{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:JXU72AMMWYJAEMZ5YG4E42MD2Z","short_pith_number":"pith:JXU72AMM","canonical_record":{"source":{"id":"2412.00781","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-01T12:11:17Z","cross_cats_sorted":[],"title_canon_sha256":"054130ae77fcd907a8bde9f4b36507b6c14d16631a3d51882f8dffb890228e4a","abstract_canon_sha256":"69b847c6e6f0ff89a277925265dd62f675cca22d84fecae4a5a511a93b88b01c"},"schema_version":"1.0"},"canonical_sha256":"4de9fd018cb61202333dc1b84e6983d676bc8d08e7e5c0505ce5d7dd414121bf","source":{"kind":"arxiv","id":"2412.00781","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.00781","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"arxiv_version","alias_value":"2412.00781v4","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.00781","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"pith_short_12","alias_value":"JXU72AMMWYJA","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"pith_short_16","alias_value":"JXU72AMMWYJAEMZ5","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"pith_short_8","alias_value":"JXU72AMM","created_at":"2026-06-08T01:03:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:JXU72AMMWYJAEMZ5YG4E42MD2Z","target":"record","payload":{"canonical_record":{"source":{"id":"2412.00781","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-01T12:11:17Z","cross_cats_sorted":[],"title_canon_sha256":"054130ae77fcd907a8bde9f4b36507b6c14d16631a3d51882f8dffb890228e4a","abstract_canon_sha256":"69b847c6e6f0ff89a277925265dd62f675cca22d84fecae4a5a511a93b88b01c"},"schema_version":"1.0"},"canonical_sha256":"4de9fd018cb61202333dc1b84e6983d676bc8d08e7e5c0505ce5d7dd414121bf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:03:42.445846Z","signature_b64":"UIbQNSfAI8ezw5NPklAzf7/tf/PlPUlWSXrbXHP4+dqxf/sJH2RKsqnlrfXw/3QbWJfa1rNeEgoFJf0sYfB8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4de9fd018cb61202333dc1b84e6983d676bc8d08e7e5c0505ce5d7dd414121bf","last_reissued_at":"2026-06-08T01:03:42.444749Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:03:42.444749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.00781","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-08T01:03:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4JR97DmI+Lyb6y1EV+ByMvBFbqaOkHe+JBMiZOb0255dwm1/ADqHIqHv/lYTAvAkDK1T2uCYKNAZEVTwLBK1BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T08:27:00.302817Z"},"content_sha256":"294b720101f06e29757aacb7e9bd156e42b59c99b6b19c321b88c2032c541443","schema_version":"1.0","event_id":"sha256:294b720101f06e29757aacb7e9bd156e42b59c99b6b19c321b88c2032c541443"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:JXU72AMMWYJAEMZ5YG4E42MD2Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Structure of the free interfaces near triple junction singularities in harmonic maps and optimal partition problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bozhidar Velichkov, Roberto Ognibene","submitted_at":"2024-12-01T12:11:17Z","abstract_excerpt":"We consider energy-minimizing harmonic maps into trees and we prove the regularity of the singular part of the free interface near triple junction points. Precisely, by proving a new epiperimetric inequality, we show that around any point of frequency $3/2$, the free interface is composed of three $C^{1,\\alpha}$-smooth $(d-1)$-dimensional manifolds (composed of points of frequency $1$) with common $C^{1,\\alpha}$-regular boundary (made of points of frequency $3/2$) that meet along this boundary at 120 degree angles. Our results also apply to spectral optimal partition problems for the Dirichlet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.00781","kind":"arxiv","version":4},"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/2412.00781/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":"2026-06-08T01:03:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RgZTH/1X9w4tFf1awd/FfYK/xW0AFxpjw43/8fpDP1Qf5lZDHVU7neclRsdC719nHS7Clgwrh8SHk1qC2ckbBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T08:27:00.303616Z"},"content_sha256":"d2d30ab75ece542a410dcfa6459f6a3571a016d2c00ebf03966b0beaee7ef74a","schema_version":"1.0","event_id":"sha256:d2d30ab75ece542a410dcfa6459f6a3571a016d2c00ebf03966b0beaee7ef74a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JXU72AMMWYJAEMZ5YG4E42MD2Z/bundle.json","state_url":"https://pith.science