benchmarks: add projective-plane Zarankiewicz certificate

Author: Grace Lee Rui Yue

benchmarks/datasets/conjecture-probes-v1/members/zarankiewicz-projective-plane-certificate.toml

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+schema_version="2"
+task_id="zarankiewicz-projective-plane-certificate"
+task_name="jacobian/zarankiewicz-projective-plane-certificate"
+evaluation_kind="regression"
+domain="mathematical-sciences"
+primary_domain="graph-theory"
+field="extremal-combinatorics"
+provenance_class="authored-conjecture-probe"
+provenance_ref="spreadsheet:unresolved-conjecture/C-044@2026-08-07"
+assurance_ceiling="CHECKED"
+required_provider="core"
+environment_profile="core-python-minimal-verifier"
+verifier_contract_version="1"
+evaluation_owner="jacobian/conjecture-probes-v1"

benchmarks/datasets/conjecture-probes-v1/zarankiewicz-projective-plane-certificate/README.md

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+# Zarankiewicz projective-plane certificate
+
+This public Regression benchmark turns retained conjecture row C-044 into a
+closed finite task. The agent must construct the order-three projective-plane
+incidence graph and an exact extremal certificate for
+`z(13,13;2,2)=52`.
+
+The primary reasoning objective is finite extremal-certificate construction.
+The verifier does not trust a named construction: it canonicalizes projective
+classes, recomputes incidence over `F3`, exhaustively rejects every `K2,2`, and
+checks the pair-count upper bound excluding 53 edges.
+
+## Curation and difficulty
+
+Provisional difficulty is Hard: the response coordinates finite-field
+normalization, a complete 52-edge incidence relation, two-sided pair
+intersection counts, and the convexity/pair-budget upper bound. Weaker agents
+are expected to omit projective representatives or confuse affine and
+projective incidence; stronger agents should produce a replayable certificate.
+
+The shortcut audit rejects tiny planes, copied edge counts without coordinates,
+partial incidence tables, and label-only extremality claims. Although the
+projective-plane construction is public, the complete typed certificate is
+independently recomputed rather than answer-matched. This task is distinct from
+existing coloring, minor, homology, and generic incidence-determinant tasks.
+
+## Provenance and boundary
+
+- Inventory source: retained `Unresolved Conjectures` row C-044 (Zarankiewicz).
+- Mathematical source: Kővári, Sós, and Turán, *On a problem of K. Zarankiewicz*,
+  Colloquium Math. 3 (1954), 50–57.
+- Construction source: the standard point-line incidence structure of
+  `PG(2,3)`; all finite claims are reconstructed locally.
+
+Full reward means `CHECKED` for this one finite graph and its exact finite
+upper bound. It does not prove any unresolved asymptotic or general
+Zarankiewicz statement.

benchmarks/datasets/conjecture-probes-v1/zarankiewicz-projective-plane-certificate/environment/Dockerfile

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+FROM python:3.12-slim@sha256:57cd7c3a7a273101a6485ba99423ee568157882804b1124b4dd04266317710de
+COPY input.json submission_schema.json /app/
+WORKDIR /app

benchmarks/datasets/conjecture-probes-v1/zarankiewicz-projective-plane-certificate/environment/input.json

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+{"task_id":"jacobian/zarankiewicz-projective-plane-certificate","conjecture_id":"C-044","field_order":3,"part_size":13,"required_degree":4,"required_edge_count":52,"scope":"pg2-f3-zarankiewicz-k22-extremal-certificate-v1"}

benchmarks/datasets/conjecture-probes-v1/zarankiewicz-projective-plane-certificate/environment/submission_schema.json

