{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:TLIHDDNLZ542NSRQMKD55OR3KV","short_pith_number":"pith:TLIHDDNL","canonical_record":{"source":{"id":"0704.3113","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2007-04-24T04:12:38Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"5813db45cc9aa8af3c9032fea7812d7c22309372d0364df47408a2cc7b7581d4","abstract_canon_sha256":"901c36bc30c5834c8613ce95bd7c6c95ef0aadece7bee8b51f61b4f94664d513"},"schema_version":"1.0"},"canonical_sha256":"9ad0718dabcf79a6ca306287deba3b556340ea07ed1e743764a25d221422c79c","source":{"kind":"arxiv","id":"0704.3113","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0704.3113","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"arxiv_version","alias_value":"0704.3113v1","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0704.3113","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"pith_short_12","alias_value":"TLIHDDNLZ542","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"pith_short_16","alias_value":"TLIHDDNLZ542NSRQ","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"pith_short_8","alias_value":"TLIHDDNL","created_at":"2026-07-04T15:01:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:TLIHDDNLZ542NSRQMKD55OR3KV","target":"record","payload":{"canonical_record":{"source":{"id":"0704.3113","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2007-04-24T04:12:38Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"5813db45cc9aa8af3c9032fea7812d7c22309372d0364df47408a2cc7b7581d4","abstract_canon_sha256":"901c36bc30c5834c8613ce95bd7c6c95ef0aadece7bee8b51f61b4f94664d513"},"schema_version":"1.0"},"canonical_sha256":"9ad0718dabcf79a6ca306287deba3b556340ea07ed1e743764a25d221422c79c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:01:02.476295Z","signature_b64":"Suj4KC5MuvxDgWaCWMKBWka/H5fqGpWqS8htV2yAnZE0hSZw3l9IeG72x9UQEdEuPpyCJkOA/hxod7Oqzv7qDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ad0718dabcf79a6ca306287deba3b556340ea07ed1e743764a25d221422c79c","last_reissued_at":"2026-07-04T15:01:02.475925Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:01:02.475925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0704.3113","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":"2026-07-04T15:01:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yOZmEh7+gRMhCuPPxFsgN2xFfY4fmVFvLTJZq3gZBeFpKsjO6Es30P9SbD1lRd5a0/sdqbzJKy4cfk3f7xA4CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T10:36:17.600060Z"},"content_sha256":"0a09f1f822a4c4ccd87ce740cffcc79f3c31c64361cc270063b0ff0a14f832b0","schema_version":"1.0","event_id":"sha256:0a09f1f822a4c4ccd87ce740cffcc79f3c31c64361cc270063b0ff0a14f832b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:TLIHDDNLZ542NSRQMKD55OR3KV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Self similar expanding solutions of the planar network flow","license":"","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Mariel Saez, Rafe Mazzeo","submitted_at":"2007-04-24T04:12:38Z","abstract_excerpt":"We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schn\\\"urer and Schulze which treats the case of three half-lines. There are multiple solutions, and these are parametrized by combinatorial objects, namely Steiner trees with respect to a complete negatively curved metric on the unit ball which span $k$ specified points on the boundary at infinity. We also provide a sharp formulation of the regularity of these solutions at $t=0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.3113","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/0704.3113/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":"2026-07-04T15:01:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ju5iH5PWge7d/lffKJOkpUowXLplS5j6GCegxtMv30AszOrIV49M+xDKwvxPedBfN7UYljgbzqMnKtZzw5wtBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T10:36:17.600487Z"},"content_sha256":"9855667174f57d75e76da0a04ed926e146d64b5bfee2c3efe13095e10ec9eec9","schema_version":"1.0","event_id":"sha256:9855667174f57d75e76da0a04ed926e146d64b5bfee2c3efe13095e10ec9eec9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TLIHDDNLZ542NSRQMKD55OR3KV/bundle.json","state_url":"https://pith.science/pith/TLIHDDNLZ542