1
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 1
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 3
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Rejected pending canonical statement adjudication; this is exactly the kind of superficially finite problem that would make a bad contract.",
      "tags": [
        "geometry",
        "ambiguous statement",
        "finite remainder"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/506"
        }
      ]
    },
    {
      "id": "bank:erdos-547-tree-ramsey-finite-remainder",
      "native_id": "erdos-547-tree-ramsey-finite-remainder",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #547 — tree Ramsey finite remainder",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "finite-exhaustion",
      "summary": "Every tree T on n vertices has diagonal Ramsey number at most 2n-2.",
      "claim_boundary": "The phrase 'all sufficiently large n' does not define the instances a verifier must cover.",
      "artifact": "Explicit cutoff, tree isomorph generation, and proof-logged Ramsey checks.",
      "rail": "finite-exhaustion",
      "verification": "finite-exhaustion",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "decidable",
      "status_effect": "Potential affirmative resolution.",
      "score": 59,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 2
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 3
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Deferred until a numeric cutoff and independently reproducible tree/Ramsey encodings are available.",
      "tags": [
        "graph theory",
        "Ramsey theory",
        "finite exhaustion"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/547"
        }
      ]
    },
    {
      "id": "bank:erdos-551-cycle-clique-ramsey-finite-remainder",
      "native_id": "erdos-551-cycle-clique-ramsey-finite-remainder",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #551 — cycle-versus-clique Ramsey finite remainder",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "finite-exhaustion",
      "summary": "R(C_k,K_n)=(k-1)(n-1)+1 for k>=n>=3 except n=k=3.",
      "claim_boundary": "Exact Ramsey equality is two-sided: a coloring proves only the lower bound and UNSAT proves only the upper bound.",
      "artifact": "Explicit pair bounds plus lower-bound colorings and upper-bound UNSAT certificates for each remaining (n,k).",
      "rail": "finite-exhaustion",
      "verification": "finite-exhaustion",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "decidable",
      "status_effect": "Potential full resolution.",
      "score": 68,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 3
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Good eventual certificate structure, but the explicit finite parameter domain is missing.",
      "tags": [
        "graph theory",
        "Ramsey theory",
        "two-sided certificate"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/551"
        }
      ]
    },
    {
      "id": "bank:erdos-556-three-color-cycle-finite-remainder",
      "native_id": "erdos-556-three-color-cycle-finite-remainder",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #556 — three-color cycle Ramsey finite remainder",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "finite-exhaustion",
      "summary": "The three-color Ramsey number of C_n is at most 4n-3.",
      "claim_boundary": "A SAT solver's UNSAT answer is not acceptable without a replayable proof and exact graph-coloring encoding.",
      "artifact": "Explicit n cutoff and per-n proof-logged UNSAT for three-colorings avoiding monochromatic C_n.",
      "rail": "finite-exhaustion",
      "verification": "finite-exhaustion",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "decidable",
      "status_effect": "Potential affirmative resolution.",
      "score": 68,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 3
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "The explicit finite interval is absent and the likely SAT instances are extremely large.",
      "tags": [
        "graph theory",
        "Ramsey theory",
        "SAT certificate"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/556"
        }
      ]
    },
    {
      "id": "bank:erdos-580-tree-embedding-finite-remainder",
      "native_id": "erdos-580-tree-embedding-finite-remainder",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #580 — dense-half tree embedding finite remainder",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "finite-exhaustion",
      "summary": "If at least half the vertices of an n-vertex graph have degree at least n/2, the graph contains every tree on at most n/2 vertices.",
      "claim_boundary": "The source does not give a usable numeric cutoff; embedding sampled trees cannot settle the universal claim.",
      "artifact": "Explicit cutoff, graph and tree generators modulo isomorphism, and complete embedding certificates.",
      "rail": "finite-exhaustion",
      "verification": "finite-exhaustion",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "decidable",
      "status_effect": "Potential affirmative resolution.",
      "score": 59,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 2
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 3
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Deferred because neither the finite range nor a tractable completeness certificate is presently specified.",
      "tags": [
        "graph theory",
        "trees",
        "finite exhaustion"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/580"
        }
      ]
    },
    {
      "id": "bank:erdos-617-balanced-coloring-counterexample",
      "native_id": "erdos-617-balanced-coloring-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #617 — a balanced complete-graph coloring",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "For r>=3, every r-coloring of K_(r^2+1) has an (r+1)-vertex set missing at least one color.",
      "claim_boundary": "Random sampling of vertex subsets cannot establish the universal balanced-coloring property.",
      "artifact": "Canonical upper-triangle edge coloring plus complete coverage evidence for every color in every (r+1)-vertex subset.",
      "rail": "proof-carrying-witness",
      "verification": "proof-carrying-witness",
      "profile": "proof-carrying-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 83,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 19,
      "selection_reason": "A disproof has a finite exact meaning, though its exhaustive subset check may be computationally demanding.",
      "tags": [
        "graph theory",
        "edge coloring",
        "counterexample"
      ],
      "acceptance": {
        "artifact": "Both implementations agree on r, graph order, subset count, coloring hash, and zero missing-color subsets.",
        "claim": "A graph theorist confirms that the negated statement and edge/subset indexing exactly match the pinned theorem.",
        "adjudication": "A dated literature and computational search establishes that the coloring is not already known."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/617"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/617.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-647-divisor-shift-witness",
      "native_id": "erdos-647-divisor-shift-witness",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #647 — a new divisor-shift witness beyond 24",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-existential-witness",
      "summary": "There exists n>24 such that max_(m<n)(m+tau(m)) <= n+2.",
      "claim_boundary": "The known n=24 boundary case is excluded; Fin n means m ranges from zero through n-1 in the pinned formal statement.",
      "artifact": "n, with exact tau(m) verification for every m<n by independent sieve and factor methods.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "verifiable",
      "status_effect": "Affirmative resolution only.",
      "score": 93,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 4
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 7,
      "selection_reason": "The full witness is one integer and a transparent exhaustive arithmetic predicate.",
      "tags": [
        "number theory",
        "divisor function",
        "existence"
      ],
      "acceptance": {
        "artifact": "Both implementations agree on n, every tau(m), the maximum, and all equality cases.",
        "claim": "A number theorist and Lean reviewer confirm Fin n indexing and sigma-zero semantics.",
        "adjudication": "A dated search confirms n is new and that no prior affirmative resolution exists."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/647"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/647.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-672-perfect-power-ap",
      "native_id": "erdos-672-perfect-power-ap",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #672 — a coprime arithmetic progression with perfect-power product",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "For every length k>=4 and exponent l>1, the product of a positive length-k arithmetic progression with coprime start and step is not an l-th power.",
      "claim_boundary": "The selected target follows the formal k>=4 theorem; lower-length variants, including known exceptions or partial results, do not count.",
      "artifact": "k,l,n,d,q with exact gcd and product identity.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "verifiable",
      "status_effect": "Disproof only.",
      "score": 91,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 10,
      "selection_reason": "A counterexample is five integers and exact arithmetic, with an unusually small correspondence surface.",
      "tags": [
        "number theory",
        "arithmetic progressions",
        "perfect powers"
      ],
      "acceptance": {
        "artifact": "Both exact arithmetic rails and the Lean replay accept identical k,l,n,d,q.",
        "claim": "A number theorist confirms the progression, gcd, product, and quantifier correspondence.",
        "adjudication": "A dated search confirms the example is new and actually changes the status."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/672"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/672.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-699-binomial-gcd-counterexample",
      "native_id": "erdos-699-binomial-gcd-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #699 — a binomial-GCD counterexample",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-counterexample-witness",
      "summary": "For 1<=i<j<=n/2, some prime p>=i divides gcd(C(n,i),C(n,j)).",
      "claim_boundary": "Showing no sampled large prime divides the GCD is insufficient; completeness follows from reconstructing the full GCD factorization.",