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+{
+  "$defs": {
+    "degrees": {
+      "items": {
+        "const": 4
+      },
+      "maxItems": 13,
+      "minItems": 13,
+      "type": "array"
+    },
+    "edge": {
+      "items": {
+        "maximum": 12,
+        "minimum": 0,
+        "type": "integer"
+      },
+      "maxItems": 2,
+      "minItems": 2,
+      "type": "array"
+    },
+    "pair_count": {
+      "additionalProperties": false,
+      "properties": {
+        "common_neighbors": {
+          "const": 1
+        },
+        "pair": {
+          "$ref": "#/$defs/edge"
+        }
+      },
+      "required": [
+        "pair",
+        "common_neighbors"
+      ],
+      "type": "object"
+    },
+    "pair_counts": {
+      "items": {
+        "$ref": "#/$defs/pair_count"
+      },
+      "maxItems": 78,
+      "minItems": 78,
+      "type": "array"
+    },
+    "triple": {
+      "items": {
+        "maximum": 2,
+        "minimum": 0,
+        "type": "integer"
+      },
+      "maxItems": 3,
+      "minItems": 3,
+      "type": "array"
+    },
+    "triples": {
+      "items": {
+        "$ref": "#/$defs/triple"
+      },
+      "maxItems": 13,
+      "minItems": 13,
+      "type": "array",
+      "uniqueItems": true
+    }
+  },
+  "$schema": "https://json-schema.org/draft/2020-12/schema",
+  "additionalProperties": false,
+  "properties": {
+    "claimed_assurance": {
+      "enum": [
+        "UNVERIFIED",
+        "COMPUTED",
+        "CHECKED",
+        "VERIFIED"
+      ]
+    },
+    "completeness": {
+      "const": "COMPLETE"
+    },
+    "conclusion": {
+      "const": "PG2_F3_ZARANKIEWICZ_EXTREMAL_CERTIFICATE"
+    },
+    "evidence": {
+      "items": {
+        "additionalProperties": false,
+        "properties": {
+          "path": {
+            "const": "evidence/answer.txt"
+          },
+          "sha256": {
+            "pattern": "^sha256:[0-9a-f]{64}$",
+            "type": "string"
+          }
+        },
+        "required": [
+          "path",
+          "sha256"
+        ],
+        "type": "object"
+      },
+      "maxItems": 1,
+      "minItems": 1,
+      "type": "array"
+    },
+    "limitations": {
+      "const": [
+        "ONE_FINITE_PG2_F3_INSTANCE",
+        "EXACT_K22_FREE_PAIR_COUNT_REPLAY",
+        "NO_GENERAL_ZARANKIEWICZ_CONCLUSION"
+      ]
+    },
+    "result": {
+      "additionalProperties": false,
+      "properties": {
+        "edge_count": {
+          "const": 52
+        },
+        "edges": {
+          "items": {
+            "$ref": "#/$defs/edge"
+          },
+          "maxItems": 52,
+          "minItems": 52,
+          "type": "array",
+          "uniqueItems": true
+        },
+        "excluded_edge_count": {
+          "const": 53
+        },
+        "left_degrees": {
+          "$ref": "#/$defs/degrees"
+        },
+        "left_pair_common_counts": {
+          "$ref": "#/$defs/pair_counts"
+        },
+        "lines": {
+          "$ref": "#/$defs/triples"
+        },
+        "pair_budget": {
+          "const": 78
+        },
+        "points": {
+          "$ref": "#/$defs/triples"
+        },
+        "right_degrees": {
+          "$ref": "#/$defs/degrees"
+        },
+        "right_pair_common_counts": {
+          "$ref": "#/$defs/pair_counts"
+        }
+      },
+      "required": [
+        "points",
+        "lines",
+        "edges",
+        "left_degrees",
+        "right_degrees",
+        "left_pair_common_counts",
+        "right_pair_common_counts",
+        "edge_count",
+        "pair_budget",
+        "excluded_edge_count"
+      ],
+      "type": "object"
+    },
+    "scope": {
+      "const": "pg2-f3-zarankiewicz-k22-extremal-certificate-v1",
+      "type": "string"
+    },
+    "task_id": {
+      "const": "jacobian/zarankiewicz-projective-plane-certificate"
+    }
+  },
+  "required": [
+    "task_id",
+    "conclusion",
+    "result",
+    "claimed_assurance",
+    "scope",
+    "completeness",
+    "evidence",
+    "limitations"
+  ],
+  "type": "object"
+}

benchmarks/datasets/conjecture-probes-v1/zarankiewicz-projective-plane-certificate/instruction.md

