{"schema_version":"source-reader-v1","record":{"source_id":"erdos-unit-distances-proof","versions":[{"work":{"authors":["OpenAI"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"self_reported","ai_note":"The manuscript credits an internal OpenAI model with the original argument; subsequent human editing and external mathematical checking are separate roles.","artifacts":[],"published":{"value":"2026-05-20"}},"id":"erdos-unit-distances:source:proof:v1","source_id":"erdos-unit-distances-proof","title":"Planar Point Sets with Many Unit Distances","owner":"OpenAI","url":"https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf","subject_scope":"Theorem 1.1 constructs arbitrarily large planar sets with a fixed positive exponent gain in their unordered unit-distance count.","revision":"2026-05-20","content_sha256":"62b5de8cacdfbc0f13ae1940aa81659cd233e6945b936612fbeab0ab8ab3d383","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf","attribution":"OpenAI","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}],"entries":[{"id":"erdos-unit-distances:entry:proof:v1","source_version_id":"erdos-unit-distances:source:proof:v1","item_id":"proof","url":"https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf","wording":"Theorem 1.1 constructs arbitrarily large planar sets with a fixed positive exponent gain in their unordered unit-distance count.","wording_kind":"paraphrase","locator":"Theorem 1.1 and Statement on AI Use","supersedes":null}],"problems":[{"slug":"erdos-unit-distances","title":"Erdős unit-distance problem","publication_version":6}],"selected_version_id":"erdos-unit-distances:source:proof:v1","available_versions":[{"id":"erdos-unit-distances:source:proof:v1","revision":"2026-05-20","observed_on":"2026-10-13","title":"Planar Point Sets with Many Unit Distances","publication":"preprint","content_sha256":"62b5de8cacdfbc0f13ae1940aa81659cd233e6945b936612fbeab0ab8ab3d383","used_in_current_publications":true}],"usages":[{"id":"source-use-48495c400ea3e35d027f","entry_id":"erdos-unit-distances:entry:proof:v1","problem":{"slug":"erdos-unit-distances","title":"Erdős unit-distance problem","publication_version":6},"kind":"assessment","title":"Fixed positive exponent gain proved","claim":{"id":"claim-unit-distances-positive-exponent","version":2,"slug":"unit-distances-positive-exponent","title":"A fixed power gain for infinitely many sizes"},"fragment":"claim-unit-distances-positive-exponent","assessment":{"id":"erdos-unit-distances:report:gain:v1","entry_id":"erdos-unit-distances:entry:proof:v1","state":"resolved","source_label":"Fixed positive exponent gain proved","scope":"For infinitely many n; unordered unit-distance pairs.","source_date":"2026-05-20","observed_on":"2026-10-13","evidence_url":"https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf","supersedes":null},"superseded":false,"formalization":null,"update":null},{"id":"source-use-e15b3f6e9f9be3700002","entry_id":"erdos-unit-distances:entry:proof:v1","problem":{"slug":"erdos-unit-distances","title":"Erdős unit-distance problem","publication_version":6},"kind":"milestone","title":"Erdős introduces the planar distance question","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-eb5e3e6876180b13ca86","entry_id":"erdos-unit-distances:entry:proof:v1","problem":{"slug":"erdos-unit-distances","title":"Erdős unit-distance problem","publication_version":6},"kind":"milestone","title":"A model-generated construction refutes near-linear growth","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-d5716fc7bb56e743bb43","entry_id":"erdos-unit-distances:entry:proof:v1","problem":{"slug":"erdos-unit-distances","title":"Erdős unit-distance problem","publication_version":6},"kind":"question","title":"Erdős unit-distance problem","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-a23b8d196ca84acf5966","entry_id":"erdos-unit-distances:entry:proof:v1","problem":{"slug":"erdos-unit-distances","title":"Erdős unit-distance problem","publication_version":6},"kind":"reading","title":"Read the mathematical argument; inspect its Claim for assessment and scope.","claim":null,"fragment":"provenance","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-6d0f6f251a9a54cd4a19","entry_id":"erdos-unit-distances:entry:proof:v1","problem":{"slug":"erdos-unit-distances","title":"Erdős unit-distance problem","publication_version":6},"kind":"statement","title":"The near-linear unit-distance conjecture","claim":{"id":"claim-unit-distances-near-linear","version":1,"slug":"unit-distances-near-linear","title":"The near-linear unit-distance conjecture"},"fragment":"claim-unit-distances-near-linear","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-90d20e4b6a4b9f3bd796","entry_id":"erdos-unit-distances:entry:proof:v1","problem":{"slug":"erdos-unit-distances","title":"Erdős unit-distance problem","publication_version":6},"kind":"statement","title":"The classical upper bound","claim":{"id":"claim-unit-distances-four-thirds-upper","version":2,"slug":"unit-distances-four-thirds-upper","title":"The classical upper bound"},"fragment":"claim-unit-distances-four-thirds-upper","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-ea3e4a3b90c05d363a72","entry_id":"erdos-unit-distances:entry:proof:v1","problem":{"slug":"erdos-unit-distances","title":"Erdős unit-distance problem","publication_version":6},"kind":"statement","title":"A fixed power gain for infinitely many sizes","claim":{"id":"claim-unit-distances-positive-exponent","version":2,"slug":"unit-distances-positive-exponent","title":"A fixed power gain for infinitely many sizes"},"fragment":"claim-unit-distances-positive-exponent","assessment":null,"superseded":false,"formalization":null,"update":null}],"linked_entries":[],"linked_sources":[]}}