      "artifact": "n,i,j plus a complete certified factorization of the exact GCD.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 90,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 12,
      "selection_reason": "Complete GCD factorization turns the negated existential into an exact finite certificate.",
      "tags": [
        "number theory",
        "binomial coefficients",
        "factorization"
      ],
      "acceptance": {
        "artifact": "Both binomial/GCD algorithms and all primality certificates agree on the complete factorization.",
        "claim": "A number theorist confirms the boundary inequalities and negation of the existential prime.",
        "adjudication": "A dated search confirms the triple is new."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/699"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/699.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-723-projective-plane-12",
      "native_id": "erdos-723-projective-plane-12",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #723 — a projective plane of order 12",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-existential-witness",
      "summary": "The prime-power conjecture asks whether every finite projective-plane order is a prime power; order 12 is the first highlighted open case.",
      "claim_boundary": "Failure of a construction search does not prove nonexistence; an existence witness alone does not classify all orders.",
      "artifact": "Canonical 157-by-157 incidence relation with exact line/point intersection checks.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "A plane of order 12 disproves the prime-power conjecture.",
      "score": 80,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 0
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Conceptually excellent but assigned feasibility zero: this celebrated finite case is far beyond a responsible near-term contract despite its easy checker.",
      "tags": [
        "finite geometry",
        "projective planes",
        "existence"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/723"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/723.lean"
        }
      ]
    },
    {
      "id": "bank:quantum-purification-uniqueness",
      "native_id": "quantum-purification-uniqueness",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "State purifications are unique up to an environment unitary",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "physlib-formal-gap",
      "summary": "Two finite-dimensional pure states with the same reduced state on the left subsystem differ by a unitary acting only on the right subsystem.",
      "claim_boundary": "The environment types are identical and the conclusion uses a unitary. Unequal or nonminimal Stinespring environments generally require an isometry and are not covered.",
      "artifact": "One proof of a frozen theorem over Physlib's Ket, MState.pure, traceRight, and a frozen right-unitary action.",
      "rail": "lean-fixed-surface",
      "verification": "formal-proof",
      "profile": "lean-fixed-surface-v2",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "library_gap",
      "status_effect": "A reviewed Physlib theorem contribution, not a new mathematical discovery.",
      "score": 87,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 4
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 4
        }
      },
      "rank": 15,
      "selection_reason": "The state-level theorem is exact, useful to the native API, and avoids overclaiming the more delicate channel-dilation result.",
      "tags": [
        "purification",
        "unitary equivalence",
        "Lean"
      ],
      "acceptance": {
        "artifact": "Comparator, scanner, pinned build, axiom audit, and independent kernel replay all accept the exact declaration.",
        "claim": "A quantum-information reviewer confirms that the frozen helper acts on the purifying subsystem and that identical environment dimension suffices for a unitary conclusion.",
        "adjudication": "A dated search of Physlib and adjacent Lean quantum libraries confirms that an equivalent theorem is not already available at promotion time."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/States/Mixed/MState.lean"
        }
      ]
    },
    {
      "id": "bank:quantum-partial-trace-spectra",
      "native_id": "quantum-partial-trace-spectra",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Nonzero spectra of complementary pure-state reductions",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "physlib-formal-gap",
      "summary": "For a finite bipartite pure state, the left and right reduced density matrices have the same multiset of nonzero eigenvalues.",
      "claim_boundary": "The full spectra need not be equal when subsystem dimensions differ because their zero multiplicities differ; only nonzero eigenvalues are claimed.",
      "artifact": "A fixed Lean proof equating filtered multisets of spectrum probabilities.",
      "rail": "lean-fixed-surface",
      "verification": "formal-proof",
      "profile": "lean-fixed-surface-v2",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "explicit_todo",
      "status_effect": "A reusable Schmidt-spectrum theorem in Physlib.",
      "score": 90,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 13,
      "selection_reason": "It is an explicit upstream gap with a crisp correction to the informal TODO and strong downstream entropy/entanglement value.",
      "tags": [
        "partial trace",
        "spectrum",
        "Schmidt decomposition"
      ],
      "acceptance": {
        "artifact": "The immutable theorem compiles under both kernel rails with only permitted foundational axioms.",
        "claim": "A reviewer confirms that multiplicities are retained, zero multiplicities alone are removed, and subsystem order matches Physlib.",
        "adjudication": "A current library search confirms the result is not already derivable through a public theorem with equivalent content."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/States/Mixed/MState.lean#L604"
        }
      ]
    },
    {
      "id": "bank:quantum-separable-convexity",
      "native_id": "quantum-separable-convexity",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Convexity of Physlib separable states",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "physlib-formal-gap",
      "summary": "Every convex mixture of two bipartite separable mixed states is separable.",
      "claim_boundary": "The target is binary convex closure in the current finite-dimensional definition; it does not formalize topological closure, infinite ensembles, or entanglement witnesses.",
      "artifact": "A fixed theorem constructing the exact density state of pρ+(1-p)σ and proving IsSeparable.",
      "rail": "lean-fixed-surface",
      "verification": "formal-proof",
      "profile": "lean-fixed-surface-v2",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "explicit_todo",
      "status_effect": "Closure of the existing IsSeparable predicate under binary convex combinations.",
      "score": 89,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 4
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 14,
      "selection_reason": "The exact upstream TODO has a compact statement, constructive proof path, and broad reuse.",
      "tags": [
        "separability",
        "convexity",
        "mixed states"
      ],
      "acceptance": {
        "artifact": "All formal rails accept the exact binary-closure theorem and immutable constructor.",
        "claim": "A quantum reviewer verifies that Physlib's IsSeparable ensemble definition is preserved with the correct weights, including endpoints.",
        "adjudication": "Upstream review confirms that no equivalent public convexity theorem exists and accepts the API placement."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/States/Mixed/MState.lean#L994"
        }
      ]
    },
    {
      "id": "bank:quantum-fubini-study-metric",
      "native_id": "quantum-fubini-study-metric",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Triangle inequality for Physlib's fidelity angle",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "physlib-formal-gap",
      "summary": "The angle arccos(fidelity(ρ,σ)), using Physlib's unsquared fidelity, satisfies the triangle inequality on finite-dimensional mixed states.",
      "claim_boundary": "Physlib fidelity is the unsquared root fidelity. This target is not the squared-fidelity angle and does not assert monotonicity, geodesic formulas, or the pure-state projective-space construction.",
      "artifact": "A proof of the exact triangle inequality plus an immutable wrapper checking the remaining metric axioms from pinned lemmas.",
      "rail": "lean-fixed-surface",
      "verification": "formal-proof",
      "profile": "lean-fixed-surface-v2",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "explicit_todo",
      "status_effect": "Completion of the fidelity-angle metric increment in the pinned API.",
      "score": 86,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 3
        }
      },
      "rank": 18,
      "selection_reason": "The source names the exact missing result, but adjacent libraries and convention-sensitive terminology demand stronger nonduplication and correspondence review.",
      "tags": [
        "fidelity",
        "metric",
        "mixed states"
      ],
      "acceptance": {
        "artifact": "Both formal rails accept the triangle theorem and immutable metric wrapper with the permitted axiom set.",
        "claim": "Review confirms that Physlib's definition is root fidelity and that the result applies to every finite-dimensional MState under exactly the frozen assumptions.",
        "adjudication": "A dated comparison with Physlib, Lean-Quantum, Lean-QIT, and Mathlib finds no equivalent theorem already available under an imported interface."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/States/Mixed/Fidelity.lean#L136"
        }
      ]
    },
    {
      "id": "bank:quantum-von-neumann-unitary-invariance",
      "native_id": "quantum-von-neumann-unitary-invariance",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Von Neumann entropy invariance under arbitrary unitaries",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "physlib-formal-gap",
      "summary": "Conjugating a finite-dimensional mixed state by any unitary leaves its von Neumann entropy unchanged.",
      "claim_boundary": "The theorem concerns exact finite-dimensional unitary conjugation, not arbitrary channels, approximate unitaries, or entropy monotonicity.",
      "artifact": "One theorem over the existing U_conj action and Sᵥₙ definition.",
      "rail": "lean-fixed-surface",
