{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ECYUOA37WTBRDRGRH57LTRSZ6U","short_pith_number":"pith:ECYUOA37","canonical_record":{"source":{"id":"2605.12307","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-12T15:55:52Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"2dfd70d1fc366d20adb049cbdce4e67258df3463a13dbcc076918b134396c762","abstract_canon_sha256":"5ad8f9829bfced69eae67c3ea4edf7e95e23c6ea52832f9805cedec396687422"},"schema_version":"1.0"},"canonical_sha256":"20b147037fb4c311c4d13f7eb9c659f51c675ad03881488fa2535a669c2e7ea2","source":{"kind":"arxiv","id":"2605.12307","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12307","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12307v3","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12307","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"pith_short_12","alias_value":"ECYUOA37WTBR","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"pith_short_16","alias_value":"ECYUOA37WTBRDRGR","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"pith_short_8","alias_value":"ECYUOA37","created_at":"2026-05-20T00:04:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ECYUOA37WTBRDRGRH57LTRSZ6U","target":"record","payload":{"canonical_record":{"source":{"id":"2605.12307","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-12T15:55:52Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"2dfd70d1fc366d20adb049cbdce4e67258df3463a13dbcc076918b134396c762","abstract_canon_sha256":"5ad8f9829bfced69eae67c3ea4edf7e95e23c6ea52832f9805cedec396687422"},"schema_version":"1.0"},"canonical_sha256":"20b147037fb4c311c4d13f7eb9c659f51c675ad03881488fa2535a669c2e7ea2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:04:36.456746Z","signature_b64":"RY4ins5+KrudFEpJBgMaaJr7YqGS0xf3y7QVgacen7ZwS8Xeet3nWzicCA+NeyyrBei36IPEIOugqMeaY4u0Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20b147037fb4c311c4d13f7eb9c659f51c675ad03881488fa2535a669c2e7ea2","last_reissued_at":"2026-05-20T00:04:36.455689Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:04:36.455689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.12307","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T00:04:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iR5ssMBHv88bJf1xsWyrBxqj94NoKWmWQgVclkzSVhDYtx9OFupLPPI4YOcI8qfB1v3zSkh8BdUZRwqGRRUrCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T20:36:23.988097Z"},"content_sha256":"77e8d8f6b4a64622d47be104861e4938fb7c71e1e22142115d1c3969facb07dd","schema_version":"1.0","event_id":"sha256:77e8d8f6b4a64622d47be104861e4938fb7c71e1e22142115d1c3969facb07dd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ECYUOA37WTBRDRGRH57LTRSZ6U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized pseudo-product structures and finite type distributions via abnormal extremals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Real-analytic distributions controllable by regular abnormal extremals have finite-dimensional symmetry algebras.","cross_cats":["math.OC"],"primary_cat":"math.DG","authors_text":"Boris Doubrov, Igor zelenko","submitted_at":"2026-05-12T15:55:52Z","abstract_excerpt":"We generalize the classical Tanaka result on the finiteness of symmetry algebra for non-degenerate pseudo-product structures to the case when the completely-integrable distributions defining the pseudo-product structure are no longer concentrated in the degree $-1$. In order to do this, we modify the notion of universal prolongation of graded nilpotent Lie algebras and generalize the original finiteness criterion of Tanaka. Using this result, we demonstrate that in real analytic category, distributions that are controllable by regular abnormal extremal trajectories, also known as singularly tr"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"In the real analytic category, distributions that are controllable by regular abnormal extremal trajectories, also known as singularly transitive, have finite-dimensional symmetries.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The distributions are real analytic and the abnormal extremals are regular; the modified universal prolongation construction preserves the finiteness property without introducing infinite-dimensional symmetries.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalized Tanaka prolongation criterion shows that singularly transitive distributions have finite-dimensional symmetries in the real analytic category.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Real-analytic distributions controllable by regular abnormal extremals have finite-dimensional symmetry algebras.