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+# Exact Zarankiewicz certificate from the projective plane over F3
+
+Construct the point-line incidence graph of the projective plane over
+`F3`. Submit the 13 normalized projective points, the 13 normalized projective
+lines, all 52 incidence edges, and exact pair-count data proving that the graph
+is 4-regular and contains no `K_{2,2}`. Normalization means that the first
+nonzero coordinate of every triple is `1` modulo 3.
+
+The verifier reconstructs projective equivalence classes, recomputes every
+dot-product incidence, checks degrees and duplicate-free coverage, exhaustively
+checks all pairs on both sides, and replays the extremal upper bound. For a
+`K_{2,2}`-free bipartite graph with 13 vertices on each side and left degrees
+`d_i`, the 13 right-vertex pairs can be used at most once, so
+`sum_i binom(d_i,2) <= binom(13,2)`. Convexity then rules out 53 edges, while
+the submitted 52-edge construction attains the bound.
+
+Evidence is matching JSON with exactly `schema_version`, `task_id`, `result`,
+and `limitations`. This finite certificate establishes only
+`z(13,13;2,2)=52`; it does not settle the general Zarankiewicz problem.
+
+<!-- BEGIN PUBLIC CONTRACT SUBMISSION BLOCK -->
+## Submission
+
+Exact PG(2,3) incidence and finite extremal replay only; no general Zarankiewicz conclusion.
+
+Write `/app/submission.json` to the exact schema in `environment/submission_schema.json`. The submission envelope requires `task_id`, `conclusion`, `result`, `claimed_assurance`, `scope`, `completeness`, `evidence`, and `limitations`.
+
+- **Conclusion:** exactly `PG2_F3_ZARANKIEWICZ_EXTREMAL_CERTIFICATE`
+- **Assurance:** scoreable values are `UNVERIFIED`, `COMPUTED`, `CHECKED` (ceiling `CHECKED`); the submission schema accepts any of `UNVERIFIED`, `COMPUTED`, `CHECKED`, `VERIFIED` but only scoreable assurances receive credit.
+- **Scope:** the exact value declared in `submission_schema.json`
+- **Completeness:** `COMPLETE`.
+- **Evidence:** 1-1 item(s); allowed path(s): `evidence/answer.txt`; digest must match `^sha256:[0-9a-f]{64}$`.
+- **Evidence media types:** `text/plain`.
+- **Required artifact filenames:** `evidence/answer.txt`.
+<!-- END PUBLIC CONTRACT SUBMISSION BLOCK -->

benchmarks/datasets/conjecture-probes-v1/zarankiewicz-projective-plane-certificate/solution/solve.py

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+from __future__ import annotations
+
+import argparse
+import hashlib
+import itertools
+import json
+from pathlib import Path
+
+TASK_ID = "jacobian/zarankiewicz-projective-plane-certificate"
+LIMITATIONS = [
+    "ONE_FINITE_PG2_F3_INSTANCE",
+    "EXACT_K22_FREE_PAIR_COUNT_REPLAY",
+    "NO_GENERAL_ZARANKIEWICZ_CONCLUSION",
+]
+
+
+def projective_triples() -> list[list[int]]:
+    triples = []
+    for value in itertools.product(range(3), repeat=3):
+        if value == (0, 0, 0):
+            continue
+        first = next(x for x in value if x)
+        inverse = 1 if first == 1 else 2
+        normalized = tuple((inverse * x) % 3 for x in value)
+        if normalized == value:
+            triples.append(list(value))
+    return sorted(triples)
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--root", type=Path, default=Path("/app"))
+    root = parser.parse_args().root
+    points = projective_triples()
+    lines = projective_triples()
+    edges = [
+        [i, j]
+        for i, point in enumerate(points)
+        for j, line in enumerate(lines)
+        if sum(a * b for a, b in zip(point, line, strict=True)) % 3 == 0
+    ]
+    pair_rows = [
+        {"pair": [i, j], "common_neighbors": 1}
+        for i, j in itertools.combinations(range(13), 2)
+    ]
+    result = {
+        "points": points,
+        "lines": lines,
+        "edges": edges,
+        "left_degrees": [4] * 13,
+        "right_degrees": [4] * 13,
+        "left_pair_common_counts": pair_rows,
+        "right_pair_common_counts": pair_rows,
+        "edge_count": 52,
+        "pair_budget": 78,
+        "excluded_edge_count": 53,
+    }
+    payload = {
+        "schema_version": "1",
+        "task_id": TASK_ID,
+        "result": result,
+        "limitations": LIMITATIONS,
+    }
+    evidence = root / "evidence/answer.txt"
+    evidence.parent.mkdir(parents=True, exist_ok=True)
+    evidence.write_text(
+        json.dumps(payload, sort_keys=True, separators=(",", ":")) + "\n"
+    )
+    submission = {
+        "task_id": TASK_ID,
+        "conclusion": "PG2_F3_ZARANKIEWICZ_EXTREMAL_CERTIFICATE",
+        "result": result,
+        "claimed_assurance": "CHECKED",
+        "scope": "pg2-f3-zarankiewicz-k22-extremal-certificate-v1",
+        "completeness": "COMPLETE",
+        "evidence": [
+            {
+                "path": "evidence/answer.txt",
+                "sha256": "sha256:" + hashlib.sha256(evidence.read_bytes()).hexdigest(),
+            }
+        ],
+        "limitations": LIMITATIONS,
+    }
+    (root / "submission.json").write_text(json.dumps(submission, sort_keys=True) + "\n")
+
+
+if __name__ == "__main__":
+    main()

benchmarks/datasets/conjecture-probes-v1/zarankiewicz-projective-plane-certificate/solution/solve.sh

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+#!/bin/sh
+set -eu
+python /solution/solve.py