      "verification": "formal-proof",
      "profile": "lean-fixed-surface-v2",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "explicit_todo",
      "status_effect": "General unitary-invariance API completion.",
      "score": 87,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 4
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 4
        }
      },
      "rank": 16,
      "selection_reason": "It is a clean explicit TODO with an already-proved spectrum-invariance bridge, making the target useful and realistically verifiable.",
      "tags": [
        "von Neumann entropy",
        "unitary invariance",
        "Lean"
      ],
      "acceptance": {
        "artifact": "The exact arbitrary-unitary declaration passes comparison, compilation, axiom audit, and independent kernel replay.",
        "claim": "A quantum reviewer confirms the action is unitary conjugation and the theorem is not merely basis relabeling.",
        "adjudication": "Current upstream and adjacent-library search establishes the API increment remains missing."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/Entropy/VonNeumann.lean#L147"
        }
      ]
    },
    {
      "id": "bank:quantum-capacity-replacement-zero",
      "native_id": "quantum-capacity-replacement-zero",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Zero quantum capacity of replacement channels",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "physlib-capacity-gap",
      "summary": "A channel that discards its input and prepares one fixed state has zero quantum capacity.",
      "claim_boundary": "The current capacity file contains sorryful dependencies, so a locally completed end theorem would not yet have an acceptable axiom closure.",
      "artifact": "A theorem only after all used AchievesRate and capacity lemmas are sorry-free.",
      "rail": "lean-fixed-surface-after-foundation-audit",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "documented_goal_on_sorryful_foundation",
      "status_effect": "Standard library theorem.",
      "score": 78,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 3
        }
      },
      "rank": null,
      "selection_reason": "Deferred until the capacity dependency cone is proved and compared with Lean-QIT's operational interfaces.",
      "tags": [
        "quantum capacity",
        "replacement channel",
        "dependency audit"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/Capacity/Capacity.lean"
        }
      ]
    },
    {
      "id": "bank:quantum-capacity-identity",
      "native_id": "quantum-capacity-identity",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Quantum capacity of the identity channel",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "physlib-capacity-gap",
      "summary": "The identity channel on a D-dimensional system has quantum capacity log base two of D.",
      "claim_boundary": "The empty-type and logarithm conventions must be fixed explicitly, and existing upper-bound lemmas are currently sorryful.",
      "artifact": "An equality theorem with dimension/nonempty cases fixed.",
      "rail": "lean-fixed-surface-after-foundation-audit",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "documented_goal_on_sorryful_foundation",
      "status_effect": "Standard capacity calibration.",
      "score": 75,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 3
        }
      },
      "rank": null,
      "selection_reason": "High-value but not contract-ready until foundational admissions and dimension conventions are resolved.",
      "tags": [
        "quantum capacity",
        "identity channel",
        "log dimension"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/Capacity/Capacity.lean"
        }
      ]
    },
    {
      "id": "bank:quantum-capacity-composition-monotone",
      "native_id": "quantum-capacity-composition-monotone",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Quantum capacity cannot increase under channel composition",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "physlib-capacity-gap",
      "summary": "Pre- or post-processing a quantum channel by another channel cannot increase its quantum capacity.",
      "claim_boundary": "Composition orientation and both one-sided inequalities must be stated separately; the current capacity foundations prevent a clean axiom audit.",
      "artifact": "Separate frozen pre-processing and post-processing inequalities.",
      "rail": "lean-fixed-surface-after-foundation-audit",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "documented_goal_on_sorryful_foundation",
      "status_effect": "Capacity API monotonicity.",
      "score": 75,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 3
        }
      },
      "rank": null,
      "selection_reason": "Deferred behind the same foundational and cross-library correspondence work as the other capacity targets.",
      "tags": [
        "quantum capacity",
        "composition",
        "monotonicity"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/Capacity/Capacity.lean"
        }
      ]
    },
    {
      "id": "bank:quantum-mub6-four-bases",
      "native_id": "quantum-mub6-four-bases",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Four mutually unbiased bases in dimension six",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "open-quantum-existence",
      "summary": "There exist four pairwise mutually unbiased orthonormal bases of complex dimension six.",
      "claim_boundary": "The source's open maximality theorem IsMaxMUBCount 6 k is not the target. This contract neither proves μ(6)=4 nor rules out five, six, or seven bases.",
      "artifact": "A kernel-checked proof of OpenQuantumProblem13.HasMUBs 6 4 with all unitary and overlap equations exact.",
      "rail": "lean-fixed-surface",
      "verification": "formal-proof",
      "profile": "lean-fixed-surface-v2",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "open_existence_increment",
      "status_effect": "Affirmative existence increment μ(6)≥4 only.",
      "score": 83,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 0
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 4
        }
      },
      "rank": 20,
      "selection_reason": "Despite near-zero expected feasibility, it is a uniquely clean, exact, high-value existential increment with a current canonical theorem vocabulary.",
      "tags": [
        "mutually unbiased bases",
        "dimension six",
        "existence"
      ],
      "acceptance": {
        "artifact": "Both formal rails and the independent matrix checker accept the same exact four-basis witness.",
        "claim": "A quantum-information specialist confirms that the pinned formal predicate exactly represents four pairwise MUBs in C^6.",
        "adjudication": "A dated search and consultation with the OQP/Formal Conjectures maintainers confirm novelty and the μ(6) lower-bound update."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/OpenQuantumProblems/13.lean"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/OpenQuantumProblems/13.lean"
        }
      ]
    },
    {
      "id": "bank:quantum-sic56-witness",
      "native_id": "quantum-sic56-witness",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "A SIC-POVM in dimension 56",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "open-quantum-existence",
      "summary": "There exists a family of 56 squared normalized complex state vectors with pairwise squared overlap 1/57.",
      "claim_boundary": "Floating-point fiducials are insufficient, and the current formal source's selected benchmark status must be rechecked against the live SIC literature.",
      "artifact": "An exact algebraic-coordinate Lean witness for HasSICPOVM 56.",
      "rail": "lean-fixed-surface",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "selected_open_benchmark_in_formal_source",
      "status_effect": "One dimension-specific affirmative existence result.",
      "score": 81,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 0
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 3
        }
      },
      "rank": null,
      "selection_reason": "Held as the first reserve: exact algebraic representation and current novelty are less secure than the selected MUB increment.",
      "tags": [
        "SIC-POVM",
        "equiangular lines",
        "algebraic witness"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/OpenQuantumProblems/23.lean"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/OpenQuantumProblems/23.lean"
        }
      ]
    },
    {
      "id": "bank:quantum-no-cloning-physlib",
      "native_id": "quantum-no-cloning-physlib",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "No-cloning theorem in Physlib",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "duplicate-audit",
      "summary": "A unitary cannot clone two distinct nonorthogonal pure states from the same fiducial state.",
      "claim_boundary": "A differently worded or weaker no-cloning target would duplicate existing work rather than add a rigorous problem.",
      "artifact": "None; the theorem is already proved.",
      "rail": "reject-duplicate",
      "verification": "deferred-or-rejected",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "already_formalized",
      "status_effect": "No effect.",
      "score": 74,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 1
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 5
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 0
        }
      },
      "rank": null,
      "selection_reason": "Rejected as a confirmed duplicate; this is a negative control for the research pipeline.",
      "tags": [
        "no cloning",
        "duplicate",
        "prior art"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/Operators/Unitary.lean#L57"
        }
      ]
    },
    {
      "id": "bank:quantum-no-deleting",
      "native_id": "quantum-no-deleting",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "No-deleting theorem",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "foundational-formalization",
      "summary": "No reversible operation can delete one copy of an arbitrary unknown pure state while leaving the other copy unchanged.",
      "claim_boundary": "Many informal no-deleting statements hide ancilla or state-dependence assumptions; no canonical Physlib theorem surface is currently pinned.",
      "artifact": "Would require a frozen blank-state, ancilla, reversibility, and universal-state formulation.",
      "rail": "lean-surface-needed",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "candidate_without_pinned_surface",
      "status_effect": "Standard theorem formalization.",
      "score": 62,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 3
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 3
