# An unsatisfiable CNF

KL-FCS-006 · SAT / SMT · version 1.0.0

## Problem

Decide a ∧ ¬a.

## Context

Make satisfiability evidence explicit: a model for a positive answer, or complete finite coverage for a negative one.

## Definitions

- **CNF**: A conjunction of clauses, each a disjunction of signed literals.
- **Satisfying model**: An assignment making every clause true.
- **Bit-vector theory**: Fixed-width values with arithmetic modulo 2ʷ; distinct from unbounded integers.

## Checked result

Satisfiable: no.

When a is false, the first clause fails. When a is true, the second clause fails. Enumeration covers the entire domain.

## Checker reasoning

1. Interpret each literal using its variable number and sign.
2. Enumerate the entire declared Boolean or bit-vector domain.
3. Check the supplied model against every constraint.
4. For unsatisfiability, require complete coverage with no satisfying assignment.

## Dataset construction

{
  "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."
}

## Formal payload

```json
{
  "specification": {
    "variables": 1,
    "clauses": [
      [
        1
      ],
      [
        -1
      ]
    ]
  },
  "claim": {
    "satisfiable": false
  },
  "witness": {
    "method": "exhaustive enumeration"
  }
}
```

## Complexity

Boolean enumeration examines 2ⁿ assignments. A width-w unary bit-vector instance examines 2ʷ values.

## Limits

Exhaustive enumeration is the certificate method here. This is not a DRAT/LRAT proof or a scalable UNSAT benchmark.

## Common error and further work

A solver’s unverified UNSAT verdict is not a certificate. Fixed-width overflow is part of the theory.

Add checked UNSAT proof formats and solver-backed artifacts with pinned versions.

## Verification

Complete Boolean enumeration · 2 assignments. 2 checker units.
Replay with `python3 tools/verify.py`. Mechanical status: checked; human review
has not yet been recorded. Custom Python verification, not a proof-assistant
claim. Checker 1.0.0 and exact source hashes are in `verification.json`.

## Provenance and references

Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.
No external dataset or model-generation experiment. Reuse-license selection
remains pending.

- [Conceptual reference](https://smt-lib.org/theories-FixedSizeBitVectors.shtml)