{
  "format_version": 3,
  "claim_formal": {
    "subject": "Goldbach conjecture",
    "property": "counterexamples in range [4, 10^6]",
    "operator": "==",
    "threshold": 0,
    "claim_type": "open_problem",
    "operator_note": "The Goldbach conjecture asserts that every even integer > 2 is the sum of two primes. This is an unresolved conjecture \u2014 no proof or disproof exists as of 2026. Computational verification up to a finite bound provides evidence but cannot prove the claim for all even integers. The literature records verification up to 4 x 10^18 (Oliveira e Silva, 2013). Our script verifies up to 10^6 with two independent methods. Verdict is always UNDETERMINED regardless of computational outcome."
  },
  "claim_natural": "The Goldbach conjecture holds for every even integer greater than 2.",
  "evidence": {
    "A1": {
      "type": "computed",
      "label": "Primary check: counterexamples in [4, 10^6] via trial division",
      "sub_claim": null,
      "method": "goldbach_trial_division() \u2014 isprime() per candidate",
      "result": "0 counterexamples in [4, 1000000]",
      "depends_on": []
    },
    "A2": {
      "type": "computed",
      "label": "Cross-check: counterexamples in [4, 10^6] via prime-pair sieve",
      "sub_claim": null,
      "method": "goldbach_prime_sieve() \u2014 primerange sieve + set lookup",
      "result": "0 counterexamples in [4, 1000000]",
      "depends_on": []
    }
  },
  "cross_checks": [
    {
      "description": "Trial division vs prime sieve methods",
      "value_a": 0,
      "value_b": 0,
      "agree": true,
      "independence": "Mathematically independent: Method 1 uses per-number isprime() calls; Method 2 pre-generates a prime set via sieve and uses set membership. Different algorithms, different data structures.",
      "fact_ids": []
    }
  ],
  "adversarial_checks": [
    {
      "question": "Has any counterexample to Goldbach's conjecture ever been found?",
      "verification_performed": "Searched: 'Goldbach conjecture counterexample found disproved'. Reviewed Wikipedia, Physics Forums, Medium articles, and LessWrong. No counterexample has ever been reported.",
      "finding": "No counterexample exists in the literature or computational records.",
      "breaks_proof": false
    },
    {
      "question": "Has the Goldbach conjecture been formally proved or disproved?",
      "verification_performed": "Searched: 'Goldbach conjecture proof solved 2024 2025 2026'. Found several claimed proofs in non-peer-reviewed venues (SSRN, SCIRP, preprints.org, ScienceOpen) but none accepted by the mainstream mathematical community. The conjecture remains open as of March 2026.",
      "finding": "No universally accepted proof or disproof exists. The conjecture is listed as an open problem by all major references.",
      "breaks_proof": false
    },
    {
      "question": "What is the current computational verification bound?",
      "verification_performed": "Searched: 'Goldbach conjecture verified computational bound 2026'. Oliveira e Silva verified up to 4 x 10^18 (2013). The Gridbach project (2025) extended this further. A March 2025 preprint reports empirical verification beyond 4 quintillion.",
      "finding": "Verified computationally up to at least 4 x 10^18. Our 10^6 bound is far below the literature record but uses two independent methods.",
      "breaks_proof": false
    }
  ],
  "verdict": {
    "value": "UNDETERMINED",
    "qualified": false,
    "qualifier": null,
    "reason": null
  },
  "key_results": {
    "verified_up_to": 1000000,
    "counterexamples_found": 0,
    "methods_agree": true,
    "literature_bound": "4 x 10^18 (Oliveira e Silva, 2013)"
  },
  "generator": {
    "name": "proof-engine",
    "version": "0.10.0",
    "repo": "https://github.com/yaniv-golan/proof-engine",
    "generated_at": "2026-03-28"
  },
  "verdict_reason": "No counterexamples found up to 1000000 by two independent methods. However, this is a universal conjecture over infinitely many even integers. Computational verification of a finite range cannot constitute a proof. The Goldbach conjecture remains an open problem.",
  "proof_py_url": "/proofs/the-goldbach-conjecture-holds-for-every-even-integ/proof.py",
  "citation": {
    "doi": null,
    "concept_doi": null,
    "url": "https://proofengine.info/proofs/the-goldbach-conjecture-holds-for-every-even-integ/",
    "author": "Proof Engine",
    "cite_bib_url": "/proofs/the-goldbach-conjecture-holds-for-every-even-integ/cite.bib",
    "cite_ris_url": "/proofs/the-goldbach-conjecture-holds-for-every-even-integ/cite.ris"
  },
  "depends_on": []
}