{
  "fact_registry": {
    "A1": {
      "label": "Prime count via Sieve of Eratosthenes",
      "method": "Sieve of Eratosthenes",
      "result": "168"
    },
    "A2": {
      "label": "Prime count via trial division (independent cross-check)",
      "method": "Trial division",
      "result": "168"
    },
    "A3": {
      "label": "Percentage of primes in [1, 1000]",
      "method": "prime_count / 1000 * 100",
      "result": "16.8%"
    }
  },
  "claim_formal": {
    "subject": "positive integers from 1 to 1000",
    "property": "percentage that are prime",
    "operator": "<",
    "operator_note": "'Fewer than 20 percent' means the count of primes in [1, 1000] divided by 1000, multiplied by 100, must be strictly less than 20. Equivalently, the prime count must be strictly less than 200.",
    "threshold": 20
  },
  "claim_natural": "Fewer than 20 percent of the positive integers from 1 to 1000 are prime.",
  "cross_checks": [
    {
      "description": "Sieve of Eratosthenes vs trial division prime count",
      "values_compared": [
        "168",
        "168"
      ],
      "agreement": true
    }
  ],
  "adversarial_checks": [
    {
      "question": "Could the sieve or trial division have a bug that undercounts primes?",
      "verification_performed": "Cross-checked two independent algorithms (Sieve of Eratosthenes and trial division). Both produce identical prime lists. Additionally, the result of 168 primes up to 1000 is a well-known value in number theory, denoted \u03c0(1000) = 168.",
      "finding": "Both methods agree on 168 primes. This matches the known value \u03c0(1000) = 168.",
      "breaks_proof": false
    },
    {
      "question": "Is there an interpretation where '20 percent' could mean something other than 200 out of 1000?",
      "verification_performed": "Considered whether 'fewer than 20 percent' could be non-strict (\u2264 vs <). The phrase 'fewer than' is unambiguously strict inequality. Even under \u2264, 168 \u2264 200 holds, so the claim would still be true.",
      "finding": "No alternative interpretation changes the result. 168 < 200 under any reading.",
      "breaks_proof": false
    }
  ],
  "verdict": "PROVED",
  "key_results": {
    "prime_count": 168,
    "total_integers": 1000,
    "percentage": 16.8,
    "threshold": 20,
    "operator": "<",
    "claim_holds": true
  },
  "generator": {
    "name": "proof-engine",
    "version": "0.10.0",
    "repo": "https://github.com/yaniv-golan/proof-engine",
    "generated_at": "2026-03-28"
  },
  "proof_py_url": "/proof-engine/proofs/fewer-than-20-percent-of-the-positive-integers-fro/proof.py"
}