{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:M7A2PPBFQYIPG77D237YPMFDUL","short_pith_number":"pith:M7A2PPBF","canonical_record":{"source":{"id":"math/0607425","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.OC","submitted_at":"2006-07-18T15:22:10Z","cross_cats_sorted":[],"title_canon_sha256":"8b457e1fcabcef030f274d37a21be044c604dfaa69d19b1916b043db4f5ae628","abstract_canon_sha256":"4e39741ef97da5de5097493d1033cc0baa2b933ef13c6899188d30152f99fde1"},"schema_version":"1.0"},"canonical_sha256":"67c1a7bc258610f37fe3d6ff87b0a3a2e5a08e797d070abf941d7e0ce146fed8","source":{"kind":"arxiv","id":"math/0607425","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0607425","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/0607425v1","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607425","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"pith_short_12","alias_value":"M7A2PPBFQYIP","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"M7A2PPBFQYIPG77D","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"M7A2PPBF","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:M7A2PPBFQYIPG77D237YPMFDUL","target":"record","payload":{"canonical_record":{"source":{"id":"math/0607425","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.OC","submitted_at":"2006-07-18T15:22:10Z","cross_cats_sorted":[],"title_canon_sha256":"8b457e1fcabcef030f274d37a21be044c604dfaa69d19b1916b043db4f5ae628","abstract_canon_sha256":"4e39741ef97da5de5097493d1033cc0baa2b933ef13c6899188d30152f99fde1"},"schema_version":"1.0"},"canonical_sha256":"67c1a7bc258610f37fe3d6ff87b0a3a2e5a08e797d070abf941d7e0ce146fed8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:50.027310Z","signature_b64":"rH1CH+e500EN36rR+kCBHCPEyjwzAtVheNfKadRuBIQiNAUkUqPEEZMEifX872helB3oB42uRlF/VJJ2Z/ElDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67c1a7bc258610f37fe3d6ff87b0a3a2e5a08e797d070abf941d7e0ce146fed8","last_reissued_at":"2026-05-18T01:08:50.026673Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:50.026673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0607425","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":"2026-05-18T01:08:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yky3z7EJVmKMu4+hqig2FNzNh9pDtZRVuy/P/t5/lghvTddSR11OaFK/imekEaLW86hPFOLNMVsHrt7NpR+TBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T06:33:36.343263Z"},"content_sha256":"6dcaeb3d6a0a12f8a2b6c49edded57fa0289a3137c11d8903c215d125f35e97f","schema_version":"1.0","event_id":"sha256:6dcaeb3d6a0a12f8a2b6c49edded57fa0289a3137c11d8903c215d125f35e97f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:M7A2PPBFQYIPG77D237YPMFDUL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotics of accessibility sets along an abnormal trajectory","license":"","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Emmanuel Tr\\'elat (IMB)","submitted_at":"2006-07-18T15:22:10Z","abstract_excerpt":"We describe precisely, under generic conditions, the contact of the accessibility set at time $T$ with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer $\\gamma$ into two sectors, bordered by the first Pontryagin's cone along $\\gamma$, called the $\\xLinfty$-sector and the $\\xLtwo$-sector. Moreover we find again necessary and sufficient conditions of optimality of an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607425","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":""},"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":"2026-05-18T01:08:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mu9NS1idf4kBkhSKEzQhBOVWfZC6jhVTOS4IDiQb9WPQNUTt5flwwb4bUMEnGqXW6/yHU/bkUAQ2k7q3OAJ2Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T06:33:36.343625Z"},"content_sha256":"ea2ed44305911de4e21a4a83924c04ef170fd6b4568afe1d321939faf4810480","schema_version":"1.0","event_id":"sha256:ea2ed44305911de4e21a4a83924c04ef170fd6b4568afe1d321939faf4810480"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M7A2PPBFQYIPG77D237YPMFDUL/bundle.json","state_url":"https