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"85195e71f2c2d546a26b6c0e4d508ec0feec549f4f37d55c40c679643899c09f"},"source":{"id":"2605.12307","kind":"arxiv","version":3},"verdict":{"id":"f00149a2-b4b9-4cb8-be06-2ab929f30efa","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:41:18.624145Z","strongest_claim":"In the real analytic category, distributions that are controllable by regular abnormal extremal trajectories, also known as singularly transitive, have finite-dimensional symmetries.","one_line_summary":"Generalized Tanaka prolongation criterion shows that singularly transitive distributions have finite-dimensional symmetries in the real analytic category.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The distributions are real analytic and the abnormal extremals are regular; the modified universal prolongation construction preserves the finiteness property without introducing infinite-dimensional symmetries.","pith_extraction_headline":"Real-analytic distributions controllable by regular abnormal extremals have finite-dimensional symmetry algebras."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.12307/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-19T22:41:58.287638Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T10:39:23.691691Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T08:31:16.618628Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T07:40:37.003424Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"1bab36de86ed7d611bf0cc6901a88557c09b6d2f1fcd6d0180a886ffa725902e"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"e01ede72722ad9c5f6887d7fe56b7ded000f7b7eff96cef4556eb1114a421f38"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"f00149a2-b4b9-4cb8-be06-2ab929f30efa"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T00:04:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d9cNY2NRZnef+V0Gz0zlrzhH9DHxM03HCAapVlPUeyZxENfUkcJULCEGubYoZyWvIUZpiRrw7KAaH9lyn7pjAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T20:36:23.988631Z"},"content_sha256":"3c0649514096d6d1ba07dc6e8fa1dba1c35a07521c6805f5575ca3be9a825be0","schema_version":"1.0","event_id":"sha256:3c0649514096d6d1ba07dc6e8fa1dba1c35a07521c6805f5575ca3be9a825be0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ECYUOA37WTBRDRGRH57LTRSZ6U/bundle.json","state_url":"https://pith.science/pith/ECYUOA37WTBRDRGRH57LTRSZ6U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ECYUOA37WTBRDRGRH57LTRSZ6U/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":"2026-05-30T20:36:23Z","links":{"resolver":"https://pith.science/pith/ECYUOA37WTBRDRGRH57LTRSZ6U","bundle":"https://pith.science/pith/ECYUOA37WTBRDRGRH57LTRSZ6U/bundle.json","state":"https://pith.science/pith/ECYUOA37WTBRDRGRH57LTRSZ6U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ECYUOA37WTBRDRGRH57LTRSZ6U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ECYUOA37WTBRDRGRH57LTRSZ6U","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":"5ad8f9829bfced69eae67c3ea4edf7e95e23c6ea52832f9805cedec396687422","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-12T15:55:52Z","title_canon_sha256":"2dfd70d1fc366d20adb049cbdce4e67258df3463a13dbcc076918b134396c762"},"schema_version":"1.0","source":{"id":"2605.12307","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12307","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12307v3","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12307","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"pith_short_12","alias_value":"ECYUOA37WTBR","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"pith_short_16","alias_value":"ECYUOA37WTBRDRGR","created_at":"2026-05-20T00:04:36Z"},{"alias_kind":"pith_short_8","alias_value":"ECYUOA37","created_at":"2026-05-20T00:04:36Z"}],"graph_snapshots":[{"event_id":"sha256:3c0649514096d6d1ba07dc6e8fa1dba1c35a07521c6805f5575ca3be9a825be0","target":"graph","created_at":"2026-05-20T00:04:36Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"In the real analytic category, distributions that are controllable by regular abnormal extremal trajectories, also known as singularly transitive, have finite-dimensional symmetries."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The distributions are real analytic and the abnormal extremals are regular; the modified universal prolongation construction preserves the finiteness property without introducing infinite-dimensional symmetries."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Generalized Tanaka prolongation criterion shows that singularly transitive distributions have finite-dimensional symmetries in the real analytic category."