benchmarks: add exact Navier–Stokes polynomial certificate

benchmarks/datasets/conjecture-probes-v1/members/navier-stokes-polynomial-certificate.toml

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+schema_version = "2"
+task_id = "navier-stokes-polynomial-certificate"
+task_name = "jacobian/navier-stokes-polynomial-certificate"
+evaluation_kind = "conjecture-probe"
+domain = "mathematical-sciences"
+primary_domain = "analysis"
+field = "fluid-dynamics"
+provenance_class = "authored-conjecture-probe"
+provenance_ref = "spreadsheet:unresolved-conjecture/C-006@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/navier-stokes-polynomial-certificate/README.md

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+# Navier–Stokes polynomial certificate
+
+An Assurance-family conjecture probe derived from retained spreadsheet row
+C-006. Its single reasoning objective is exact symbolic PDE certification:
+construct an affine divergence-free flow and quadratic pressure, then expose
+the complete residual coefficients for independent replay.
+
+The task rejects zero-vorticity shortcuts, bounds all rational coefficients,
+accepts alternative exact flows, and never promotes one two-dimensional steady
+solution into evidence about the open three-dimensional regularity problem.
+
+Difficulty is provisionally Hard because success requires coordinating two
+polynomial vector equations, incompressibility, canonical rational arithmetic,
+and the assurance boundary. Empirical calibration is not yet available.
+
+Source: retained row C-006; Clay Mathematics Institute,
+<https://www.claymath.org/millennium-problems/navier-stokes-equation/>, status
+checked 2026-08-06. The immutable authored fixture is identified by the
+SHA-256 digest in `task.toml`; the benchmark does not freeze or reproduce Clay
+page text.

benchmarks/datasets/conjecture-probes-v1/navier-stokes-polynomial-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/navier-stokes-polynomial-certificate/environment/input.json

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+{"task_id":"jacobian/navier-stokes-polynomial-certificate","conjecture_id":"C-006","coefficient_domain":"QQ","variables":["x","y"],"velocity_monomials":["1","x","y"],"pressure_monomials":["1","x","y","x^2","xy","y^2"],"coefficient_numerator_bound":50,"coefficient_denominator_bound":20,"required_vorticity":"NONZERO","scope":"steady-affine-2d-polynomial-fields-v1"}

benchmarks/datasets/conjecture-probes-v1/navier-stokes-polynomial-certificate/environment/submission_schema.json

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+{
+  "$defs": {
+    "rat": {
+      "pattern": "^(?:0|-?[1-9][0-9]*)(?:/[1-9][0-9]*)?$",
+      "type": "string"
+    }
+  },
+  "$schema": "https://json-schema.org/draft/2020-12/schema",
+  "additionalProperties": false,
+  "properties": {
+    "claimed_assurance": {
+      "enum": [
+        "UNVERIFIED",
+        "COMPUTED",
+        "CHECKED",
+        "VERIFIED"
+      ]
+    },
+    "completeness": {
+      "const": "COMPLETE"
+    },
+    "conclusion": {
+      "const": "STEADY_INCOMPRESSIBLE_POLYNOMIAL_CERTIFICATE"
+    },
+    "evidence": {
+      "items": {
+        "additionalProperties": false,
+        "properties": {
+          "path": {
+            "const": "evidence/answer.txt"
+          },
+          "sha256": {
+            "pattern": "^sha256:[0-9a-f]{64}$",
+            "type": "string"
+          }
+        },
+        "required": [
+          "path",
+          "sha256"
+        ],
+        "type": "object"
+      },
+      "maxItems": 1,
+      "minItems": 1,
+      "type": "array"
+    },
+    "limitations": {
+      "const": [
+        "ONE_EXACT_2D_STEADY_POLYNOMIAL_FIELD",
+        "NO_GLOBAL_NAVIER_STOKES_REGULARITY_CONCLUSION"
+      ]
+    },
+    "result": {
+      "additionalProperties": false,
+      "properties": {
+        "divergence": {
+          "items": {
+            "$ref": "#/$defs/rat"
+          },
+          "maxItems": 1,
+          "minItems": 1,
+          "type": "array"
+        },
+        "momentum_x": {
+          "items": {
+            "$ref": "#/$defs/rat"
+          },
+          "maxItems": 3,
+          "minItems": 3,
+          "type": "array"
+        },
+        "momentum_y": {
+          "items": {
+            "$ref": "#/$defs/rat"
+          },
+          "maxItems": 3,
+          "minItems": 3,
+          "type": "array"
+        },
+        "pressure": {
+          "items": {
+            "$ref": "#/$defs/rat"
+          },
+          "maxItems": 6,
+          "minItems": 6,
+          "type": "array"
+        },
+        "velocity": {
+          "items": {
+            "items": {
+              "$ref": "#/$defs/rat"
+            },
+            "maxItems": 3,
+            "minItems": 3,
+            "type": "array"
+          },
+          "maxItems": 2,
+          "minItems": 2,
+          "type": "array"
+        },
+        "vorticity": {
+          "$ref": "#/$defs/rat"
+        }
+      },
+      "required": [
+        "velocity",
+        "pressure",
+        "divergence",
+        "momentum_x",
+        "momentum_y",
+        "vorticity"
+      ],
+      "type": "object"
+    },
+    "scope": {
+      "const": "steady-affine-2d-polynomial-fields-v1",
+      "type": "string"
+    },
+    "task_id": {
+      "const": "jacobian/navier-stokes-polynomial-certificate"
+    }
+  },
+  "required": [
+    "task_id",
+    "conclusion",
+    "result",
+    "claimed_assurance",
+    "scope",
+    "completeness",
+    "evidence",
+    "limitations"
+  ],
+  "type": "object"
+}

benchmarks/datasets/conjecture-probes-v1/navier-stokes-polynomial-certificate/instruction.md