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 3
        }
      },
      "rank": null,
      "selection_reason": "Deferred until a quantum reviewer authors and validates the exact reversible-process statement.",
      "tags": [
        "no deleting",
        "reversibility",
        "statement design"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib"
        }
      ]
    },
    {
      "id": "bank:quantum-teleportation-correctness",
      "native_id": "quantum-teleportation-correctness",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Exact correctness of qubit teleportation",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "protocol-correctness",
      "summary": "For every input qubit, the standard Bell-pair, Bell-measurement, classical-control, and Pauli-correction protocol returns the input state on the receiver system.",
      "claim_boundary": "The result depends on exact measurement, classical branch, and subsystem-order semantics; proving only the unitary premeasurement circuit is not teleportation correctness.",
      "artifact": "A fixed channel-level circuit and equality to the identity channel.",
      "rail": "lean-circuit-interface-needed",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "interface_fragmented",
      "status_effect": "Protocol-correctness theorem.",
      "score": 65,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 2
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 2
        }
      },
      "rank": null,
      "selection_reason": "Deferred because no single pinned library currently supplies the complete protocol interface without duplicating adjacent work.",
      "tags": [
        "teleportation",
        "measurement",
        "classical control"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib"
        }
      ]
    },
    {
      "id": "bank:quantum-superdense-coding",
      "native_id": "quantum-superdense-coding",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Exact correctness of superdense coding",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "protocol-correctness",
      "summary": "With one shared Bell pair, the standard four encodings and Bell-basis decoder transmit two classical bits by one qubit.",
      "claim_boundary": "A four-message protocol theorem does not by itself prove an entanglement-assisted classical-capacity formula.",
      "artifact": "Four exact branch theorems for a frozen encoder/decoder circuit.",
      "rail": "lean-circuit-interface-needed",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "documented_capacity-adjacent-goal",
      "status_effect": "Finite protocol correctness only.",
      "score": 65,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 2
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 2
        }
      },
      "rank": null,
      "selection_reason": "Useful, but a stable cross-library circuit and classical-output interface must be chosen before the claim can be fixed.",
      "tags": [
        "superdense coding",
        "Bell states",
        "protocol correctness"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib/blob/dd43e9e65791468c067d8e47222fde69951020ae/QuantumInfo/Capacity/Capacity.lean"
        }
      ]
    },
    {
      "id": "bank:quantum-peres-mermin-contextuality",
      "native_id": "quantum-peres-mermin-contextuality",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Peres–Mermin square contextuality certificate",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "finite-contextuality-proof",
      "summary": "The nine observables of the Peres–Mermin square obey the required commuting product relations, while no noncontextual ±1 value assignment satisfies all row and column parities.",
      "claim_boundary": "The operator convention, sign pattern, and distinction between algebraic contradiction and an experimental noncontextuality inequality must be explicit.",
      "artifact": "Frozen 4-by-4 operators, commuting/product proofs, and a complete 2^9 assignment rejection certificate.",
      "rail": "hybrid-lean-and-finite-exhaustion",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "candidate_without_native_surface",
      "status_effect": "Formalization of a standard state-independent contextuality proof.",
      "score": 76,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 3
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 3
        }
      },
      "rank": null,
      "selection_reason": "Strong reserve candidate, but it lacks a current canonical theorem surface and has lower research leverage than the selected native TODOs.",
      "tags": [
        "contextuality",
        "observables",
        "finite certificate"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib"
        }
      ]
    },
    {
      "id": "bank:quantum-five-qubit-code",
      "native_id": "quantum-five-qubit-code",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Exact correction proof for the five-qubit code",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "quantum-error-correction",
      "summary": "The standard [[5,1,3]] stabilizer code corrects every single-qubit Pauli error.",
      "claim_boundary": "Correcting the finite Pauli basis must be connected formally to the promised one-qubit error class; a syndrome table alone is insufficient.",
      "artifact": "Pinned generators, logical operators, syndrome table, recovery map, and exact Knill–Laflamme or branchwise proofs.",
      "rail": "hybrid-exact-stabilizer-and-lean",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "candidate_without_native_surface",
      "status_effect": "Standard code-correctness formalization.",
      "score": 73,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 5
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 3
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 3
        }
      },
      "rank": null,
      "selection_reason": "Deferred until the stabilizer/recovery interface and adjacent Lean-Quantum overlap are pinned.",
      "tags": [
        "quantum error correction",
        "stabilizer code",
        "syndrome"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://github.com/leanprover-community/physlib"
        }
      ]
    },
    {
      "id": "bank:erdos-742-murty-simon-finite-remainder",
      "native_id": "erdos-742-murty-simon-finite-remainder",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #742 — Murty–Simon finite remainder",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "finite-exhaustion",
      "summary": "Every diameter-two-critical n-vertex graph has at most floor(n^2/4) edges.",
      "claim_boundary": "The pinned formal file says only 'sufficiently large'; the finite interval is not contract-ready.",
      "artifact": "Explicit Füredi cutoff plus graph-isomorph generation and proof-logged checks for every missing n.",
      "rail": "finite-exhaustion",
      "verification": "finite-exhaustion",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "decidable",
      "status_effect": "Potential affirmative resolution.",
      "score": 74,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 3
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Deferred until the analytic theorem yields an explicit cutoff and the diameter-critical encoding is independently audited.",
      "tags": [
        "graph theory",
        "diameter",
        "finite exhaustion"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/742"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/742.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-779-prime-perturbation-counterexample",
      "native_id": "erdos-779-prime-perturbation-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #779 — prime perturbations of primorials",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-number-theory-counterexample",
      "summary": "For each n>=1, if P is the product of the first n+1 primes, some prime p strictly between the nth prime and P makes P+p prime.",
      "claim_boundary": "Extending the verified range is useful data but does not resolve the universal statement and should be a separate frontier contract.",
      "artifact": "n, exact P, and a complete interval-prime enumeration with compositeness certificates for P+p.",
      "rail": "proof-carrying-counterexample",
      "verification": "proof-carrying-witness",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 70,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 3
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "The absence certificate spans a potentially enormous prime interval; a separate verified-range frontier would need a dated baseline.",
      "tags": [
        "number theory",
        "primes",
        "complete interval"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/779"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/779.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-835-johnson-coloring-witness",
      "native_id": "erdos-835-johnson-coloring-witness",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "recommended",
      "stage_label": "Top 20",
      "title": "Erdős #835 — a k+1 coloring of J(2k,k)",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-existential-witness",
      "summary": "There exists k>2 such that the k-subsets of a 2k-set can be colored with k+1 colors and every (k+1)-subset sees all colors; equivalently chi(J(2k,k))=k+1.",
      "claim_boundary": "A coloring with more than k+1 colors, a partial coloring, or a lower bound on chromatic number does not prove the existential.",
      "artifact": "k and a color for every k-subset in combinadic order; exhaustive local checks.",
      "rail": "exact-witness",
      "verification": "exact-witness",
      "profile": "exact-witness-v1",
      "maturity": "protocol_ready_verifier_unimplemented",
      "status": "verifiable",
      "status_effect": "Affirmative resolution only.",
      "score": 86,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 5
        },
        "research_value": {
          "label": "Research value",
          "value": 5
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": 17,
      "selection_reason": "The artifact can be large, but its combinadic encoding and local all-colors predicate are exact and independently enumerable.",
      "tags": [
        "graph theory",
        "Johnson graph",
        "coloring"
      ],
      "acceptance": {
        "artifact": "Both enumeration algorithms and the concrete Lean replay accept the same k, array hash, and qualifying-subset count.",
        "claim": "A graph theorist confirms Property k establishes the stated affirmative answer and checks the chromatic-number correspondence dependency.",
        "adjudication": "A dated search confirms the coloring is not already known and handles newer excluded k ranges."
