{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VCK6O66CTAODQFRFKLDX34B3BY","short_pith_number":"pith:VCK6O66C","canonical_record":{"source":{"id":"2606.11585","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-10T02:24:04Z","cross_cats_sorted":["cs.CL","nlin.AO"],"title_canon_sha256":"a24f2ddf9975ca4973dc9a3f0300b415ffb8b83285818fc5961a59087fbbe707","abstract_canon_sha256":"230484547298ffe8503c16c2a9db6cd4a7a281d7089bed6643e460b7f68e383b"},"schema_version":"1.0"},"canonical_sha256":"a895e77bc2981c38162552c77df03b0e155ae29836d559d4b0c23087f4c6ee33","source":{"kind":"arxiv","id":"2606.11585","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11585","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11585v1","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11585","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"pith_short_12","alias_value":"VCK6O66CTAOD","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"pith_short_16","alias_value":"VCK6O66CTAODQFRF","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"pith_short_8","alias_value":"VCK6O66C","created_at":"2026-06-11T01:09:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VCK6O66CTAODQFRFKLDX34B3BY","target":"record","payload":{"canonical_record":{"source":{"id":"2606.11585","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-10T02:24:04Z","cross_cats_sorted":["cs.CL","nlin.AO"],"title_canon_sha256":"a24f2ddf9975ca4973dc9a3f0300b415ffb8b83285818fc5961a59087fbbe707","abstract_canon_sha256":"230484547298ffe8503c16c2a9db6cd4a7a281d7089bed6643e460b7f68e383b"},"schema_version":"1.0"},"canonical_sha256":"a895e77bc2981c38162552c77df03b0e155ae29836d559d4b0c23087f4c6ee33","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:09:57.555999Z","signature_b64":"XzAnvnxy6EvuvRMk1mVJ0MnxZj/BbVPDuny/lGLKSA9F63rasGZmjKr6liCSAARKb1mqagfD2zabor8KOyL0CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a895e77bc2981c38162552c77df03b0e155ae29836d559d4b0c23087f4c6ee33","last_reissued_at":"2026-06-11T01:09:57.555217Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:09:57.555217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.11585","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-11T01:09:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cNkkx2Dk8awWinQJifunHl6G9FcKp0F7VTU9QeihdPOX/fJ10PIIqP3ep5OVNAKUHa2DpO1w4Lf06OHtnaFADw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T13:35:44.633372Z"},"content_sha256":"3e6995517ed8db49bb2837a0e30d5c1f811c0f99b498b5252f62b060599f8364","schema_version":"1.0","event_id":"sha256:3e6995517ed8db49bb2837a0e30d5c1f811c0f99b498b5252f62b060599f8364"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VCK6O66CTAODQFRFKLDX34B3BY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kuramoto Attention: Synchronizing Self-Attention on the Torus","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CL","nlin.AO"],"primary_cat":"cs.LG","authors_text":"Joshua Nunley","submitted_at":"2026-06-10T02:24:04Z","abstract_excerpt":"We introduce Kuramoto attention, a self-attention layer in which each hidden coordinate is an angle. The layer scores tokens by gated cosine similarity, attends over previous phase states, and updates each token by the tangent component of the attention-weighted circular mean. Because the values are the raw phase states, this update is exactly the Kuramoto coupling term $\\sum_u A_{t,u}\\sin(\\theta_u-\\theta_t)$, with the attention matrix acting as an adaptive, content-dependent coupling kernel. Equivalently, the gated score is a learned metric on the torus that selects which tokens couple, and t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11585","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.11585/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-11T01:09:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"63XtlkyCze8VFo4keIQn/s+OpB1+tN+9qBYGCexPQs82qWvwAHYJ5z0nU93tU6OOd6vog0kIRefw0DV80MojDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T13:35:44.633814Z"},"content_sha256":"781081a3c2b6fd0d0c928416da71f6e18282e0011625e5c2f99f0e77e720d46e","schema_version":"1.0","event_id":"sha256:781081a3c2b6fd0d0c928416da71f6e18282e0011625e5c2f99f0e77e720d46e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VCK6O66CTAODQFRFKLDX34B3BY/bundle.json","state_url":"https://pith.science/pith/VCK6O66CTAODQFRFKLDX34