{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NFAVD6YV6TQMT5CVVLD4FUAWN3","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":"e35b95011cd6596ac138bed5b0298165a0ee43aa680ecbfaaf488a0ff62c8869","cross_cats_sorted":["math.OC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T17:04:41Z","title_canon_sha256":"1731f8acc8e23050af0efedd17393bd3e661f8d2b5ea8d557237a518b8356e63"},"schema_version":"1.0","source":{"id":"2605.13783","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13783","created_at":"2026-05-18T02:44:15Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13783v1","created_at":"2026-05-18T02:44:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13783","created_at":"2026-05-18T02:44:15Z"},{"alias_kind":"pith_short_12","alias_value":"NFAVD6YV6TQM","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"NFAVD6YV6TQMT5CV","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"NFAVD6YV","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:a01a814f6ff7f438e2d2c47044ebe5bf904cb9266b6bb5b59559caa5f548c185","target":"graph","created_at":"2026-05-18T02:44:15Z","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 prove that the synchronized branch is a unique smooth family of Nash equilibria emerging from the uniform state at the bifurcation: at each supercritical interaction strength the synchronized equilibrium is unique up to rotation of the torus, and converges smoothly to the uniform distribution as the interaction parameter decreases to the critical threshold. Both follow from our main technical result: the scalar self-consistency map is strictly concave."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The proof depends on sharp shape estimates for the value function and a pointwise geometric-mean monotonicity that determines the sign of the cubic moment; these estimates are derived under the specific stationary Kuramoto interaction and may fail for other interaction kernels or non-stationary settings."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"The synchronized stationary equilibria in the Kuramoto mean field game are unique up to rotation for all supercritical interaction strengths and form a smooth branch converging to the uniform state at the critical threshold, proven by showing the self-consistency map is strictly concave."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"In the stationary Kuramoto mean field game the synchronized Nash equilibria form a unique smooth branch that emerges from the uniform state at the critical interaction strength."}],"snapshot_sha256":"1f2cb535bc31f141b33b367d6620239c42e79879f4773bdc2b51f668bf51f1db"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"2fa9889f596c0460682ea59acfae42bc16edba3880c311c07f0aaba35241222f"},"paper":{"abstract_excerpt":"The stationary Kuramoto mean field game models a population of phase oscillators that form synchronized Nash equilibria above a critical interaction strength. We prove that the synchronized branch is a unique smooth family of Nash equilibria emerging from the uniform state at the bifurcation: at each supercritical interaction strength the synchronized equilibrium is unique up to rotation of the torus, and converges smoothly to the uniform distribution as the interaction parameter decreases to the critical threshold. Both follow from our main technical result: the scalar self-consistency map is","authors_text":"Sebastian Munoz","cross_cats":["math.OC"],"headline":"In the stationary Kuramoto mean field game the synchronized Nash equilibria form a unique smooth branch that emerges from the uniform state at the critical interaction strength.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T17:04:41Z","title":"Uniqueness of synchronized stationary equilibria in the Kuramoto mean field game"},"references":{"count":16,"internal_anchors":0,"resolved_work":16,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"J. A. Acebr\\'on, L. L. Bonilla, C. J. P\\'erez Vicente, F. Ritort, and R. Spigler. The K uramoto model: a simple paradigm for synchronization phenomena. Reviews of Modern Physics 77 (2005), no.\\ 1, 137","work_id":"6b2e05e1-f754-48a2-870c-ed51eb9b4ee4","year":2005},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"L. Bertini, G. Giacomin, and C. Poquet. Synchronization and random long time dynamics for mean-field plane rotators. Probability Theory and Related Fields 160 (2014), no.\\ 3--4, 593--653","work_id":"d8d0c082-070f-46bb-a38b-ad72f6d6bdc7","year":2014},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"R. Carmona and F. Delarue. Probabilistic Theory of Mean Field Games with Applications, I--II. Probability Theory and Stochastic Modelling, vols.\\ 83--84. Springer, 2018","work_id":"53ca1268-9d08-4e40-919e-6691f67ffb3d","year":2018},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"R. Carmona, Q. Cormier, and H. M. Soner. Synchronization in a K uramoto mean field game. Communications in Partial Differential Equations 48 (2023), no.\\ 9, 1214--1244. arXiv:2210.12912","work_id":"4ffdcd16-f718-4586-a84a-e2616144db2a","year":2023},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"R. Carmona, Q. Cormier, and H. M. Soner. K uramoto mean field game with intrinsic frequencies. Preprint, 2025. arXiv:2509.18000","work_id":"de816287-b2b3-4d93-8425-289ee8b68fd3","year":2025}],"snapshot_sha256":"c46faf63e3d57cb2241bc4f054ff0789320f66b6e198957d848e2581ff5aa0a1"},"source":{"id":"2605.13783","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T17:41:52.986629Z","id":"c00ccecd-4043-493c-90a5-216868d59cc5","model_set":{"reader":"grok-4.3"},"one_line_summary":"The synchronized stationary equilibria in the Kuramoto mean field game are unique up to rotation for all supercritical interaction strengths and form a smooth branch converging to the uniform state at the critical threshold, proven by showing the self-consistency map is strictly concave.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"In the stationary Kuramoto mean field game the synchronized Nash equilibria form a unique smooth branch that emerges from the uniform state at the critical interaction strength.","strongest_claim":"We prove that the synchronized branch is a unique smooth family of Nash equilibria emerging from the uniform state at the bifurcation: at each supercritical interaction strength the synchronized equilibrium is unique up to rotation of the torus, and converges smoothly to the uniform distribution as the interaction parameter decreases to the critical threshold. Both follow from our main technical result: the scalar self-consistency map is strictly concave.","weakest_assumption":"The proof depends on sharp shape estimates for the value function and a pointwise geometric-mean monotonicity that determines the sign of the cubic moment; these estimates are derived under the specific stationary Kuramoto interaction and may fail for other interaction kernels or non-stationary settings."}},"verdict_id":"c00ccecd-4043-493c-90a5-216868d59cc5"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6472d208f259cb820264ad4c5e9318c387914673196496809def95dd844f853e","target":"record","created_at":"2026-05-18T02:44:15Z","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":"e35b95011cd6596ac138bed5b0298165a0ee43aa680ecbfaaf488a0ff62c8869","cross_cats_sorted":["math.OC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T17:04:41Z","title_canon_sha256":"1731f8acc8e23050af0efedd17393bd3e661f8d2b5ea8d557237a518b8356e63"},"schema_version":"1.0","source":{"id":"2605.13783","kind":"arxiv","version":1}},"canonical_sha256":"694151fb15f4e0c9f455aac7c2d0166ef66b603ba3241a9185c4f83ebf981687","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"694151fb15f4e0c9f455aac7c2d0166ef66b603ba3241a9185c4f83ebf981687","first_computed_at":"2026-05-18T02:44:15.711653Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:15.711653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AmQToao/EvKu7YXBO86AUeGIC7noYFyHVAUJ2RN3tkLDGjXeMP2AjfGO6Ker0IHB83tlKjs2TB+8pCk4yq/iDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:15.712171Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13783","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6472d208f259cb820264ad4c5e9318c387914673196496809def95dd844f853e","sha256:a01a814f6ff7f438e2d2c47044ebe5bf904cb9266b6bb5b59559caa5f548c185"],"state_sha256":"915e1be7549d1fb2fd5d4f67900ba706a90c1102b08e98afee0d7becf23359c8"}