{
  "id": "KL-FCS-006",
  "version": "1.0.0",
  "domain": "SAT / SMT",
  "kind": "sat",
  "title": "An unsatisfiable CNF",
  "problem": "Decide a ∧ ¬a.",
  "specification": {
    "variables": 1,
    "clauses": [
      [
        1
      ],
      [
        -1
      ]
    ]
  },
  "claim": {
    "satisfiable": false
  },
  "witness": {
    "method": "exhaustive enumeration"
  },
  "verification_scope": "Complete Boolean enumeration · 2 assignments",
  "explanation": "When a is false, the first clause fails. When a is true, the second clause fails. Enumeration covers the entire domain.",
  "limitations": "Exhaustive enumeration is the certificate method here. This is not a DRAT/LRAT proof or a scalable UNSAT benchmark.",
  "verification_status": "mechanically-checked",
  "review_status": "awaiting-independent-review",
  "provenance": {
    "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
    "external_dataset": null,
    "model_run": null
  },
  "references": [
    "https://smt-lib.org/theories-FixedSizeBitVectors.shtml"
  ],
  "license_status": "not-yet-selected",
  "dataset": {
    "family": "sat-smt",
    "task": "Decide a ∧ ¬a.",
    "input_encoding": "Structured JSON; field meanings are stated in the specification.",
    "coverage": "Complete Boolean enumeration · 2 assignments",
    "acceptance": [
      "Interpret each literal using its variable number and sign.",
      "Enumerate the entire declared Boolean or bit-vector domain.",
      "Check the supplied model against every constraint.",
      "For unsatisfiability, require complete coverage with no satisfying assignment."
    ],
    "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
    "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
  },
  "lesson": {
    "motivation": "Make satisfiability evidence explicit: a model for a positive answer, or complete finite coverage for a negative one.",
    "definitions": [
      {
        "term": "CNF",
        "definition": "A conjunction of clauses, each a disjunction of signed literals."
      },
      {
        "term": "Satisfying model",
        "definition": "An assignment making every clause true."
      },
      {
        "term": "Bit-vector theory",
        "definition": "Fixed-width values with arithmetic modulo 2ʷ; distinct from unbounded integers."
      }
    ],
    "reasoning": [
      "Interpret each literal using its variable number and sign.",
      "Enumerate the entire declared Boolean or bit-vector domain.",
      "Check the supplied model against every constraint.",
      "For unsatisfiability, require complete coverage with no satisfying assignment."
    ],
    "worked_example": "When a is false, the first clause fails. When a is true, the second clause fails. Enumeration covers the entire domain.",
    "complexity": "Boolean enumeration examines 2ⁿ assignments. A width-w unary bit-vector instance examines 2ʷ values.",
    "common_error": "A solver’s unverified UNSAT verdict is not a certificate. Fixed-width overflow is part of the theory.",
    "further_work": "Add checked UNSAT proof formats and solver-backed artifacts with pinned versions."
  },
  "related_ids": [
    "KL-FCS-005",
    "KL-FCS-007"
  ]
}