NSRQMKD55OR3KV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TLIHDDNLZ542NSRQMKD55OR3KV/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":"2026-07-05T10:36:17Z","links":{"resolver":"https://pith.science/pith/TLIHDDNLZ542NSRQMKD55OR3KV","bundle":"https://pith.science/pith/TLIHDDNLZ542NSRQMKD55OR3KV/bundle.json","state":"https://pith.science/pith/TLIHDDNLZ542NSRQMKD55OR3KV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TLIHDDNLZ542NSRQMKD55OR3KV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:TLIHDDNLZ542NSRQMKD55OR3KV","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":"901c36bc30c5834c8613ce95bd7c6c95ef0aadece7bee8b51f61b4f94664d513","cross_cats_sorted":["math.AP"],"license":"","primary_cat":"math.DG","submitted_at":"2007-04-24T04:12:38Z","title_canon_sha256":"5813db45cc9aa8af3c9032fea7812d7c22309372d0364df47408a2cc7b7581d4"},"schema_version":"1.0","source":{"id":"0704.3113","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0704.3113","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"arxiv_version","alias_value":"0704.3113v1","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0704.3113","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"pith_short_12","alias_value":"TLIHDDNLZ542","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"pith_short_16","alias_value":"TLIHDDNLZ542NSRQ","created_at":"2026-07-04T15:01:02Z"},{"alias_kind":"pith_short_8","alias_value":"TLIHDDNL","created_at":"2026-07-04T15:01:02Z"}],"graph_snapshots":[{"event_id":"sha256:9855667174f57d75e76da0a04ed926e146d64b5bfee2c3efe13095e10ec9eec9","target":"graph","created_at":"2026-07-04T15:01:02Z","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/0704.3113/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schn\\\"urer and Schulze which treats the case of three half-lines. There are multiple solutions, and these are parametrized by combinatorial objects, namely Steiner trees with respect to a complete negatively curved metric on the unit ball which span $k$ specified points on the boundary at infinity. We also provide a sharp formulation of the regularity of these solutions at $t=0$.","authors_text":"Mariel Saez, Rafe Mazzeo","cross_cats":["math.AP"],"headline":"","license":"","primary_cat":"math.DG","submitted_at":"2007-04-24T04:12:38Z","title":"Self similar expanding solutions of the planar network flow"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.3113","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":"sha256:0a09f1f822a4c4ccd87ce740cffcc79f3c31c64361cc270063b0ff0a14f832b0","target":"record","created_at":"2026-07-04T15:01:02Z","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":"901c36bc30c5834c8613ce95bd7c6c95ef0aadece7bee8b51f61b4f94664d513","cross_cats_sorted":["math.AP"],"license":"","primary_cat":"math.DG","submitted_at":"2007-04-24T04:12:38Z","title_canon_sha256":"5813db45cc9aa8af3c9032fea7812d7c22309372d0364df47408a2cc7b7581d4"},"schema_version":"1.0","source":{"id":"0704.3113","kind":"arxiv","version":1}},"canonical_sha256":"9ad0718dabcf79a6ca306287deba3b556340ea07ed1e743764a25d221422c79c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9ad0718dabcf79a6ca306287deba3b556340ea07ed1e743764a25d221422c79c","first_computed_at":"2026-07-04T15:01:02.475925Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:01:02.475925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Suj4KC5MuvxDgWaCWMKBWka/H5fqGpWqS8htV2yAnZE0hSZw3l9IeG72x9UQEdEuPpyCJkOA/hxod7Oqzv7qDg==","signature_status":"signed_v1","signed_at":"2026-07-04T15:01:02.476295Z","signed_message":"canonical_sha256_bytes"},"source_id":"0704.3113","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0a09f1f822a4c4ccd87ce740cffcc79f3c31c64361cc270063b0ff0a14f832b0","sha256:9855667174f57d75e76da0a04ed926e146d64b5bfee2c3efe13095e10ec9eec9"],"state_sha256":"ef428682a07b796835b3f6cdee3a314d27980314265ecc8b2e85165b4c774a57"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xqfGk+lPpNGW6xGtK06jz/rY+wcJznbzawpDu62OcyESRxMF0Khe17QcN3FyCAMNb1KIIjh5agM+C1xqad0BBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T10:36:17.603355Z","bundle_sha256":"dea8473e8390bd57991b502574563c31aed6c5e263a141a0cdcb435771f12ec3"}}