benchmarks: add finite Hadwiger minor certificate

Author: Grace Lee Rui Yue

benchmarks/datasets/conjecture-probes-v1/hadwiger-triangle-free-minor-certificate/README.md

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+# Hadwiger triangle-free minor certificate
+
+Regression-family task from retained row C-008. Its sole objective is a fully
+checkable finite coloring/minor relationship: exhaustive rejection of all
+three-colorings plus a connected `K4` branch-set model. Triangle-free and
+minimum-degree requirements prevent a trivial complete-graph witness.
+
+The verifier accepts vertex relabelings and alternative qualifying graphs.
+Difficulty is provisional Hard due to simultaneous construction, coloring,
+exhaustive lower-bound verification, and minor connectivity/adjoinment.
+
+Source context: <https://mathworld.wolfram.com/HadwigerConjecture.html>, status
+checked 2026-08-06. The authored finite fixture does not prove the conjecture.

benchmarks/datasets/conjecture-probes-v1/hadwiger-triangle-free-minor-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/hadwiger-triangle-free-minor-certificate/environment/input.json

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+{"task_id":"jacobian/hadwiger-triangle-free-minor-certificate","conjecture_id":"C-008","vertex_count":11,"edge_count":20,"required_chromatic_number":4,"required_minor_order":4,"require_triangle_free":true,"scope":"triangle-free-11-vertex-hadwiger-instance-v1"}

benchmarks/datasets/conjecture-probes-v1/hadwiger-triangle-free-minor-certificate/environment/submission_schema.json

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+{
+  "$defs": {
+    "branch": {
+      "items": {
+        "maximum": 10,
+        "minimum": 0,
+        "type": "integer"
+      },
+      "maxItems": 8,
+      "minItems": 1,
+      "type": "array",
+      "uniqueItems": true
+    },
+    "edge": {
+      "items": {
+        "maximum": 10,
+        "minimum": 0,
+        "type": "integer"
+      },
+      "maxItems": 2,
+      "minItems": 2,
+      "type": "array"
+    }
+  },
+  "$schema": "https://json-schema.org/draft/2020-12/schema",
+  "additionalProperties": false,
+  "properties": {
+    "claimed_assurance": {
+      "enum": [
+        "UNVERIFIED",
+        "COMPUTED",
+        "CHECKED",
+        "VERIFIED"
+      ]
+    },
+    "completeness": {
+      "const": "COMPLETE"
+    },
+    "conclusion": {
+      "const": "FINITE_HADWIGER_K4_CERTIFICATE"
+    },
+    "evidence": {
+      "items": {
+        "additionalProperties": false,
+        "properties": {
+          "path": {
+            "const": "evidence/answer.txt"
+          },
+          "sha256": {
+            "pattern": "^sha256:[0-9a-f]{64}$",
+            "type": "string"
+          }
+        },
+        "required": [
+          "path",
+          "sha256"
+        ],
+        "type": "object"
+      },
+      "maxItems": 1,
+      "minItems": 1,
+      "type": "array"
+    },
+    "limitations": {
+      "const": [
+        "ONE_TRIANGLE_FREE_11_VERTEX_GRAPH",
+        "EXHAUSTIVE_THREE_COLOR_REJECTION",
+        "NO_GLOBAL_HADWIGER_CONCLUSION"
+      ]
+    },
+    "result": {
+      "additionalProperties": false,
+      "properties": {
+        "branch_sets": {
+          "items": {
+            "$ref": "#/$defs/branch"
+          },
+          "maxItems": 4,
+          "minItems": 4,
+          "type": "array"
+        },
+        "chromatic_number": {
+          "const": 4
+        },
+        "edges": {
+          "items": {
+            "$ref": "#/$defs/edge"
+          },
+          "maxItems": 20,
+          "minItems": 20,
+          "type": "array",
+          "uniqueItems": true
+        },
+        "four_coloring": {
+          "items": {
+            "maximum": 3,
+            "minimum": 0,
+            "type": "integer"
+          },
+          "maxItems": 11,
+          "minItems": 11,
+          "type": "array"
+        },
+        "minor_order": {
+          "const": 4
+        }
+      },
+      "required": [
+        "edges",
+        "four_coloring",
+        "branch_sets",
+        "chromatic_number",
+        "minor_order"
+      ],
+      "type": "object"
+    },
+    "scope": {
+      "const": "triangle-free-11-vertex-hadwiger-instance-v1",
+      "type": "string"
+    },
+    "task_id": {
+      "const": "jacobian/hadwiger-triangle-free-minor-certificate"
+    }
+  },
+  "required": [
+    "task_id",
+    "conclusion",
+    "result",
+    "claimed_assurance",
+    "scope",
+    "completeness",
+    "evidence",
+    "limitations"
+  ],
+  "type": "object"
+}