/pith/JXU72AMMWYJAEMZ5YG4E42MD2Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JXU72AMMWYJAEMZ5YG4E42MD2Z/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":"2026-06-10T08:27:00Z","links":{"resolver":"https://pith.science/pith/JXU72AMMWYJAEMZ5YG4E42MD2Z","bundle":"https://pith.science/pith/JXU72AMMWYJAEMZ5YG4E42MD2Z/bundle.json","state":"https://pith.science/pith/JXU72AMMWYJAEMZ5YG4E42MD2Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JXU72AMMWYJAEMZ5YG4E42MD2Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:JXU72AMMWYJAEMZ5YG4E42MD2Z","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":"69b847c6e6f0ff89a277925265dd62f675cca22d84fecae4a5a511a93b88b01c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-01T12:11:17Z","title_canon_sha256":"054130ae77fcd907a8bde9f4b36507b6c14d16631a3d51882f8dffb890228e4a"},"schema_version":"1.0","source":{"id":"2412.00781","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.00781","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"arxiv_version","alias_value":"2412.00781v4","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.00781","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"pith_short_12","alias_value":"JXU72AMMWYJA","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"pith_short_16","alias_value":"JXU72AMMWYJAEMZ5","created_at":"2026-06-08T01:03:42Z"},{"alias_kind":"pith_short_8","alias_value":"JXU72AMM","created_at":"2026-06-08T01:03:42Z"}],"graph_snapshots":[{"event_id":"sha256:d2d30ab75ece542a410dcfa6459f6a3571a016d2c00ebf03966b0beaee7ef74a","target":"graph","created_at":"2026-06-08T01:03:42Z","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/2412.00781/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider energy-minimizing harmonic maps into trees and we prove the regularity of the singular part of the free interface near triple junction points. Precisely, by proving a new epiperimetric inequality, we show that around any point of frequency $3/2$, the free interface is composed of three $C^{1,\\alpha}$-smooth $(d-1)$-dimensional manifolds (composed of points of frequency $1$) with common $C^{1,\\alpha}$-regular boundary (made of points of frequency $3/2$) that meet along this boundary at 120 degree angles. Our results also apply to spectral optimal partition problems for the Dirichlet","authors_text":"Bozhidar Velichkov, Roberto Ognibene","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-01T12:11:17Z","title":"Structure of the free interfaces near triple junction singularities in harmonic maps and optimal partition problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.00781","kind":"arxiv","version":4},"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":"sha256:294b720101f06e29757aacb7e9bd156e42b59c99b6b19c321b88c2032c541443","target":"record","created_at":"2026-06-08T01:03:42Z","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":"69b847c6e6f0ff89a277925265dd62f675cca22d84fecae4a5a511a93b88b01c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-01T12:11:17Z","title_canon_sha256":"054130ae77fcd907a8bde9f4b36507b6c14d16631a3d51882f8dffb890228e4a"},"schema_version":"1.0","source":{"id":"2412.00781","kind":"arxiv","version":4}},"canonical_sha256":"4de9fd018cb61202333dc1b84e6983d676bc8d08e7e5c0505ce5d7dd414121bf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4de9fd018cb61202333dc1b84e6983d676bc8d08e7e5c0505ce5d7dd414121bf","first_computed_at":"2026-06-08T01:03:42.444749Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:03:42.444749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UIbQNSfAI8ezw5NPklAzf7/tf/PlPUlWSXrbXHP4+dqxf/sJH2RKsqnlrfXw/3QbWJfa1rNeEgoFJf0sYfB8Cw==","signature_status":"signed_v1","signed_at":"2026-06-08T01:03:42.445846Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.00781","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:294b720101f06e29757aacb7e9bd156e42b59c99b6b19c321b88c2032c541443","sha256:d2d30ab75ece542a410dcfa6459f6a3571a016d2c00ebf03966b0beaee7ef74a"],"state_sha256":"0c1deeca1f8dcc6ecf109d3b9e33fc7bcb598d26457d4e2cf624e1863d36b93d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Os8o6bSHkCgq3K44666aliv/Vb+nFjStKKzqzUd7BaSook+ATFM868LAEWCVjIO3KlEHTYLGhWDyyeMuRNGACg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T08:27:00.308111Z","bundle_sha256":"be98b0450f08d56912819e90b2b291a722990833e0bd037f6070b06ef19cbf41"}}