benchmarks: add Yang–Mills gauge-invariance certificate

benchmarks/datasets/conjecture-probes-v1/members/yang-mills-gauge-invariance-certificate.toml

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+schema_version="2"
+task_id="yang-mills-gauge-invariance-certificate"
+task_name="jacobian/yang-mills-gauge-invariance-certificate"
+evaluation_kind="conjecture-probe"
+domain="mathematical-sciences"
+primary_domain="geometry-topology"
+field="gauge-theory"
+provenance_class="authored-conjecture-probe"
+provenance_ref="spreadsheet:unresolved-conjecture/C-040@2026-08-06"
+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/yang-mills-gauge-invariance-certificate/README.md

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+# Yang–Mills finite gauge-invariance certificate
+
+Assurance-family task from retained row C-040. It tests one objective: exact
+noncommutative gauge-covariance replay. The verifier accepts alternative
+bounded rational unit quaternions and checks every Hamilton product rather
+than matching an Oracle trace. Identity/scalar shortcuts are forbidden.
+
+Difficulty is provisional Hard due to eight coordinated SU(2) choices, four
+left/right transformations, two ordered products, conjugacy, and strict scope
+calibration. The result is not evidence for continuum Yang–Mills construction
+or a mass gap.
+
+Source: Clay Mathematics Institute,
+<https://www.claymath.org/millennium-problems/yang-mills-and-mass-gap/>, status
+checked 2026-08-06. The authored fixture is digest-pinned.

benchmarks/datasets/conjecture-probes-v1/yang-mills-gauge-invariance-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/yang-mills-gauge-invariance-certificate/environment/input.json

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+{"task_id":"jacobian/yang-mills-gauge-invariance-certificate","conjecture_id":"C-040","group_model":"SU2_UNIT_QUATERNIONS_OVER_QQ","oriented_cycle":[0,1,2,3,0],"coefficient_numerator_bound":20,"coefficient_denominator_bound":20,"scope":"one-rational-su2-plaquette-v1"}

benchmarks/datasets/conjecture-probes-v1/yang-mills-gauge-invariance-certificate/environment/submission_schema.json

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+{
+  "$defs": {
+    "quat": {
+      "items": {
+        "$ref": "#/$defs/rat"
+      },
+      "maxItems": 4,
+      "minItems": 4,
+      "type": "array"
+    },
+    "quats4": {
+      "items": {
+        "$ref": "#/$defs/quat"
+      },
+      "maxItems": 4,
+      "minItems": 4,
+      "type": "array"
+    },
+    "rat": {
+      "pattern": "^(?:0|-?[1-9][0-9]*)(?:/[1-9][0-9]*)?$",
+      "type": "string"
+    }
+  },
+  "$schema": "https://json-schema.org/draft/2020-12/schema",
+  "additionalProperties": false,
+  "properties": {
+    "claimed_assurance": {
+      "enum": [
+        "UNVERIFIED",
+        "COMPUTED",
+        "CHECKED",
+        "VERIFIED"
+      ]
+    },
+    "completeness": {
+      "const": "COMPLETE"
+    },
+    "conclusion": {
+      "const": "FINITE_SU2_PLAQUETTE_GAUGE_INVARIANCE"
+    },
+    "evidence": {
+      "items": {
+        "additionalProperties": false,
+        "properties": {
+          "path": {
+            "const": "evidence/answer.txt"
+          },
+          "sha256": {
+            "pattern": "^sha256:[0-9a-f]{64}$",
+            "type": "string"
+          }
+        },
+        "required": [
+          "path",
+          "sha256"
+        ],
+        "type": "object"
+      },
+      "maxItems": 1,
+      "minItems": 1,
+      "type": "array"
+    },
+    "limitations": {
+      "const": [
+        "ONE_FINITE_RATIONAL_SU2_PLAQUETTE",
+        "NO_CONTINUUM_YANG_MILLS_CONSTRUCTION",
+        "NO_MASS_GAP_CONCLUSION"
+      ]
+    },
+    "result": {
+      "additionalProperties": false,
+      "properties": {
+        "conjugated_plaquette": {
+          "$ref": "#/$defs/quat"
+        },
+        "gauges": {
+          "$ref": "#/$defs/quats4"
+        },
+        "links": {
+          "$ref": "#/$defs/quats4"
+        },
+        "plaquette": {
+          "$ref": "#/$defs/quat"
+        },
+        "scalar_trace_invariant": {
+          "const": true
+        },
+        "transformed_links": {
+          "$ref": "#/$defs/quats4"
+        },
+        "transformed_plaquette": {
+          "$ref": "#/$defs/quat"
+        }
+      },
+      "required": [
+        "links",
+        "gauges",
+        "transformed_links",
+        "plaquette",
+        "transformed_plaquette",
+        "conjugated_plaquette",
+        "scalar_trace_invariant"
+      ],
+      "type": "object"
+    },
+    "scope": {
+      "const": "one-rational-su2-plaquette-v1",
+      "type": "string"
+    },
+    "task_id": {
+      "const": "jacobian/yang-mills-gauge-invariance-certificate"
+    }
+  },
+  "required": [
+    "task_id",
+    "conclusion",
+    "result",
+    "claimed_assurance",
+    "scope",
+    "completeness",
+    "evidence",
+    "limitations"
+  ],
+  "type": "object"
+}

