{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "helfgott-ternary-goldbach-with-finite-verification-2013", "title": "Helfgott: ternary Goldbach with Helfgott-Platt finite verification", "summary": "Every odd integer greater than five is a sum of three primes. Explicit major- and minor-arc estimates and an optimized large sieve cover all sufficiently large integers; a rigorously verified finite range overlaps that analytic threshold and completes the result.", "boundary": "This is the three-primes theorem, not binary Goldbach for all even integers. Numerical verification alone is finite; global closure uses analytic estimates and their documented computational inputs. 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Helfgott", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "The ternary Goldbach conjecture is true", "url": "https://arxiv.org/abs/1312.7748v2" } }, { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "David J. Platt", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "verifier" ], "source": { "citation": "Numerical Verification of the Ternary Goldbach Conjecture up to8.875e30", "url": "https://arxiv.org/abs/1305.3062v2" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "The ternary Goldbach conjecture is true", "url": "https://arxiv.org/abs/1312.7748v2" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-1", "kind": "source_reference", "source": { "citation": "Numerical Verification of the Ternary Goldbach Conjecture up to8.875e30", "url": "https://arxiv.org/abs/1305.3062v2" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "This is the three-primes theorem, not binary Goldbach for all even integers. Numerical verification alone is finite; global closure uses analytic estimates and their documented computational inputs. No assumption of the full Riemann or generalized Riemann hypothesis is introduced.", "domain": "Analytic and computational number theory" }, "sources": [ { "citation": "The ternary Goldbach conjecture is true", "url": "https://arxiv.org/abs/1312.7748v2" }, { "citation": "Numerical Verification of the Ternary Goldbach Conjecture up to8.875e30", "url": "https://arxiv.org/abs/1305.3062v2" } ], "statement": "# Helfgott: ternary Goldbach with Helfgott-Platt finite verification\n\n## Every odd integer above five\n\nThe three-primes, or ternary Goldbach, theorem says that every odd integer greater than five can be expressed as a sum of three primes. It differs from the two-primes conjecture about even integers. Small examples are easy: 7 = 2 + 2 + 3 and 11 = 3 + 3 + 5. The challenge is to justify the word every, rather than to collect convincing examples.\n\n## Two ranges must actually meet\n\nImagine a classroom proof that covers every integer above one million, combined with an exact computation covering all relevant integers up to two million. The ranges overlap, so nothing is left between them. If the computation stopped at nine hundred thousand, a gap would remain. These invented thresholds explain the logic of combining an analytic theorem with a finite verification; they are not the thresholds in the original papers.\n\n## Analysis handles the unbounded part\n\nHelfgott's work develops explicit estimates using exponential sums, sieve ideas and the circle method. Explicit means that the size conditions can be made concrete enough to connect with a verified finite range. Computation alone remains finite; the analytic estimates are what extend the argument to arbitrarily large integers. Your finite exercise should state its range before you begin. Decomposing every eligible odd number up to a chosen limit still leaves infinitely many larger numbers, so its role differs from the analytic argument.\n\n## Credit the two original works accurately\n\nThe main three-primes proof has Helfgott as its author. The finite verification paper is coauthored by Helfgott and David Platt. This combined record preserves both roles without pretending that Platt coauthored the sole-author analytic paper or counting the completed theorem twice. Earlier analytic methods retain predecessor credit. The result does not assume the full Riemann hypothesis and does not prove binary Goldbach.\n\n## A helpful continuation task\n\nDraw a number line showing the actual analytic threshold and verified range from the sources, with the units and versions recorded. Check that the overlap is genuine. If you cannot establish a threshold from the available text, label it unknown. A careful dependency diagram is useful evidence; another finite scan of small numbers is not a replacement for global closure.\n\n## Scientific scope\n\nThis is the three-primes theorem, not binary Goldbach for all even integers. Numerical verification alone is finite; global closure uses analytic estimates and their documented computational inputs. No assumption of the full Riemann or generalized Riemann hypothesis is introduced.\n\n## Sources for this explanation\n\n- [The ternary Goldbach conjecture is true](https://arxiv.org/abs/1312.7748v2): Pinned original contribution and stated hypotheses or experimental limits.\n- [Numerical Verification of the Ternary Goldbach Conjecture up to8.875e30](https://arxiv.org/abs/1305.3062v2): Pinned original contribution and stated hypotheses or experimental limits.