benchmarks/datasets/conjecture-probes-v1/hadwiger-triangle-free-minor-certificate/instruction.md

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+# Construct a nontrivial finite Hadwiger certificate
+
+Submit a simple connected triangle-free graph on vertices `0..10` with exactly
+20 edges and minimum degree at least 3. Supply a proper four-coloring and four
+pairwise-disjoint nonempty branch sets witnessing a `K4` minor: every branch
+set must induce a connected subgraph and every pair must have a crossing edge.
+
+The verifier independently rejects every possible three-coloring by exact
+backtracking, checks the submitted four-coloring, and validates the complete
+minor model. Complete graphs, triangles, isolated padding, and label-only
+chromatic claims are rejected.
+
+Evidence is matching JSON with exactly `schema_version`, `task_id`, `result`,
+and `limitations`, at most 2 MiB. This checks one finite graph only and does not
+prove Hadwiger's conjecture.
+
+<!-- BEGIN PUBLIC CONTRACT SUBMISSION BLOCK -->
+## Submission
+
+Exact finite coloring and minor replay only; no global Hadwiger 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 `FINITE_HADWIGER_K4_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/hadwiger-triangle-free-minor-certificate/solution/solve.py

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+from __future__ import annotations
+
+import argparse
+import hashlib
+import json
+from pathlib import Path
+
+TASK_ID = "jacobian/hadwiger-triangle-free-minor-certificate"
+LIMITATIONS = [
+    "ONE_TRIANGLE_FREE_11_VERTEX_GRAPH",
+    "EXHAUSTIVE_THREE_COLOR_REJECTION",
+    "NO_GLOBAL_HADWIGER_CONCLUSION",
+]
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--root", type=Path, default=Path("/app"))
+    root = parser.parse_args().root
+    edges = [
+        [0, 1],
+        [0, 4],
+        [0, 6],
+        [0, 9],
+        [1, 2],
+        [1, 5],
+        [1, 7],
+        [2, 3],
+        [2, 6],
+        [2, 8],
+        [3, 4],
+        [3, 7],
+        [3, 9],
+        [4, 5],
+        [4, 8],
+        [5, 10],
+        [6, 10],
+        [7, 10],
+        [8, 10],
+        [9, 10],
+    ]
+    result = {
+        "edges": edges,
+        "four_coloring": [0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3],
+        "branch_sets": [[0], [1], [2, 6], [3, 4, 5]],
+        "chromatic_number": 4,
+        "minor_order": 4,
+    }
+    payload = {
+        "schema_version": "1",
+        "task_id": TASK_ID,
+        "result": result,
+        "limitations": LIMITATIONS,
+    }
+    e = root / "evidence/answer.txt"
+    e.parent.mkdir(parents=True, exist_ok=True)
+    e.write_text(json.dumps(payload, sort_keys=True, separators=(",", ":")) + "\n")
+    s = {
+        "task_id": TASK_ID,
+        "conclusion": "FINITE_HADWIGER_K4_CERTIFICATE",
+        "result": result,
+        "claimed_assurance": "CHECKED",
+        "scope": "triangle-free-11-vertex-hadwiger-instance-v1",
+        "completeness": "COMPLETE",
+        "evidence": [
+            {
+                "path": "evidence/answer.txt",
+                "sha256": "sha256:" + hashlib.sha256(e.read_bytes()).hexdigest(),
+            }
+        ],
+        "limitations": LIMITATIONS,
+    }
+    (root / "submission.json").write_text(json.dumps(s, sort_keys=True) + "\n")
+
+
+if __name__ == "__main__":
+    main()