B3BY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VCK6O66CTAODQFRFKLDX34B3BY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-04T13:35:44Z","links":{"resolver":"https://pith.science/pith/VCK6O66CTAODQFRFKLDX34B3BY","bundle":"https://pith.science/pith/VCK6O66CTAODQFRFKLDX34B3BY/bundle.json","state":"https://pith.science/pith/VCK6O66CTAODQFRFKLDX34B3BY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VCK6O66CTAODQFRFKLDX34B3BY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VCK6O66CTAODQFRFKLDX34B3BY","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":"230484547298ffe8503c16c2a9db6cd4a7a281d7089bed6643e460b7f68e383b","cross_cats_sorted":["cs.CL","nlin.AO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-10T02:24:04Z","title_canon_sha256":"a24f2ddf9975ca4973dc9a3f0300b415ffb8b83285818fc5961a59087fbbe707"},"schema_version":"1.0","source":{"id":"2606.11585","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11585","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11585v1","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11585","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"pith_short_12","alias_value":"VCK6O66CTAOD","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"pith_short_16","alias_value":"VCK6O66CTAODQFRF","created_at":"2026-06-11T01:09:57Z"},{"alias_kind":"pith_short_8","alias_value":"VCK6O66C","created_at":"2026-06-11T01:09:57Z"}],"graph_snapshots":[{"event_id":"sha256:781081a3c2b6fd0d0c928416da71f6e18282e0011625e5c2f99f0e77e720d46e","target":"graph","created_at":"2026-06-11T01:09:57Z","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":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.11585/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce Kuramoto attention, a self-attention layer in which each hidden coordinate is an angle. The layer scores tokens by gated cosine similarity, attends over previous phase states, and updates each token by the tangent component of the attention-weighted circular mean. Because the values are the raw phase states, this update is exactly the Kuramoto coupling term $\\sum_u A_{t,u}\\sin(\\theta_u-\\theta_t)$, with the attention matrix acting as an adaptive, content-dependent coupling kernel. Equivalently, the gated score is a learned metric on the torus that selects which tokens couple, and t","authors_text":"Joshua Nunley","cross_cats":["cs.CL","nlin.AO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-10T02:24:04Z","title":"Kuramoto Attention: Synchronizing Self-Attention on the Torus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11585","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3e6995517ed8db49bb2837a0e30d5c1f811c0f99b498b5252f62b060599f8364","target":"record","created_at":"2026-06-11T01:09:57Z","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":"230484547298ffe8503c16c2a9db6cd4a7a281d7089bed6643e460b7f68e383b","cross_cats_sorted":["cs.CL","nlin.AO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-10T02:24:04Z","title_canon_sha256":"a24f2ddf9975ca4973dc9a3f0300b415ffb8b83285818fc5961a59087fbbe707"},"schema_version":"1.0","source":{"id":"2606.11585","kind":"arxiv","version":1}},"canonical_sha256":"a895e77bc2981c38162552c77df03b0e155ae29836d559d4b0c23087f4c6ee33","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a895e77bc2981c38162552c77df03b0e155ae29836d559d4b0c23087f4c6ee33","first_computed_at":"2026-06-11T01:09:57.555217Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T01:09:57.555217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XzAnvnxy6EvuvRMk1mVJ0MnxZj/BbVPDuny/lGLKSA9F63rasGZmjKr6liCSAARKb1mqagfD2zabor8KOyL0CQ==","signature_status":"signed_v1","signed_at":"2026-06-11T01:09:57.555999Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11585","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e6995517ed8db49bb2837a0e30d5c1f811c0f99b498b5252f62b060599f8364","sha256:781081a3c2b6fd0d0c928416da71f6e18282e0011625e5c2f99f0e77e720d46e"],"state_sha256":"da90fa67da36d5113f86c69649b61339fbec7994055ceac09c8d800819e1ec07"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/eN00yKZLQK+c8ubL2qX5NL9n/wDdx+7BXr3Kysyx9A1OmiawlcicGky1llUe2BJba0X9gN40fsO1yty8vOfAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T13:35:44.636424Z","bundle_sha256":"5eb7e2438ef4314d40cf468a4ac3ee1b3954cce73906bba1a251662ac5312434"}}