      },
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/835"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/835.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-848-squarefree-product-finite-remainder",
      "native_id": "erdos-848-squarefree-product-finite-remainder",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #848 — reconcile and certify the finite squarefree-product remainder",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "status-conflict",
      "summary": "For each N, the largest A in {0,...,N-1} with every ab+1 nonsquarefree has the size of the residue class 7 mod 25.",
      "claim_boundary": "A formalized asymptotic theorem plus finite experiments is not automatically a proof for every N; conversely, duplicating an existing full proof is not a new research contract.",
      "artifact": "No new contract until the full-proof claim, asymptotic theorem, and finite verification are independently separated and reviewed.",
      "rail": "adjudication_first",
      "verification": "deferred-or-rejected",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "conflicting_sources",
      "status_effect": "None.",
      "score": 42,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 2
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 2
        },
        "research_value": {
          "label": "Research value",
          "value": 2
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 2
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 0
        }
      },
      "rank": null,
      "selection_reason": "Rejected pending independent adjudication of conflicting status and adjacent claimed proof; a new contract risks duplication and overclaiming.",
      "tags": [
        "number theory",
        "status conflict",
        "prior art"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/848"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/848.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-982-convex-distance-counterexample",
      "native_id": "erdos-982-convex-distance-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #982 — convex polygon distance counterexample",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-geometric-counterexample",
      "summary": "Some vertex of every convex n-gon determines at least floor(n/2) distinct distances to the other vertices.",
      "claim_boundary": "Numerically clustered distances do not count as equal; every distance equality and inequality must be exact.",
      "artifact": "Algebraic coordinates, convex-polygon certificate, and exact distance-equivalence partitions.",
      "rail": "formal-algebraic-witness",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Disproof only.",
      "score": 70,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 3
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Deferred until an audited algebraic-coordinate schema and exact convexity checker are available.",
      "tags": [
        "geometry",
        "distances",
        "algebraic numbers"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/982"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/982.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-1041-lemniscate-path-counterexample",
      "native_id": "erdos-1041-lemniscate-path-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #1041 — short paths in polynomial lemniscates",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "formal-proof",
      "summary": "For a monic polynomial with roots in the unit disk, two roots can be joined inside |f(z)|<1 by a path of length below two.",
      "claim_boundary": "The pinned target's Hausdorff-measure notion of path range length needs expert correspondence review before promotion.",
      "artifact": "A complete formal proof or a rigorously formalized counterexample with path-length and sublevel-set semantics.",
      "rail": "lean-fixed-surface",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Full resolution.",
      "score": 62,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 4
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 2
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 0
        },
        "source_strength": {
          "label": "Source strength",
          "value": 4
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 5
        }
      },
      "rank": null,
      "selection_reason": "Rejected for now because statement correspondence and formal analytic infrastructure are both unresolved.",
      "tags": [
        "analysis",
        "polynomials",
        "statement review"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/1041"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/1041.lean"
        }
      ]
    },
    {
      "id": "bank:erdos-1082-distance-counterexample",
      "native_id": "erdos-1082-distance-counterexample",
      "kind": "bank-candidate",
      "source": "asymmetry-bank",
      "source_label": "Asymmetry candidate bank",
      "stage": "evaluated",
      "stage_label": "Evaluated reserve",
      "title": "Erdős #1082(i) — distinct-distance counterexample with no three collinear",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "erdos-geometric-counterexample",
      "summary": "Every finite planar set with no three collinear determines at least floor(n/2) distinct pairwise distances.",
      "claim_boundary": "Part (ii) already has a formalized counterexample and is not this target; the total-distance and single-vantage-point claims must not be conflated.",
      "artifact": "Algebraic coordinates, exact noncollinearity, and exact squared-distance class count.",
      "rail": "formal-algebraic-witness",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "evaluated_not_selected",
      "status": "falsifiable",
      "status_effect": "Disproof of part (i) only.",
      "score": 68,
      "score_components": {
        "artifact_determinism": {
          "label": "Artifact determinism",
          "value": 3
        },
        "claim_exactness": {
          "label": "Claim exactness",
          "value": 4
        },
        "research_value": {
          "label": "Research value",
          "value": 4
        },
        "feasibility": {
          "label": "Feasibility",
          "value": 1
        },
        "source_strength": {
          "label": "Source strength",
          "value": 5
        },
        "nonduplication": {
          "label": "Non-duplication",
          "value": 4
        }
      },
      "rank": null,
      "selection_reason": "Good claim boundary, but exact algebraic geometry is less mature than the selected finite arithmetic contracts.",
      "tags": [
        "geometry",
        "distances",
        "noncollinearity"
      ],
      "acceptance": null,
      "market": null,
      "provenance": [
        {
          "label": "Canonical source",
          "href": "https://www.erdosproblems.com/1082"
        },
        {
          "label": "Pinned formal surface",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/1082.lean"
        }
      ]
    },
    {
      "id": "contract:erdos-406",
      "native_id": "erdos-406",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "Resolve Erdős #406 — ternary powers of two",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "mathematics",
      "summary": "Decide the canonical question: are there only finitely many powers of 2 whose base-3 representation uses no digit 2? The accepted artifact must prove finiteness or infinitude. Finding one more example would refute the stronger “only 1, 4, 256” conjecture, but would not resolve this contract.",
      "claim_boundary": "The former finite-witness tier has been retired here: any new exponent must exceed the enormous checked range, is not cheaply verified by expanding 2ⁿ, and would refute only the stronger enumeration claim.",
      "artifact": "One complete Target.lean file. Outside the answer and proof holes, every byte of the target is fixed.",
      "rail": "FORMAL · LEAN 4",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "EQUIVALENT TO #406",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "mathematics",
        "FULL RESOLUTION",
        "FORMAL · LEAN 4"
      ],
      "acceptance": {
        "artifact": "The package checker fixes the declaration, permits answer(True) or answer(False), and rejects tactic-block escapes, sorry, admit, native_decide, run_tac, unsafe declarations, and project axioms.",
        "claim": "Lean 4.27.0 compiles against hashed canonical source and mathlib pins; a separate audit file confirms that the axiom closure contains only propext, Classical.choice, and Quot.sound.",
        "adjudication": "An independent Lean reviewer and a number theorist reproduce the build, confirm fidelity to the historical question, and clear the literature/status update."
      },
      "market": {
        "contract_id": "contract_20548c09c45a",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:41.098307"
      },
      "provenance": [
        {
          "label": "Protocol V1",
          "href": "/contracts/erdos-406/contract.json"
        },
        {
          "label": "Target",
          "href": "/contracts/erdos-406/Target.lean"
        },
        {
          "label": "Package Checker",
          "href": "/contracts/erdos-406/package-check.js"
        },
        {
          "label": "Tests",
          "href": "/contracts/erdos-406/package-check.test.js"
        },
        {
          "label": "Canonical Source",
          "href": "https://github.com/google-deepmind/formal-conjectures/blob/5a60e068cceb4edffa992dd0bdbda8c6c17185c5/FormalConjectures/ErdosProblems/406.lean"
        },
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:vdw-2-7",
      "native_id": "vdw-2-7",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "Move the lower-bound frontier for w(2,7)",
      "field": "mathematics",
      "field_label": "Mathematics",
      "family": "mathematics",
      "summary": "Submit a 2-coloring of {1,…,n}, with n ≥ 3704, containing no monochromatic 7-term arithmetic progression. An accepted coloring establishes exactly w(2,7) > n; it improves a lower bound and does not claim the exact value.",
      "claim_boundary": "An exact value w(2,7)=N is deliberately outside this contract: it would additionally require a separately specified and encoding-audited UNSAT certificate at N.",
      "artifact": "Exact UTF-8 bytes {\"coloring\":\"0101…\"} plus one LF. Character i colors integer i+1; n is the exact string length.",
      "rail": "EXACT WITNESS",
      "verification": "exact-witness",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "CERTIFIES w(2,7) > n",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "mathematics",
        "LOWER-BOUND RECORD",
        "EXACT WITNESS"
      ],
      "acceptance": {
        "artifact": "The canonicalizer rejects whitespace, extra fields, duplicate encodings, invalid UTF-8, and non-binary characters; it records hashes of the exact bytes and coloring.",
        "claim": "Both independent checkers enumerate every a,a+d,…,a+6d within {1,…,n}; neither may find a monochromatic progression and their progression counts must agree.",
        "adjudication": "The artifact is one byte-canonical JSON bit-string. An orchestrator hashes the exact file and requires two independently written deterministic checkers, enumerating every 7-term progression in different orders, to agree on every acceptance output. Acceptance also requires a clean replay and a dated frontier check against the published 3703 baseline."
      },
      "market": {
        "contract_id": "contract_9bc5e45ef11e",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:52.390026"
      },
      "provenance": [
        {
          "label": "Protocol V1",
          "href": "/contracts/vdw-2-7/contract.json"
        },
        {
          "label": "Verifier",
          "href": "/contracts/vdw-2-7/verify.js"
        },
        {
          "label": "Checker A",
          "href": "/contracts/vdw-2-7/check.js"
        },
        {
          "label": "Checker B",
          "href": "/contracts/vdw-2-7/check-independent.js"
        },
        {
          "label": "Baseline Source",
          "href": "https://arxiv.org/abs/1603.03301"
        },
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:chsh-expectation",
      "native_id": "chsh-expectation",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "The classical CHSH expectation bound — exact scope",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "lean × quantum",
      "summary": "Prove in Lean that four measurable deterministic ±1 response functions on one probability space satisfy the stronger absolute bound |E[A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁]| ≤ 2. This is the algebraic core used by local-hidden-variable arguments; the contract deliberately does not claim that this theorem alone formalizes every locality assumption or certifies an experiment.",
      "claim_boundary": "The old issuer-graded 1-TFT posting remains an external audit record, but its immutable one-sided and auto-settlement terms are not this protocol; no external bond or funds were changed.",
      "artifact": "One complete Target.lean file. Every byte outside the single marked proof hole is fixed.",
      "rail": "FORMAL · LEAN 4.32",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "DETERMINISTIC-RESPONSE CORE",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "lean × quantum",
        "ABSOLUTE EXPECTATION BOUND",
        "FORMAL · LEAN 4.32"
      ],
      "acceptance": {
        "artifact": "The package checker fixes all imports, definitions, hypotheses, signs, the absolute value, and the bound 2. It rejects indentation escapes, placeholders, project axioms, opaque or unsafe declarations, native_decide, run_tac, extern, and implemented_by.",
        "claim": "Lean recompiles at the pinned Mathlib commit and CHSH-source hash; a separate audit file accepts only propext, Classical.choice, and Quot.sound. A Lean reviewer confirms the exact integral theorem—not a weakened substitute—is in the dependency closure.",
        "adjudication": "A quantum-information reviewer confirms that the description stops at measurable deterministic response functions. Experimental statistics, measurement independence, locality modeling, stochastic reduction, and quantum violation remain outside this artifact."