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+# Construct an exact steady incompressible polynomial flow
+
+Construct a non-irrotational steady two-dimensional polynomial velocity field
+`u=(u1,u2)` with affine components and a quadratic pressure `p`, all over
+`QQ`, such that `div(u)=0` and `(u·grad)u + grad(p) - Δu = 0`.
+
+Use the monomial orders frozen in `/app/input.json`. Submit every coefficient
+as a canonical rational string (`0`, `-3`, `5/7`; reduced denominator positive)
+and include the independently checkable coefficient vectors for divergence,
+both momentum residuals, and scalar vorticity `∂x u2 - ∂y u1`. The vorticity
+must be nonzero, so the zero field and pure-gradient shortcuts are rejected.
+
+`evidence/answer.txt` must be a JSON object with exactly `schema_version`,
+`task_id`, `result`, and `limitations`, matching the result and limitations in
+the submission. 
+This finite symbolic certificate concerns one exact steady polynomial flow.
+It neither proves nor disproves global existence or smoothness for the
+three-dimensional Navier–Stokes equations. Claim only `CHECKED` for the frozen
+symbolic contract.
+
+<!-- BEGIN PUBLIC CONTRACT SUBMISSION BLOCK -->
+## Submission
+
+Exact symbolic replay of one 2D polynomial field; no conclusion about the open 3D regularity problem.
+
+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 `STEADY_INCOMPRESSIBLE_POLYNOMIAL_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/navier-stokes-polynomial-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/navier-stokes-polynomial-certificate"
+LIMITATIONS = [
+    "ONE_EXACT_2D_STEADY_POLYNOMIAL_FIELD",
+    "NO_GLOBAL_NAVIER_STOKES_REGULARITY_CONCLUSION",
+]
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--root", type=Path, default=Path("/app"))
+    root = parser.parse_args().root
+    result = {
+        "velocity": [["0", "0", "-1"], ["0", "1", "0"]],
+        "pressure": ["0", "0", "0", "1/2", "0", "1/2"],
+        "divergence": ["0"],
+        "momentum_x": ["0", "0", "0"],
+        "momentum_y": ["0", "0", "0"],
+        "vorticity": "2",
+    }
+    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": "STEADY_INCOMPRESSIBLE_POLYNOMIAL_CERTIFICATE",
+        "result": result,
+        "claimed_assurance": "CHECKED",
+        "scope": "steady-affine-2d-polynomial-fields-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/navier-stokes-polynomial-certificate/solution/solve.sh

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

benchmarks/datasets/conjecture-probes-v1/navier-stokes-polynomial-certificate/task.toml

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+schema_version = "1.4"
+artifacts = ["/app/submission.json", "/app/evidence"]
+
+[task]
+name = "jacobian/navier-stokes-polynomial-certificate"
+version = "1.0.0"
+description = "Construct and independently replay one exact non-irrotational steady incompressible affine flow with quadratic pressure."
+keywords = ["navier-stokes", "symbolic-pde", "incompressibility", "conjecture-scope"]
+
+[metadata]
+evaluation_kind = "conjecture-probe"
+domain = "mathematical-sciences"
+primary_domain = "analysis"
+field = "fluid-dynamics"
+assurance_ceiling = "CHECKED"
+answer_visibility = "public"
+provenance_class = "authored-conjecture-probe"
+fixture_digest = "sha256:068453167467b09e3949d8f12c32548c9eef3da5355fefe6ad3c5bcd18b8a2bf"
+required_provider = "core"
+author_name = "Jacobian contributors"
+difficulty = "hard"
+category = "mathematics"
+tags = ["difficulty-provisional", "symbolic-pde", "exact-rationals", "scope-audit", "offline"]
+case_version = "conjecture-probes-v1"
+contamination_class = "authored-finite-conjecture-probe"
+derivation = "Derived from retained spreadsheet row C-006 and the Clay Navier-Stokes problem statement; the finite authored contract checks one exact 2D polynomial certificate only."
+
+[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/navier-stokes-polynomial-certificate/tests/Dockerfile

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+FROM python:3.12-slim@sha256:57cd7c3a7a273101a6485ba99423ee568157882804b1124b4dd04266317710de
+LABEL jacobian.task="jacobian/navier-stokes-polynomial-certificate" \
+      jacobian.checksum="80f2620510cebeb5f1ae177ec29c94fd3ca64427a59a1fbd73055cdae9eb2340"
+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 verifier_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/navier-stokes-polynomial-certificate"'

benchmarks/datasets/conjecture-probes-v1/navier-stokes-polynomial-certificate/tests/input.json

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+{"task_id":"jacobian/navier-stokes-polynomial-certificate","conjecture_id":"C-006","coefficient_domain":"QQ","variables":["x","y"],"velocity_monomials":["1","x","y"],"pressure_monomials":["1","x","y","x^2","xy","y^2"],"coefficient_numerator_bound":50,"coefficient_denominator_bound":20,"required_vorticity":"NONZERO","scope":"steady-affine-2d-polynomial-fields-v1"}