benchmarks/datasets/conjecture-probes-v1/hadwiger-triangle-free-minor-certificate/solution/solve.sh

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

benchmarks/datasets/conjecture-probes-v1/hadwiger-triangle-free-minor-certificate/task.toml

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+schema_version="1.4"
+artifacts=["/app/submission.json","/app/evidence"]
+[task]
+name="jacobian/hadwiger-triangle-free-minor-certificate"
+version="1.0.0"
+description="Construct a triangle-free 4-chromatic graph with an exact K4 minor certificate."
+keywords=["hadwiger","graph-coloring","clique-minor","triangle-free"]
+[metadata]
+evaluation_kind="conjecture-probe"
+domain="mathematical-sciences"
+primary_domain="graph-theory"
+field="graph-minors"
+assurance_ceiling="CHECKED"
+answer_visibility="public"
+provenance_class="authored-conjecture-probe"
+fixture_digest="sha256:fc4ba259102fdf0f361c0393e7b540738ef18969631ebfb44a499c70b41ba0a5"
+required_provider="core"
+author_name="Jacobian contributors"
+difficulty="hard"
+category="mathematics"
+tags=["difficulty-provisional","graph-coloring","clique-minor","exhaustive-verification","offline"]
+case_version="conjecture-probes-v1"
+contamination_class="authored-finite-conjecture-probe"
+derivation="Derived from retained spreadsheet row C-008; the finite graph contract is authored and independently checked."
+[agent]
+timeout_sec=600.0
+[verifier]
+timeout_sec=120.0
+environment_mode="separate"
+[environment]
+network_mode="no-network"
+cpus=1
+memory_mb=1024
+storage_mb=4096
+[verifier.environment]
+network_mode="no-network"
+cpus=1
+memory_mb=1024
+storage_mb=4096

benchmarks/datasets/conjecture-probes-v1/hadwiger-triangle-free-minor-certificate/tests/Dockerfile

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+FROM python:3.12-slim@sha256:57cd7c3a7a273101a6485ba99423ee568157882804b1124b4dd04266317710de
+LABEL jacobian.task="jacobian/hadwiger-triangle-free-minor-certificate" \
+      jacobian.checksum="97c73358d2fcc15771a1fa7401bc7dc7041549e11f1b4e9604ac6eb039b87726"
+RUN python -m pip install --no-cache-dir attrs==26.1.0 jsonschema==4.26.0 jsonschema-specifications==2025.9.1 referencing==0.37.0 rpds-py==2026.6.3 typing-extensions==4.16.0
+COPY input.json public_contract.json test.sh verifier.py verifier_support.py /tests/
+COPY input.json /app/input.json
+RUN chmod +x /tests/test.sh && python -c 'import json; assert json.load(open("/tests/input.json"))["task_id"] == "jacobian/hadwiger-triangle-free-minor-certificate"'

benchmarks/datasets/conjecture-probes-v1/hadwiger-triangle-free-minor-certificate/tests/input.json

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+{"task_id":"jacobian/hadwiger-triangle-free-minor-certificate","conjecture_id":"C-008","vertex_count":11,"edge_count":20,"required_chromatic_number":4,"required_minor_order":4,"require_triangle_free":true,"scope":"triangle-free-11-vertex-hadwiger-instance-v1"}