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Real-analytic distributions controllable by regular abnormal extremals have finite-dimensional symmetry algebras."}],"snapshot_sha256":"85195e71f2c2d546a26b6c0e4d508ec0feec549f4f37d55c40c679643899c09f"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"e01ede72722ad9c5f6887d7fe56b7ded000f7b7eff96cef4556eb1114a421f38"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"claim_evidence","ran_at":"2026-05-19T22:41:58.287638Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-19T10:39:23.691691Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T08:31:16.618628Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T07:40:37.003424Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.12307/integrity.json","findings":[],"snapshot_sha256":"1bab36de86ed7d611bf0cc6901a88557c09b6d2f1fcd6d0180a886ffa725902e","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We generalize the classical Tanaka result on the finiteness of symmetry algebra for non-degenerate pseudo-product structures to the case when the completely-integrable distributions defining the pseudo-product structure are no longer concentrated in the degree $-1$. In order to do this, we modify the notion of universal prolongation of graded nilpotent Lie algebras and generalize the original finiteness criterion of Tanaka. Using this result, we demonstrate that in real analytic category, distributions that are controllable by regular abnormal extremal trajectories, also known as singularly tr","authors_text":"Boris Doubrov, Igor zelenko","cross_cats":["math.OC"],"headline":"Real-analytic distributions controllable by regular abnormal extremals have finite-dimensional symmetry algebras.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-12T15:55:52Z","title":"Generalized pseudo-product structures and finite type distributions via abnormal extremals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.12307","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-14T20:41:18.624145Z","id":"f00149a2-b4b9-4cb8-be06-2ab929f30efa","model_set":{"reader":"grok-4.3"},"one_line_summary":"Generalized Tanaka prolongation criterion shows that singularly transitive distributions have finite-dimensional symmetries in the real analytic category.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Real-analytic distributions controllable by regular abnormal extremals have finite-dimensional symmetry algebras.","strongest_claim":"In the real analytic category, distributions that are controllable by regular abnormal extremal trajectories, also known as singularly transitive, have finite-dimensional symmetries.","weakest_assumption":"The distributions are real analytic and the abnormal extremals are regular; the modified universal prolongation construction preserves the finiteness property without introducing infinite-dimensional symmetries."}},"verdict_id":"f00149a2-b4b9-4cb8-be06-2ab929f30efa"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:77e8d8f6b4a64622d47be104861e4938fb7c71e1e22142115d1c3969facb07dd","target":"record","created_at":"2026-05-20T00:04:36Z","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":"5ad8f9829bfced69eae67c3ea4edf7e95e23c6ea52832f9805cedec396687422","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-12T15:55:52Z","title_canon_sha256":"2dfd70d1fc366d20adb049cbdce4e67258df3463a13dbcc076918b134396c762"},"schema_version":"1.0","source":{"id":"2605.12307","kind":"arxiv","version":3}},"canonical_sha256":"20b147037fb4c311c4d13f7eb9c659f51c675ad03881488fa2535a669c2e7ea2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"20b147037fb4c311c4d13f7eb9c659f51c675ad03881488fa2535a669c2e7ea2","first_computed_at":"2026-05-20T00:04:36.455689Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:36.455689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RY4ins5+KrudFEpJBgMaaJr7YqGS0xf3y7QVgacen7ZwS8Xeet3nWzicCA+NeyyrBei36IPEIOugqMeaY4u0Aw==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:36.456746Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.12307","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:77e8d8f6b4a64622d47be104861e4938fb7c71e1e22142115d1c3969facb07dd","sha256:3c0649514096d6d1ba07dc6e8fa1dba1c35a07521c6805f5575ca3be9a825be0"],"state_sha256":"eefd684831694c0a2582e81731e5c242b0c5ea0c03cd4e7e304a5eb32005221e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YPIbU8wXiQXUvs3ksNYS/8VGssoLGF137l9+PZndmuUYumWdZMAB4SPs7NCODaGTPCU3ZYp9xD5+wPPkH+5CCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T20:36:23.990968Z","bundle_sha256":"b73c09a239af459a92ad91bf48e18a8e11bb400ef60bc3deb548a24e24e1a760"}}