{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:FPBJFEO7CHPPA5KNGUBWWZWBOF","short_pith_number":"pith:FPBJFEO7","canonical_record":{"source":{"id":"1507.08079","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-29T09:44:00Z","cross_cats_sorted":["nlin.AO"],"title_canon_sha256":"b28c9c9476539f61c26b93f1b7de666e404f35594398a35cc6d8673683770a53","abstract_canon_sha256":"48b718aa7dfa54dd56aa6605287496a676a89acbf53702a210b4835915118b46"},"schema_version":"1.0"},"canonical_sha256":"2bc29291df11def0754d35036b66c1716baa1a56706082d7e3edafb6c18d2f67","source":{"kind":"arxiv","id":"1507.08079","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08079","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08079v2","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08079","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"FPBJFEO7CHPP","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FPBJFEO7CHPPA5KN","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FPBJFEO7","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:FPBJFEO7CHPPA5KNGUBWWZWBOF","target":"record","payload":{"canonical_record":{"source":{"id":"1507.08079","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-29T09:44:00Z","cross_cats_sorted":["nlin.AO"],"title_canon_sha256":"b28c9c9476539f61c26b93f1b7de666e404f35594398a35cc6d8673683770a53","abstract_canon_sha256":"48b718aa7dfa54dd56aa6605287496a676a89acbf53702a210b4835915118b46"},"schema_version":"1.0"},"canonical_sha256":"2bc29291df11def0754d35036b66c1716baa1a56706082d7e3edafb6c18d2f67","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:52.635599Z","signature_b64":"uU72xZMfY/rQrZp1AQN66M1WH5X6fAV9fYMT8TWy8ThsjEVXaFRA48/YiVKRiIR0ljf9S8eeiGEMt/KZ9i6yDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2bc29291df11def0754d35036b66c1716baa1a56706082d7e3edafb6c18d2f67","last_reissued_at":"2026-05-18T01:17:52.634905Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:52.634905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.08079","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gTMRBiInooU4fwBu7eFtnY62tdl4upTHpfVc4SGpDZGizObaYtIAa2CsJ3Ekh4iopMFhviiDyZG0MObOzwQYDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T22:15:46.755732Z"},"content_sha256":"7ae576b12073e943c0e3298171cb90cfc0ce987eaf71c9ff7e3ff7d184e0becf","schema_version":"1.0","event_id":"sha256:7ae576b12073e943c0e3298171cb90cfc0ce987eaf71c9ff7e3ff7d184e0becf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:FPBJFEO7CHPPA5KNGUBWWZWBOF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hopf normal form with $S_N$ symmetry and reduction to systems of nonlinearly coupled phase oscillators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["nlin.AO"],"primary_cat":"math.DS","authors_text":"Ana Rodrigues, Peter Ashwin","submitted_at":"2015-07-29T09:44:00Z","abstract_excerpt":"Coupled oscillator models where $N$ oscillators are identical and symmetrically coupled to all others with full permutation symmetry $S_N$ are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive - and we characterise generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08079","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T01:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gEh9/UIG33W1DSmB20KEp/4QrMFQMWywDBdPpUyGfoIQel7gTnxIl/a6mST3BB0XmMrXEItS8rTzVy2sytJABg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T22:15:46.756486Z"},"content_sha256":"6bd9f0291679f335c4792d66c4b655a0e48325b9d39c4b37a0c28cfa082fbfa6","schema_version":"1.0","event_id":"sha256:6bd9f0291679f335c4792d66c4b655a0e48325b9d39c4b37a0c28cfa082fbfa6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FPBJFEO7CHPPA5KNGUBWWZWBOF/bundle.json","state_url":"https://pith.science/pith/FPBJFEO7CHPPA5KNGUBWWZWBOF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FPBJFEO7CHPPA5KNGUBWWZWBOF/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":"2026-05-31T22:15:46Z","links":{"resolver":"https://pith.science/pith/FPBJFEO7CHPPA5KNGUBWWZWBOF","bundle":"https://pith.science/pith/FPBJFEO7CHPPA5KNGUBWWZWBOF/bundle.json","state":"https://pith.science/pith/FPBJFEO7CHPPA5KNGUBWWZWBOF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FPBJFEO7CHPPA5KNGUBWWZWBOF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FPBJFEO7CHPPA5KNGUBWWZWBOF","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":"48b718aa7dfa54dd56aa6605287496a676a89acbf53702a210b4835915118b46","cross_cats_sorted":["nlin.AO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-29T09:44:00Z","title_canon_sha256":"b28c9c9476539f61c26b93f1b7de666e404f35594398a35cc6d8673683770a53"},"schema_version":"1.0","source":{"id":"1507.08079","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08079","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08079v2","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08079","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"FPBJFEO7CHPP","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FPBJFEO7CHPPA5KN","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FPBJFEO7","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:6bd9f0291679f335c4792d66c4b655a0e48325b9d39c4b37a0c28cfa082fbfa6","target":"graph","created_at":"2026-05-18T01:17:52Z","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"},"paper":{"abstract_excerpt":"Coupled oscillator models where $N$ oscillators are identical and symmetrically coupled to all others with full permutation symmetry $S_N$ are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive - and we characterise generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical s","authors_text":"Ana Rodrigues, Peter Ashwin","cross_cats":["nlin.AO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-29T09:44:00Z","title":"Hopf normal form with $S_N$ symmetry and reduction to systems of nonlinearly coupled phase oscillators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08079","kind":"arxiv","version":2},"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":"sha256:7ae576b12073e943c0e3298171cb90cfc0ce987eaf71c9ff7e3ff7d184e0becf","target":"record","created_at":"2026-05-18T01:17:52Z","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":"48b718aa7dfa54dd56aa6605287496a676a89acbf53702a210b4835915118b46","cross_cats_sorted":["nlin.AO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-29T09:44:00Z","title_canon_sha256":"b28c9c9476539f61c26b93f1b7de666e404f35594398a35cc6d8673683770a53"},"schema_version":"1.0","source":{"id":"1507.08079","kind":"arxiv","version":2}},"canonical_sha256":"2bc29291df11def0754d35036b66c1716baa1a56706082d7e3edafb6c18d2f67","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2bc29291df11def0754d35036b66c1716baa1a56706082d7e3edafb6c18d2f67","first_computed_at":"2026-05-18T01:17:52.634905Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:52.634905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uU72xZMfY/rQrZp1AQN66M1WH5X6fAV9fYMT8TWy8ThsjEVXaFRA48/YiVKRiIR0ljf9S8eeiGEMt/KZ9i6yDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:52.635599Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.08079","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ae576b12073e943c0e3298171cb90cfc0ce987eaf71c9ff7e3ff7d184e0becf","sha256:6bd9f0291679f335c4792d66c4b655a0e48325b9d39c4b37a0c28cfa082fbfa6"],"state_sha256":"9d1dbd0770d8dead38aadf714f658688a60982c04032c4c3a19e9e0639115623"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9G4T7OmgNEB/Vepb3BNqyEF3Yv05UMZ1rXvWl2eGeHLWzEC8z+3RJQqMMxC+RBzAwrOBsq+9SwopyfGst0NMAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T22:15:46.760359Z","bundle_sha256":"b74d73256184a4e0d0fd8fed5a761369ace3eb37c70dbcf73bf7c383f9d420f0"}}