benchmarks/datasets/conjecture-probes-v1/yang-mills-gauge-invariance-certificate/instruction.md

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+# Replay one exact rational SU(2) plaquette gauge transformation
+
+Represent SU(2) elements as unit quaternions `(w,x,y,z)` over `QQ`. Choose four
+nonidentity link quaternions on the oriented square `0→1→2→3→0` and four
+nonidentity, pairwise-distinct vertex gauge quaternions, within the frozen
+coefficient bounds. Submit the transformed links `g_i U_ij g_j^-1`, the
+original and transformed ordered plaquette products, and `g_0 P g_0^-1`.
+
+The verifier independently checks canonical rational form, unit norms,
+Hamilton products, inverses, every transformed link, plaquette conjugacy, and
+invariance of the plaquette scalar part. Identity-only and commutative-scalar
+shortcuts are rejected.
+
+Evidence is a matching JSON object with exactly `schema_version`, `task_id`,
+`result`, and `limitations`, at most 2 MiB. This is one finite lattice-gauge
+identity. It says nothing about continuum construction or a Yang–Mills mass
+gap.
+
+<!-- BEGIN PUBLIC CONTRACT SUBMISSION BLOCK -->
+## Submission
+
+Exact finite SU(2) quaternion replay only; no continuum or mass-gap 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_SU2_PLAQUETTE_GAUGE_INVARIANCE`
+- **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/yang-mills-gauge-invariance-certificate/solution/solve.py

