{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QCH2ZBNDNAX57TTXY6HQP264EV","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c94f48b6c3b0633e895dde53cf36af82708e1ea27a882e83698ca33b30ff7be7","cross_cats_sorted":["math.CO","math.DS","math.GT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.QA","submitted_at":"2026-03-04T17:12:06Z","title_canon_sha256":"244e0392fd304055216de31b60cd5cde4e16c1afe118803113f9195d701ab0df"},"schema_version":"1.0","source":{"id":"2603.04295","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2603.04295","created_at":"2026-05-18T02:45:05Z"},{"alias_kind":"arxiv_version","alias_value":"2603.04295v2","created_at":"2026-05-18T02:45:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.04295","created_at":"2026-05-18T02:45:05Z"},{"alias_kind":"pith_short_12","alias_value":"QCH2ZBNDNAX5","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"QCH2ZBNDNAX57TTX","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"QCH2ZBND","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:8e6cc322c77c067edd6a6715aafcb5f83ac6956254ed3eb5ddc155d6b2980be0","target":"graph","created_at":"2026-05-18T02:45:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We interpret every q-rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on q-rationals, the Springborn operations, which can be seen as a quadratic version of the Farey addition."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The q-deformation of the Farey triangulation and modular surface preserves the essential incidence and adjacency relations of the classical case for all positive real q, without introducing singularities or requiring additional restrictions on q."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"q-rationals are realized as circles in the plane with Springborn operations defined geometrically as homothety centers, producing a q-deformed midpoint formula and a new q-version of Markov numbers."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"q-rational numbers correspond to circles in a deformed Farey triangulation for every positive real q."}],"snapshot_sha256":"3ca055e3ee404fa08f8b91207aa143a20a5213c8b7188c80050a022ba8321286"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the geometry of $q$-rational numbers, introduced by Morier-Genoud and Ovsienko, for positive real $q$. In particular, we construct and analyse the deformed Farey triangulation and the deformed modular surface. We interpret every $q$-rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on $q$-rationals, the Springborn operations, which can be seen as a quadratic version of the Farey addition. Geometrically, the Springborn operations correspond to taking the homothety centers of a pair of two circles. As an application, we d","authors_text":"Alexander Thomas, Olga Paris-Romaskevich, Perrine Jouteur","cross_cats":["math.CO","math.DS","math.GT"],"headline":"q-rational numbers correspond to circles in a deformed Farey triangulation for every positive real q.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.QA","submitted_at":"2026-03-04T17:12:06Z","title":"Plane geometry of $q$-rationals and Springborn Operations"},"references":{"count":31,"internal_anchors":1,"resolved_work":31,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Aigner.Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings","work_id":"b7575299-884d-4011-86f8-f3d4783d6d0e","year":2013},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"arXiv:2511.11290","work_id":"25752ac3-7b95-48a5-bc32-494865c93511","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"A. Bapat, L. Becker, and A. M. Licata.q-deformed rational numbers and the 2-Calabi-Yau cate- gory of typeA 2.Forum Math. Sigma, 11:41, 2023. arXiv/2202.07613","work_id":"d0d762be-1a71-4105-801a-4c45dde679b3","year":2023},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"L. Bittmann, P. Jouteur, E. Kantarcı O˘ guz, M. Molander, and E. Yıldırım. A mirror deformation of Markov numbers. Preprint, arXiv/2602.14802, 2026","work_id":"3c785417-0b97-4f04-b073-a1ab57cc022e","year":2026},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"A. Elzenaar, J. Gong, G. J. Martin, and J. Schillewaert. Bounding deformation spaces of Kleinian groups with two generators. Preprint, arXiv/2405.15970, 2024","work_id":"0b9f27b5-61a7-42e3-8189-eec247e6b5ae","year":2024}],"snapshot_sha256":"8c64db57605b48f0ba5d33b7377c7c0a7aa22468f81a70404357c1329a7b8c86"},"source":{"id":"2603.04295","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-15T16:39:18.255565Z","id":"07396ee3-f72d-45a5-9eb6-8e1292113d84","model_set":{"reader":"grok-4.3"},"one_line_summary":"q-rationals are realized as circles in the plane with Springborn operations defined geometrically as homothety centers, producing a q-deformed midpoint formula and a new q-version of Markov numbers.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"q-rational numbers correspond to circles in a deformed Farey triangulation for every positive real q.","strongest_claim":"We interpret every q-rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on q-rationals, the Springborn operations, which can be seen as a quadratic version of the Farey addition.","weakest_assumption":"The q-deformation of the Farey triangulation and modular surface preserves the essential incidence and adjacency relations of the classical case for all positive real q, without introducing singularities or requiring additional restrictions on q."}},"verdict_id":"07396ee3-f72d-45a5-9eb6-8e1292113d84"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:95945543587108f55a22c0ff82a9be0f4608ced507ce97f6bcdb441db7c1e49a","target":"record","created_at":"2026-05-18T02:45:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c94f48b6c3b0633e895dde53cf36af82708e1ea27a882e83698ca33b30ff7be7","cross_cats_sorted":["math.CO","math.DS","math.GT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.QA","submitted_at":"2026-03-04T17:12:06Z","title_canon_sha256":"244e0392fd304055216de31b60cd5cde4e16c1afe118803113f9195d701ab0df"},"schema_version":"1.0","source":{"id":"2603.04295","kind":"arxiv","version":2}},"canonical_sha256":"808fac85a3682fdfce77c78f07ebdc2540ae539ec83b6e6b34a15362436f8328","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"808fac85a3682fdfce77c78f07ebdc2540ae539ec83b6e6b34a15362436f8328","first_computed_at":"2026-05-18T02:45:05.032824Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:45:05.032824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lvVYZms5d3HXHLtWRqhOe3y547K3RdwUKtygizomcbkwE2R+TEE+80D6FD0EFYdBFDfUZgn9aKL0eyFqv21WCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:45:05.033340Z","signed_message":"canonical_sha256_bytes"},"source_id":"2603.04295","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95945543587108f55a22c0ff82a9be0f4608ced507ce97f6bcdb441db7c1e49a","sha256:8e6cc322c77c067edd6a6715aafcb5f83ac6956254ed3eb5ddc155d6b2980be0"],"state_sha256":"b88d8b56ddc71b48527c1d8a8d303eb9142fbbab8d7e15aa4a03e5b9e7fc9d62"}