      },
      "market": {
        "contract_id": "contract_17dbdfccc232",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:52.900356"
      },
      "provenance": [
        {
          "label": "Protocol V1",
          "href": "/contracts/chsh-expectation/contract.json"
        },
        {
          "label": "Target",
          "href": "/contracts/chsh-expectation/Target.lean"
        },
        {
          "label": "Shared Verifier",
          "href": "/contracts/lean-v1/verify.sh"
        },
        {
          "label": "Package Checker",
          "href": "/contracts/lean-v1/package-check.js"
        },
        {
          "label": "Negative Tests",
          "href": "/contracts/lean-v1/package-check.test.js"
        },
        {
          "label": "Pinned Mathlib Source",
          "href": "https://github.com/leanprover-community/mathlib4/blob/81a5d257c8e410db227a6665ed08f64fea08e997/Mathlib/Algebra/Star/CHSH.lean"
        },
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:tsirelson-tight",
      "native_id": "tsirelson-tight",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "Tsirelson tightness in Mathlib's own CHSH interface",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "lean × quantum",
      "summary": "Construct four Matrix (Fin 4) (Fin 4) ℂ witnesses satisfying all twelve fields of Mathlib's IsCHSHTuple, plus a nonzero vector on which the fixed CHSH operator acts by exactly 2√2. Mathlib v4.32.0 still records this witness as future work. A 2026 Lean rigidity project proves an adjacent Bell-operator eigenvalue in a different API, so this contract claims a Mathlib-native integration gap—not a first Lean proof.",
      "claim_boundary": "The contract does not copy the adjacent project's implementation or promise a Mathlib merge. The legacy 1-TFT issuer-graded posting remains external and untouched, but its terms are detached from this protocol.",
      "artifact": "One complete Target.lean file. Every byte outside the single marked proof hole is fixed; witnesses must inhabit the exact existential.",
      "rail": "FORMAL · LEAN 4.32",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "FILLS PINNED INTERFACE GAP",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "lean × quantum",
        "MATHLIB-NATIVE 4×4 WITNESS",
        "FORMAL · LEAN 4.32"
      ],
      "acceptance": {
        "artifact": "The checker rejects changes to Fin 4, complex matrices, IsCHSHTuple, any CHSH sign, v ≠ 0, or the exact 2·√2 eigenvalue. The kernel must check the witness and every proof field without placeholders or escape mechanisms.",
        "claim": "Lean recompiles at the pinned Mathlib commit and CHSH-source hash; an isolated audit accepts only the declared kernel axioms. A Lean reviewer confirms that the exact matrix existential—not a bespoke replacement predicate—is proved.",
        "adjudication": "A quantum-information reviewer compares the result with Mathlib's future-work note and the disclosed 2026 rigidity code. The ledger may record only completion of this Mathlib-native target; novelty and upstream acceptance are separate decisions."
      },
      "market": {
        "contract_id": "contract_c76617d65774",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:53.167937"
      },
      "provenance": [
        {
          "label": "Protocol V1",
          "href": "/contracts/tsirelson-tight/contract.json"
        },
        {
          "label": "Target",
          "href": "/contracts/tsirelson-tight/Target.lean"
        },
        {
          "label": "Shared Verifier",
          "href": "/contracts/lean-v1/verify.sh"
        },
        {
          "label": "Package Checker",
          "href": "/contracts/lean-v1/package-check.js"
        },
        {
          "label": "Negative Tests",
          "href": "/contracts/lean-v1/package-check.test.js"
        },
        {
          "label": "Mathlib Future Work",
          "href": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/CHSH.html"
        },
        {
          "label": "Adjacent Lean Work",
          "href": "https://github.com/trzstony/RigidityTheorem/tree/62468b0c9bdcbf9963aeeb424961fee9ad302c45"
        },
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:no-cloning",
      "native_id": "no-cloning",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "No-cloning, machine-checked",
      "field": "formal-quantum",
      "field_label": "Formal quantum",
      "family": "lean × quantum",
      "summary": "Prove in Lean 4 that for n ≥ 2 no unitary on ℂⁿ⊗ℂⁿ together with a unit blank state clones every unit vector — in the pair-indexed matrix formulation that is unblocked in today's Mathlib. Unitarity preserves inner products, so ⟨φ,ψ⟩ = ⟨φ,ψ⟩² for all unit pairs; e₀ and (e₀+e₁)/√2 refute it.",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "A sorry-free proof matching the published Matrix.unitaryGroup skeleton, pinned Mathlib, axiom audit clean. Kernel acceptance settles trustlessly on the VVUQ Lean rail. Encouraged foundation: leanprover-community/physlib QuantumInfo (density matrices, channels, entropy formalizations) alongside Mathlib.",
      "rail": "FORMAL · LEAN 4",
      "verification": "formal-proof",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "lean × quantum",
        "PUBLISHED OBJECTIVE",
        "FORMAL · LEAN 4"
      ],
      "acceptance": {
        "artifact": "A sorry-free proof matching the published Matrix.unitaryGroup skeleton, pinned Mathlib, axiom audit clean. Kernel acceptance settles trustlessly on the VVUQ Lean rail. Encouraged foundation: leanprover-community/physlib QuantumInfo (density matrices, channels, entropy formalizations) alongside Mathlib.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": {
        "contract_id": "contract_7842b402caa4",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:53.429631"
      },
      "provenance": [
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:px-crossing",
      "native_id": "px-crossing",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "A foundry-legal waveguide crossing: 365 nm of bandwidth, −25 dB of crosstalk, zero DRC violations",
      "field": "photonics",
      "field_label": "Photonics",
      "family": "photonic ic",
      "summary": "On 220 nm SOI, design a 90° single-mode crossing ≤5×5 µm with insertion loss ≤0.3 dB and crosstalk ≤−25 dB at every wavelength from 1260–1625 nm (O+C+L), surviving ±20 nm fabrication bias — and passing a real MPW ruleset (200 nm feature / 250 nm gap) with zero KLayout DRC violations. Published devices achieve the bandwidth, the crosstalk depth, or manufacturability — never all three.",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "Parametric gdsfactory source regenerates the GDS bit-for-bit (sha256 match); issuer reruns the tidy3d FDTD sweep (≥37 samples, all four ports), the ±20 nm bias sweep, and the DRC deck from source. Gates preregistered; no precomputed spectra accepted.",
      "rail": "SIMULATION · SOURCE + DRC",
      "verification": "reproducible-computation",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "photonic ic",
        "PUBLISHED OBJECTIVE",
        "SIMULATION · SOURCE + DRC"
      ],
      "acceptance": {
        "artifact": "Parametric gdsfactory source regenerates the GDS bit-for-bit (sha256 match); issuer reruns the tidy3d FDTD sweep (≥37 samples, all four ports), the ±20 nm bias sweep, and the DRC deck from source. Gates preregistered; no precomputed spectra accepted.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": {
        "contract_id": "contract_113aeaa84710",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:53.674416"
      },
      "provenance": [
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:px-edge-coupler",
      "native_id": "px-edge-coupler",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "A sub-0.4 dB O-band edge coupler in a 40 µm taper",
      "field": "photonics",
      "field_label": "Photonics",
      "family": "photonic ic",
      "summary": "Fiber-to-chip edge coupler on 220 nm SOI for the full O-band: ≤0.4 dB average loss (≤0.55 dB worst-case) for BOTH polarizations against a 3 µm MFD fiber, PDL ≤0.10 dB, in a taper ≤40 µm long — DRC-clean. The measured loss record (~0.54 dB) needs a much longer bi-level taper; the compact inverse-designed one loses 3.4 dB. Nobody sits at this intersection.",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "Issuer regenerates GDS from the parametric source (hash match) and reruns the 3D FDTD Gaussian-injection sweep for TE0/TM0 across 1260–1360 nm plus ±20 nm/±10 nm tolerance biases and the DRC deck. Reproduced loss must match within ±0.05 dB.",
      "rail": "SIMULATION · SOURCE + DRC",
      "verification": "reproducible-computation",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "photonic ic",
        "PUBLISHED OBJECTIVE",
        "SIMULATION · SOURCE + DRC"
      ],
      "acceptance": {
        "artifact": "Issuer regenerates GDS from the parametric source (hash match) and reruns the 3D FDTD Gaussian-injection sweep for TE0/TM0 across 1260–1360 nm plus ±20 nm/±10 nm tolerance biases and the DRC deck. Reproduced loss must match within ±0.05 dB.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": {
        "contract_id": "contract_51050a4c8d5d",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:53.964615"
      },
      "provenance": [
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:px-awg",
      "native_id": "px-awg",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "A compact flat-top 100 GHz silicon AWG with −30 dB crosstalk",
      "field": "photonics",
      "field_label": "Photonics",
      "family": "photonic ic",
      "summary": "8-channel, 100 GHz ITU-grid C-band AWG on 220 nm SOI: flat-top passband (1-dB width ≥0.40 nm, ripple ≤0.5 dB), excess loss ≤2.5 dB, adjacent crosstalk ≤−30 dB, core ≤400×500 µm, DRC-clean, robust in a σ=5 nm + ±25 nm Monte Carlo. Ultra-low-crosstalk Gaussian designs and flat-top MMI designs sit on opposite ends of a known trade-off — close it.",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "Two-level rerunnable verification: component S-parameters (FDTD/EME of star-coupler, aperture, bends) composed into a documented circuit-level spectral model; ≥100-sample Monte Carlo; KLayout DRC. Issuer reruns both levels from regenerated source; centers within ±0.02 nm.",
      "rail": "SIMULATION · SOURCE + DRC",
      "verification": "reproducible-computation",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "photonic ic",
        "PUBLISHED OBJECTIVE",
        "SIMULATION · SOURCE + DRC"
      ],
      "acceptance": {
        "artifact": "Two-level rerunnable verification: component S-parameters (FDTD/EME of star-coupler, aperture, bends) composed into a documented circuit-level spectral model; ≥100-sample Monte Carlo; KLayout DRC. Issuer reruns both levels from regenerated source; centers within ±0.02 nm.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": {
        "contract_id": "contract_cd53a3bdc0ed",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:54.276213"
      },
      "provenance": [
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:icl-mzi8",