://pith.science/pith/M7A2PPBFQYIPG77D237YPMFDUL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M7A2PPBFQYIPG77D237YPMFDUL/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":"2026-07-04T06:33:36Z","links":{"resolver":"https://pith.science/pith/M7A2PPBFQYIPG77D237YPMFDUL","bundle":"https://pith.science/pith/M7A2PPBFQYIPG77D237YPMFDUL/bundle.json","state":"https://pith.science/pith/M7A2PPBFQYIPG77D237YPMFDUL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M7A2PPBFQYIPG77D237YPMFDUL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:M7A2PPBFQYIPG77D237YPMFDUL","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":"4e39741ef97da5de5097493d1033cc0baa2b933ef13c6899188d30152f99fde1","cross_cats_sorted":[],"license":"","primary_cat":"math.OC","submitted_at":"2006-07-18T15:22:10Z","title_canon_sha256":"8b457e1fcabcef030f274d37a21be044c604dfaa69d19b1916b043db4f5ae628"},"schema_version":"1.0","source":{"id":"math/0607425","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0607425","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/0607425v1","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607425","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"pith_short_12","alias_value":"M7A2PPBFQYIP","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"M7A2PPBFQYIPG77D","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"M7A2PPBF","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:ea2ed44305911de4e21a4a83924c04ef170fd6b4568afe1d321939faf4810480","target":"graph","created_at":"2026-05-18T01:08:50Z","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":"We describe precisely, under generic conditions, the contact of the accessibility set at time $T$ with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer $\\gamma$ into two sectors, bordered by the first Pontryagin's cone along $\\gamma$, called the $\\xLinfty$-sector and the $\\xLtwo$-sector. Moreover we find again necessary and sufficient conditions of optimality of an","authors_text":"Emmanuel Tr\\'elat (IMB)","cross_cats":[],"headline":"","license":"","primary_cat":"math.OC","submitted_at":"2006-07-18T15:22:10Z","title":"Asymptotics of accessibility sets along an abnormal trajectory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607425","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":"sha256:6dcaeb3d6a0a12f8a2b6c49edded57fa0289a3137c11d8903c215d125f35e97f","target":"record","created_at":"2026-05-18T01:08:50Z","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":"4e39741ef97da5de5097493d1033cc0baa2b933ef13c6899188d30152f99fde1","cross_cats_sorted":[],"license":"","primary_cat":"math.OC","submitted_at":"2006-07-18T15:22:10Z","title_canon_sha256":"8b457e1fcabcef030f274d37a21be044c604dfaa69d19b1916b043db4f5ae628"},"schema_version":"1.0","source":{"id":"math/0607425","kind":"arxiv","version":1}},"canonical_sha256":"67c1a7bc258610f37fe3d6ff87b0a3a2e5a08e797d070abf941d7e0ce146fed8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"67c1a7bc258610f37fe3d6ff87b0a3a2e5a08e797d070abf941d7e0ce146fed8","first_computed_at":"2026-05-18T01:08:50.026673Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:50.026673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rH1CH+e500EN36rR+kCBHCPEyjwzAtVheNfKadRuBIQiNAUkUqPEEZMEifX872helB3oB42uRlF/VJJ2Z/ElDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:50.027310Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0607425","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6dcaeb3d6a0a12f8a2b6c49edded57fa0289a3137c11d8903c215d125f35e97f","sha256:ea2ed44305911de4e21a4a83924c04ef170fd6b4568afe1d321939faf4810480"],"state_sha256":"b0e8e71a5b2d65e17d15d425632a1e7c9e3605682a347ce07139041204cfb922"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JlZ1PyFSS1FxwY5fORQ4trFgT4JiQZyEtb7QVqnES47rgRP4Rc8J6couSE3TCNsXFrAoGm5nHyV6vwQqSCmTAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T06:33:36.345705Z","bundle_sha256":"28f07e35235312b315f59f379aeff16b3254c7c2defd4e8eed6c003cd10638db"}}