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+from __future__ import annotations
+
+import argparse
+import hashlib
+import json
+from fractions import Fraction
+from pathlib import Path
+
+TASK_ID = "jacobian/yang-mills-gauge-invariance-certificate"
+LIMITATIONS = [
+    "ONE_FINITE_RATIONAL_SU2_PLAQUETTE",
+    "NO_CONTINUUM_YANG_MILLS_CONSTRUCTION",
+    "NO_MASS_GAP_CONCLUSION",
+]
+
+
+def mul(a, b):
+    w, x, y, z = a
+    bw, bx, by, bz = b
+    return (
+        w * bw - x * bx - y * by - z * bz,
+        w * bx + x * bw + y * bz - z * by,
+        w * by - x * bz + y * bw + z * bx,
+        w * bz + x * by - y * bx + z * bw,
+    )
+
+
+def inv(a):
+    return (a[0], -a[1], -a[2], -a[3])
+
+
+def product(items):
+    out = (Fraction(1), Fraction(0), Fraction(0), Fraction(0))
+    for item in items:
+        out = mul(out, item)
+    return out
+
+
+def row(q):
+    return [str(x) for x in q]
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--root", type=Path, default=Path("/app"))
+    root = parser.parse_args().root
+    links = [
+        (Fraction(3, 5), Fraction(4, 5), 0, 0),
+        (Fraction(5, 13), 0, Fraction(12, 13), 0),
+        (Fraction(8, 17), 0, 0, Fraction(15, 17)),
+        (0, 1, 0, 0),
+    ]
+    gauges = [
+        (Fraction(4, 5), Fraction(3, 5), 0, 0),
+        (Fraction(12, 13), 0, Fraction(5, 13), 0),
+        (Fraction(15, 17), 0, 0, Fraction(8, 17)),
+        (0, 0, 1, 0),
+    ]
+    transformed = [
+        mul(mul(gauges[i], links[i]), inv(gauges[(i + 1) % 4])) for i in range(4)
+    ]
+    p = product(links)
+    tp = product(transformed)
+    cp = mul(mul(gauges[0], p), inv(gauges[0]))
+    result = {
+        "links": [row(q) for q in links],
+        "gauges": [row(q) for q in gauges],
+        "transformed_links": [row(q) for q in transformed],
+        "plaquette": row(p),
+        "transformed_plaquette": row(tp),
+        "conjugated_plaquette": row(cp),
+        "scalar_trace_invariant": True,
+    }
+    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_SU2_PLAQUETTE_GAUGE_INVARIANCE",
+        "result": result,
+        "claimed_assurance": "CHECKED",
+        "scope": "one-rational-su2-plaquette-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/yang-mills-gauge-invariance-certificate/solution/solve.sh

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

benchmarks/datasets/conjecture-probes-v1/yang-mills-gauge-invariance-certificate/task.toml

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+schema_version="1.4"
+artifacts=["/app/submission.json","/app/evidence"]
+[task]
+name="jacobian/yang-mills-gauge-invariance-certificate"
+version="1.0.0"
+description="Construct and replay one exact rational SU(2) plaquette gauge transformation."
+keywords=["yang-mills","su2","quaternions","gauge-invariance"]
+[metadata]
+evaluation_kind="conjecture-probe"
+domain="mathematical-sciences"
+primary_domain="geometry-topology"
+field="gauge-theory"
+assurance_ceiling="CHECKED"
+answer_visibility="public"
+provenance_class="authored-conjecture-probe"
+fixture_digest="sha256:0a463664f04926239181db2bae2525ddf9efb2cb4e242f7f058686ba9eb56a7c"
+required_provider="core"
+author_name="Jacobian contributors"
+difficulty="hard"
+category="mathematics"
+tags=["difficulty-provisional","gauge-theory","exact-rationals","quaternions","offline"]
+case_version="conjecture-probes-v1"
+contamination_class="authored-finite-conjecture-probe"
+derivation="Derived from retained spreadsheet row C-040 and the Clay Yang-Mills problem statement; the finite rational plaquette is authored."
+[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/yang-mills-gauge-invariance-certificate/tests/Dockerfile

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+FROM python:3.12-slim@sha256:57cd7c3a7a273101a6485ba99423ee568157882804b1124b4dd04266317710de
+LABEL jacobian.task="jacobian/yang-mills-gauge-invariance-certificate" \
+      jacobian.checksum="740774e0bc8ac0f144d73167657aa6762fb1b2261accb8fba45e8d7fbde4d66a"
+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/yang-mills-gauge-invariance-certificate"'

benchmarks/datasets/conjecture-probes-v1/yang-mills-gauge-invariance-certificate/tests/input.json

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+{"task_id":"jacobian/yang-mills-gauge-invariance-certificate","conjecture_id":"C-040","group_model":"SU2_UNIT_QUATERNIONS_OVER_QQ","oriented_cycle":[0,1,2,3,0],"coefficient_numerator_bound":20,"coefficient_denominator_bound":20,"scope":"one-rational-su2-plaquette-v1"}