      "native_id": "icl-mzi8",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "In-context learning through a frozen photonic mesh",
      "field": "machine-learning-photonics",
      "field_label": "ML × photonics",
      "family": "icl × photonics",
      "summary": "On a simulated 8-mode MZI mesh (56 phase shifters) under a frozen, physically-faithful noise model — shot noise + thermo-optic crosstalk — perform in-context linear regression with K=16 examples where the phase configuration is bit-identical (sha256-checked) across every forward pass: all adaptation happens in the light, never in the weights. Clear MSE ≤ 1.15× the noiseless digital linear-attention baseline on 500 sealed holdout seeds, with energy-per-token reported.",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "Everything committed by hash before submissions open: noise model, mesh topology, function class, eval harness, public seeds — plus a sealed holdout seed set (hash published, contents revealed at grading). Submission is a static 56-value phase configuration, hash-logged before the holdout unseals. Verifier reruns in a pinned container; any phase mutation across the 17 passes fails outright; bootstrap CI upper bound must clear too.",
      "rail": "CODE + PROOF · FROZEN EVALUATION",
      "verification": "reproducible-computation",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "icl × photonics",
        "PUBLISHED OBJECTIVE",
        "CODE + PROOF · FROZEN EVALUATION"
      ],
      "acceptance": {
        "artifact": "Everything committed by hash before submissions open: noise model, mesh topology, function class, eval harness, public seeds — plus a sealed holdout seed set (hash published, contents revealed at grading). Submission is a static 56-value phase configuration, hash-logged before the holdout unseals. Verifier reruns in a pinned container; any phase mutation across the 17 passes fails outright; bootstrap CI upper bound must clear too.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": {
        "contract_id": "contract_4c71c87de617",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:54.515060"
      },
      "provenance": [
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:icl-energy-bound",
      "native_id": "icl-energy-bound",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "An energy-constrained sample-complexity bound for photonic attention",
      "field": "machine-learning-photonics",
      "field_label": "ML × photonics",
      "family": "icl × photonics",
      "summary": "Prove a one-sided closed-form bound on the in-context-learning risk of single-layer linear attention when its weight matrices must be synthesized by an N-mode interferometer under a total phase-shifter energy budget E — bridging the Lu et al. asymptotic ICL theory (free matrices) with the O(1/√N) programmable-photonics energy limit (arXiv:2408.09673). No paper connects the two; either direction is a new theorem.",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "Written proof + a machine-evaluable bound(K,N,E) function. Four frozen checks: line-by-line faithfulness audit against hash-committed reference equations; the bound must hold against Monte Carlo CIs at all 10 frozen (N,K,E) grid points; a sympy script verifies the E→∞ limit recovers Lu et al.; numeric check that E→0 recovers the prior-variance floor. 9/10 grid points is a falsified conjecture, not a proof.",
      "rail": "CODE + PROOF · FROZEN EVALUATION",
      "verification": "reproducible-computation",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "icl × photonics",
        "PUBLISHED OBJECTIVE",
        "CODE + PROOF · FROZEN EVALUATION"
      ],
      "acceptance": {
        "artifact": "Written proof + a machine-evaluable bound(K,N,E) function. Four frozen checks: line-by-line faithfulness audit against hash-committed reference equations; the bound must hold against Monte Carlo CIs at all 10 frozen (N,K,E) grid points; a sympy script verifies the E→∞ limit recovers Lu et al.; numeric check that E→0 recovers the prior-variance floor. 9/10 grid points is a falsified conjecture, not a proof.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": {
        "contract_id": "contract_8211efa35792",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-26T00:12:54.755405"
      },
      "provenance": [
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:wc-01",
      "native_id": "wc-01",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published",
      "title": "Shape-aware visual verification: a deterministic metric that beats IoU",
      "field": "research-infrastructure",
      "field_label": "Research infrastructure",
      "family": "pixcell · wc-01",
      "summary": "A pure, deterministic shape-agreement metric for calibrated device rasters that improves on IoU and the boundary baseline under blinded judgments — without rewarding area-filling shortcuts. Five preregistered acceptance tests (T1–T5).",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "Issuer-graded against T1–T5 under the PixCell conductor-gate discipline: frozen perturbation suite, adversarial pairs must score worse, one rerun script reproduces all scores.",
      "rail": "ISSUER · PREREGISTERED GATES",
      "verification": "reviewed-artifact",
      "profile": null,
      "maturity": "published",
      "status": "published",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "pixcell · wc-01",
        "PUBLISHED OBJECTIVE",
        "ISSUER · PREREGISTERED GATES"
      ],
      "acceptance": {
        "artifact": "Issuer-graded against T1–T5 under the PixCell conductor-gate discipline: frozen perturbation suite, adversarial pairs must score worse, one rerun script reproduces all scores.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": null,
      "provenance": [
        {
          "label": "Claim Source",
          "href": "https://pixcell.qpaig.com/#work-contracts"
        }
      ]
    },
    {
      "id": "contract:wc-02",
      "native_id": "wc-02",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "Process-faithful 2D→3D retargeting: one program, four process stacks",
      "field": "research-infrastructure",
      "field_label": "Research infrastructure",
      "family": "pixcell · wc-02",
      "summary": "One editable primitive program retargeted across four versioned process models with explicit 3D realization — AST-identical topology, preregistered functional/footprint/process/convergence gates, independently rerun Tidy3D artifacts.",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "Gates sha256-committed before simulation; issuer reruns at least one stack from shipped inputs; mesh-convergence and energy-conservation audits recomputed from field data.",
      "rail": "ISSUER · PREREGISTERED GATES",
      "verification": "reviewed-artifact",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "pixcell · wc-02",
        "PUBLISHED OBJECTIVE",
        "ISSUER · PREREGISTERED GATES"
      ],
      "acceptance": {
        "artifact": "Gates sha256-committed before simulation; issuer reruns at least one stack from shipped inputs; mesh-convergence and energy-conservation audits recomputed from field data.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": {
        "contract_id": "contract_f55f58150008",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-25T20:27:36.599246"
      },
      "provenance": [
        {
          "label": "Claim Source",
          "href": "https://pixcell.qpaig.com/#work-contracts"
        },
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:wc-03",
      "native_id": "wc-03",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "A representation & scale dataset with protected splits",
      "field": "research-infrastructure",
      "field_label": "Research infrastructure",
      "family": "pixcell · wc-03",
      "summary": "A versioned, licensed corpus separating topology, scale evidence, and executable reconstruction; salted-hash protected splits with a leakage audit; abstention enforced where scale is unknowable; joint vs staged curricula at matched compute.",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "Machine-checkable license manifest, schema-validated annotation layers, zero split crossover in the shipped near-duplicate audit, one rerun script reproduces the headline table.",
      "rail": "ISSUER · PREREGISTERED GATES",
      "verification": "reviewed-artifact",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "pixcell · wc-03",
        "PUBLISHED OBJECTIVE",
        "ISSUER · PREREGISTERED GATES"
      ],
      "acceptance": {
        "artifact": "Machine-checkable license manifest, schema-validated annotation layers, zero split crossover in the shipped near-duplicate audit, one rerun script reproduces the headline table.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": {
        "contract_id": "contract_3a48cb45d01f",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-25T20:27:37.008296"
      },
      "provenance": [
        {
          "label": "Claim Source",
          "href": "https://pixcell.qpaig.com/#work-contracts"
        },
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "contract:wc-04",
      "native_id": "wc-04",
      "kind": "published-contract",
      "source": "asymmetry-contracts",
      "source_label": "Asymmetry published contract",
      "stage": "published",
      "stage_label": "Published · VVUQ listed",
      "title": "The smallest open model that survives PixCell's L0–L4 suite",
      "field": "research-infrastructure",
      "field_label": "Research infrastructure",
      "family": "pixcell · wc-04",
      "summary": "Freeze a candidate registry, per-tier floors, and identical budgets by hash before any evaluation; find the smallest open-weight model (by active parameters) that clears every tier and untouched paper figures; release everything.",
      "claim_boundary": "Published display record. The index preserves its stated verifier but does not infer a stronger research-status effect.",
      "artifact": "Issuer recomputes confidence bounds from shipped raw attempts via the included bootstrap script; generated programs must pass the conductor gate's purity rules; 'smallest' scoped to the dated registry hash.",
      "rail": "ISSUER · PREREGISTERED GATES",
      "verification": "reviewed-artifact",
      "profile": null,
      "maturity": "published_with_market_record",
      "status": "open",
      "status_effect": "Acceptance applies only to the stated objective and verifier.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": null,
      "tags": [
        "pixcell · wc-04",
        "PUBLISHED OBJECTIVE",
        "ISSUER · PREREGISTERED GATES"
      ],
      "acceptance": {
        "artifact": "Issuer recomputes confidence bounds from shipped raw attempts via the included bootstrap script; generated programs must pass the conductor gate's purity rules; 'smallest' scoped to the dated registry hash.",
        "claim": "Review the accepted artifact against the exact published statement and its declared tolerances.",
        "adjudication": "Any novelty or frontier update remains a separate expert decision."