\n", "summary": "Every odd integer greater than five is a sum of three primes. Explicit major- and minor-arc estimates and an optimized large sieve cover all sufficiently large integers; a rigorously verified finite range overlaps that analytic threshold and completes the result.", "title": "Helfgott: ternary Goldbach with Helfgott-Platt finite verification" }, "record_id": "urn:uuid:66a26e2c-8423-4154-a851-23f0112c2cfc", "version": 1, "source_ref": { "record_id": "urn:uuid:66a26e2c-8423-4154-a851-23f0112c2cfc", "version": 1, "content_hash": "sha256:630f92a718607d8286b9038fcd32062355ac17272183ee63ce499b4260c9c760" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:29d7705d47928cafb979076c730593f6a8c1426922b4ac425132a05af95db310" }, "target": { "record_id": "urn:uuid:66a26e2c-8423-4154-a851-23f0112c2cfc", "version": 1, "content_hash": "sha256:630f92a718607d8286b9038fcd32062355ac17272183ee63ce499b4260c9c760" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "helfgott-ternary-goldbach-with-finite-verification-2013", "entry_id": "helfgott-ternary-goldbach-with-finite-verification-2013", "revision": 1, "title": "Helfgott: ternary Goldbach with Helfgott-Platt finite verification", "context": "Analytic and computational number theory", "track": "Exact theorem and reusable mathematical method", "band": "S", "modifier": "-", "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-23", "evidence_cutoff": "2026-10-23", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Every odd integer greater than five is a sum of three primes. Explicit major- and minor-arc estimates and an optimized large sieve cover all sufficiently large integers; a rigorously verified finite range overlaps that analytic threshold and completes the result.", "scope": "Every odd integer greater than five is a sum of three primes. Explicit major- and minor-arc estimates and an optimized large sieve cover all sufficiently large integers; a rigorously verified finite range overlaps that analytic threshold and completes the result. This is the three-primes theorem, not binary Goldbach for all even integers. Numerical verification alone is finite; global closure uses analytic estimates and their documented computational inputs. No assumption of the full Riemann or generalized Riemann hypothesis is introduced.", "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. This is the three-primes theorem, not binary Goldbach for all even integers. Numerical verification alone is finite; global closure uses analytic estimates and their documented computational inputs. No assumption of the full Riemann or generalized Riemann hypothesis is introduced.", "references": [ { "title": "The ternary Goldbach conjecture is true", "url": "https://arxiv.org/abs/1312.7748v2", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." }, { "title": "Numerical Verification of the Ternary Goldbach Conjecture up to8.875e30", "url": "https://arxiv.org/abs/1305.3062v2", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." } ], "roles": [ { "who": "The named authors in the research record", "credit": "SCIENCE", "did": "Helfgott authors the analytic proof; Helfgott and Platt coauthor the finite verification. Source-specific roles remain in the native record. This is one theorem package, not a joint byline for both papers." } ], "target": { "record_id": "urn:uuid:66a26e2c-8423-4154-a851-23f0112c2cfc", "version": 1, "content_hash": "sha256:630f92a718607d8286b9038fcd32062355ac17272183ee63ce499b4260c9c760" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "350000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "helfgott-ternary-goldbach-with-finite-verification-2013", "target": { "record_id": "urn:uuid:66a26e2c-8423-4154-a851-23f0112c2cfc", "version": 1, "content_hash": "sha256:630f92a718607d8286b9038fcd32062355ac17272183ee63ce499b4260c9c760" }, "sections": [ { "title": "Every odd integer above five", "text": "The three-primes, or ternary Goldbach, theorem says that every odd integer greater than five can be expressed as a sum of three primes. It differs from the two-primes conjecture about even integers. Small examples are easy: 7 = 2 + 2 + 3 and 11 = 3 + 3 + 5. The challenge is to justify the word every, rather than to collect convincing examples." }, { "title": "Two ranges must actually meet", "text": "Imagine a classroom proof that covers every integer above one million, combined with an exact computation covering all relevant integers up to two million. The ranges overlap, so nothing is left between them. If the computation stopped at nine hundred thousand, a gap would remain. These invented thresholds explain the logic of combining an analytic theorem with a finite verification; they are not the thresholds in the original papers." }, { "title": "Analysis handles the unbounded part", "text": "Helfgott's work develops explicit estimates using exponential sums, sieve ideas and the circle method. Explicit means that the size conditions can be made concrete enough to connect with a verified finite range. Computation alone remains finite; the analytic estimates are what extend the argument to arbitrarily large integers. Your finite exercise should state its range before you begin. Decomposing every eligible odd number up to a chosen limit still leaves infinitely many larger numbers, so its role differs from the analytic argument." }, { "title": "Credit the two original works accurately", "text": "The main three-primes proof has Helfgott as its author. The finite verification paper is coauthored by Helfgott and David Platt. This combined record preserves both roles without pretending that Platt coauthored the sole-author analytic paper or counting the completed theorem twice. Earlier analytic methods retain predecessor credit. The result does not assume the full Riemann hypothesis and does not prove binary Goldbach." }, { "title": "A helpful continuation task", "text": "Draw a number line showing the actual analytic threshold and verified range from the sources, with the units and versions recorded. Check that the overlap is genuine. If you cannot establish a threshold from the available text, label it unknown. A careful dependency diagram is useful evidence; another finite scan of small numbers is not a replacement for global closure." } ], "references": [ { "title": "The ternary Goldbach conjecture is true", "url": "https://arxiv.org/abs/1312.7748v2", "locator": "Pinned original contribution and stated hypotheses or experimental limits." }, { "title": "Numerical Verification of the Ternary Goldbach Conjecture up to8.875e30", "url": "https://arxiv.org/abs/1305.3062v2", "locator": "Pinned original contribution and stated hypotheses or experimental limits." } ] }, "reading_status": "EXPLANATORY_GUIDE" }