      },
      "market": {
        "contract_id": "contract_abb35955056a",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-25T20:27:37.226080"
      },
      "provenance": [
        {
          "label": "Claim Source",
          "href": "https://pixcell.qpaig.com/#work-contracts"
        },
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "vvuq:contract_7c2aeeaa54ca",
      "native_id": "contract_7c2aeeaa54ca",
      "kind": "external-contract",
      "source": "vvuq-external",
      "source_label": "Curated external VVUQ",
      "stage": "external",
      "stage_label": "External · unscored",
      "title": "NERE fidelity at an unplotted detuning",
      "field": "quantum-research",
      "field_label": "Quantum research",
      "family": "networked-entanglement",
      "summary": "Using the protocol of the NERE paper, what is the singlet fidelity F at detuning delta/gamma = 4.8247, a value NOT plotted in Fig. 1? Answer within 1e-4 absolute.",
      "claim_boundary": "The public description names the protocol but does not pin a paper revision, model implementation, parameter conventions, or reference value. The tolerance is exact only after those are fixed.",
      "artifact": "A singlet-fidelity value at the specified detuning within 1e-4 absolute error, accompanied by the model or code used to reproduce it.",
      "rail": "Numerical reproduction",
      "verification": "numeric-reproduction",
      "profile": null,
      "maturity": "external_spec_correspondence_unreviewed",
      "status": "open",
      "status_effect": "No Asymmetry status effect; settlement remains with the external issuer.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": "Included for topical relevance and a concrete output; not evaluated under the Asymmetry suitability rubric.",
      "tags": [
        "NERE",
        "entanglement",
        "interpolation",
        "numerical"
      ],
      "acceptance": {
        "artifact": "Reproduce the requested numerical output and tolerance from an explicitly identified model or source.",
        "claim": "The public description names the protocol but does not pin a paper revision, model implementation, parameter conventions, or reference value. The tolerance is exact only after those are fixed.",
        "adjudication": "The VVUQ issuer, not Asymmetry, controls acceptance and settlement."
      },
      "market": {
        "contract_id": "contract_7c2aeeaa54ca",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-20T17:25:34.926968"
      },
      "provenance": [
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "vvuq:contract_8bdf61d1d573",
      "native_id": "contract_8bdf61d1d573",
      "kind": "external-contract",
      "source": "vvuq-external",
      "source_label": "Curated external VVUQ",
      "stage": "external",
      "stage_label": "External · unscored",
      "title": "One-round NERE Bell-pair distillation",
      "field": "quantum-research",
      "field_label": "Quantum research",
      "family": "networked-entanglement",
      "summary": "Take two Bell pairs, each at the fidelity of the 10th plotted point in Fig. 1. After ONE round of DEJMPS distillation, what are the output fidelity F_out and the success probability p_success (each to 1e-4)? Which distillation protocol gives the higher output fidelity from these inputs, DEJMPS or EPL?",
      "claim_boundary": "The source figure, point ordering, state model, and exact DEJMPS/EPL conventions are not pinned in the public description. Those correspondence choices require issuer review.",
      "artifact": "The DEJMPS output fidelity and success probability to 1e-4, plus a DEJMPS-versus-EPL comparison, with the input point and equations made reproducible.",
      "rail": "Numerical reproduction",
      "verification": "numeric-reproduction",
      "profile": null,
      "maturity": "external_spec_correspondence_unreviewed",
      "status": "open",
      "status_effect": "No Asymmetry status effect; settlement remains with the external issuer.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": "Included for topical relevance and a concrete output; not evaluated under the Asymmetry suitability rubric.",
      "tags": [
        "NERE",
        "entanglement distillation",
        "DEJMPS",
        "EPL",
        "numerical"
      ],
      "acceptance": {
        "artifact": "Reproduce the requested numerical output and tolerance from an explicitly identified model or source.",
        "claim": "The source figure, point ordering, state model, and exact DEJMPS/EPL conventions are not pinned in the public description. Those correspondence choices require issuer review.",
        "adjudication": "The VVUQ issuer, not Asymmetry, controls acceptance and settlement."
      },
      "market": {
        "contract_id": "contract_8bdf61d1d573",
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        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-20T17:25:34.753075"
      },
      "provenance": [
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    },
    {
      "id": "vvuq:contract_9ebec1ae41bb",
      "native_id": "contract_9ebec1ae41bb",
      "kind": "external-contract",
      "source": "vvuq-external",
      "source_label": "Curated external VVUQ",
      "stage": "external",
      "stage_label": "External · unscored",
      "title": "NERE threshold operating point",
      "field": "quantum-research",
      "field_label": "Quantum research",
      "family": "networked-entanglement",
      "summary": "The NERE protocol trades rate for fidelity via detuning. Find the SMALLEST detuning delta/gamma at which the singlet fidelity reaches F >= 0.999, and report the scattered intensity |alpha|^2 at that detuning (which sets the achievable rate). Detuning to 0.01 precision.",
      "claim_boundary": "The VVUQ description does not pin paper bytes, equations, code, tie-breaking, or a global-minimum certificate. Treat acceptance as issuer-reviewed numerical reproduction, not an independently settled optimum.",
      "artifact": "The smallest detuning, to 0.01, reaching the stated singlet-fidelity threshold, together with the scattered intensity at that operating point and a reproducible calculation.",
      "rail": "Numerical reproduction",
      "verification": "numeric-reproduction",
      "profile": null,
      "maturity": "external_spec_correspondence_unreviewed",
      "status": "open",
      "status_effect": "No Asymmetry status effect; settlement remains with the external issuer.",
      "score": null,
      "score_components": null,
      "rank": null,
      "selection_reason": "Included for topical relevance and a concrete output; not evaluated under the Asymmetry suitability rubric.",
      "tags": [
        "NERE",
        "entanglement",
        "optimization",
        "numerical"
      ],
      "acceptance": {
        "artifact": "Reproduce the requested numerical output and tolerance from an explicitly identified model or source.",
        "claim": "The VVUQ description does not pin paper bytes, equations, code, tie-breaking, or a global-minimum certificate. Treat acceptance as issuer-reviewed numerical reproduction, not an independently settled optimum.",
        "adjudication": "The VVUQ issuer, not Asymmetry, controls acceptance and settlement."
      },
      "market": {
        "contract_id": "contract_9ebec1ae41bb",
        "status": "open",
        "funded": true,
        "escrow_tft": 1,
        "advertised_tft": 1,
        "claims_count": 1,
        "created_at": "2026-07-21T18:35:48.478083"
      },
      "provenance": [
        {
          "label": "VVUQ market",
          "href": "https://vvuq.dirkenglund.org/bounties"
        }
      ]
    }
  ]
}