{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:JGWTLIB2F3PT4AAXVNDXBTCWET","short_pith_number":"pith:JGWTLIB2","canonical_record":{"source":{"id":"1410.8870","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-10-31T19:59:25Z","cross_cats_sorted":[],"title_canon_sha256":"51651202e418ad3e456180ea0142ea54ad7b92d4d51b7cc470270fe219ae9aa9","abstract_canon_sha256":"9f6c24e5e0e3228f7cf36eb4a1195c51c7777e9e2b5935689fb54a21a5e1e37c"},"schema_version":"1.0"},"canonical_sha256":"49ad35a03a2edf3e0017ab4770cc5624dbf2eebd9ba7e17cb8bbaf2bb2712aca","source":{"kind":"arxiv","id":"1410.8870","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.8870","created_at":"2026-05-18T02:38:53Z"},{"alias_kind":"arxiv_version","alias_value":"1410.8870v1","created_at":"2026-05-18T02:38:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.8870","created_at":"2026-05-18T02:38:53Z"},{"alias_kind":"pith_short_12","alias_value":"JGWTLIB2F3PT","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"JGWTLIB2F3PT4AAX","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"JGWTLIB2","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:JGWTLIB2F3PT4AAXVNDXBTCWET","target":"record","payload":{"canonical_record":{"source":{"id":"1410.8870","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-10-31T19:59:25Z","cross_cats_sorted":[],"title_canon_sha256":"51651202e418ad3e456180ea0142ea54ad7b92d4d51b7cc470270fe219ae9aa9","abstract_canon_sha256":"9f6c24e5e0e3228f7cf36eb4a1195c51c7777e9e2b5935689fb54a21a5e1e37c"},"schema_version":"1.0"},"canonical_sha256":"49ad35a03a2edf3e0017ab4770cc5624dbf2eebd9ba7e17cb8bbaf2bb2712aca","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:53.528605Z","signature_b64":"8XW87V00VVsQkr/gssuhAY6aXIRJRMSXf2A3YXD5W8QTVPUtkBb6lun77iH6eaoHhexoUgicWzXVFS9ze4aEAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49ad35a03a2edf3e0017ab4770cc5624dbf2eebd9ba7e17cb8bbaf2bb2712aca","last_reissued_at":"2026-05-18T02:38:53.528186Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:53.528186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.8870","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":"2026-05-18T02:38:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nWC+RBCYIiV5PIWOsOQPMG5xXaH6hKxLJiEWh+5bHPCAGrr+9FibMfJziVJeVWEH2km109S1JJEV6gsz/j74DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T08:33:46.673928Z"},"content_sha256":"c1b34c233e5be6c1c888539fc25e81e625e07406bbb6fd570eeb7cd5e2ffa2d3","schema_version":"1.0","event_id":"sha256:c1b34c233e5be6c1c888539fc25e81e625e07406bbb6fd570eeb7cd5e2ffa2d3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:JGWTLIB2F3PT4AAXVNDXBTCWET","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ergodic decompositions for folding and unfolding paths in Outer space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Alexandra Pettet, Hossein Namazi, Patrick Reynolds","submitted_at":"2014-10-31T19:59:25Z","abstract_excerpt":"We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the folding/unfolding path. It follows that non-unique ergodicity leads to distortion when projecting to the complex of free factors, and we give two applications of this fact. First, we show that if a subgroup $H$ of $Out(\\FN)$ quasi-isometrically embeds into the complex of free factors v"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8870","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":"2026-05-18T02:38:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fyYQj8aJFacHjul+qaSGlBPLqmO7CfmdYsgwB3iauo1SyGuBeR65JvYjn4q00cGuQYHvU/9C7cu0ECHJIGnQDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T08:33:46.674272Z"},"content_sha256":"8d7b67c59bc9085e896821b7ce151391afeb584c339e1f707187cd9264ec4ca7","schema_version":"1.0","event_id":"sha256:8d7b67c59bc9085e896821b7ce151391afeb584c339e1f707187cd9264ec4ca7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JGWTLIB2F3PT4AAXVNDXBTCWET/bundle.json","state_url":"https://pith.science/pith/JGWTLIB2F3PT4AAXVNDXBTCWET/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JGWTLIB2F3PT4AAXVNDXBTCWET/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":"2026-06-30T08:33:46Z","links":{"resolver":"https://pith.science/pith/JGWTLIB2F3PT4AAXVNDXBTCWET","bundle":"https://pith.science/pith/JGWTLIB2F3PT4AAXVNDXBTCWET/bundle.json","state":"https://pith.science/pith/JGWTLIB2F3PT4AAXVNDXBTCWET/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JGWTLIB2F3PT4AAXVNDXBTCWET/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:JGWTLIB2F3PT4AAXVNDXBTCWET","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":"9f6c24e5e0e3228f7cf36eb4a1195c51c7777e9e2b5935689fb54a21a5e1e37c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-10-31T19:59:25Z","title_canon_sha256":"51651202e418ad3e456180ea0142ea54ad7b92d4d51b7cc470270fe219ae9aa9"},"schema_version":"1.0","source":{"id":"1410.8870","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.8870","created_at":"2026-05-18T02:38:53Z"},{"alias_kind":"arxiv_version","alias_value":"1410.8870v1","created_at":"2026-05-18T02:38:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.8870","created_at":"2026-05-18T02:38:53Z"},{"alias_kind":"pith_short_12","alias_value":"JGWTLIB2F3PT","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"JGWTLIB2F3PT4AAX","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"JGWTLIB2","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:8d7b67c59bc9085e896821b7ce151391afeb584c339e1f707187cd9264ec4ca7","target":"graph","created_at":"2026-05-18T02:38:53Z","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 relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the folding/unfolding path. It follows that non-unique ergodicity leads to distortion when projecting to the complex of free factors, and we give two applications of this fact. First, we show that if a subgroup $H$ of $Out(\\FN)$ quasi-isometrically embeds into the complex of free factors v","authors_text":"Alexandra Pettet, Hossein Namazi, Patrick Reynolds","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-10-31T19:59:25Z","title":"Ergodic decompositions for folding and unfolding paths in Outer space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8870","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:c1b34c233e5be6c1c888539fc25e81e625e07406bbb6fd570eeb7cd5e2ffa2d3","target":"record","created_at":"2026-05-18T02:38:53Z","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":"9f6c24e5e0e3228f7cf36eb4a1195c51c7777e9e2b5935689fb54a21a5e1e37c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-10-31T19:59:25Z","title_canon_sha256":"51651202e418ad3e456180ea0142ea54ad7b92d4d51b7cc470270fe219ae9aa9"},"schema_version":"1.0","source":{"id":"1410.8870","kind":"arxiv","version":1}},"canonical_sha256":"49ad35a03a2edf3e0017ab4770cc5624dbf2eebd9ba7e17cb8bbaf2bb2712aca","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"49ad35a03a2edf3e0017ab4770cc5624dbf2eebd9ba7e17cb8bbaf2bb2712aca","first_computed_at":"2026-05-18T02:38:53.528186Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:53.528186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8XW87V00VVsQkr/gssuhAY6aXIRJRMSXf2A3YXD5W8QTVPUtkBb6lun77iH6eaoHhexoUgicWzXVFS9ze4aEAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:53.528605Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.8870","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c1b34c233e5be6c1c888539fc25e81e625e07406bbb6fd570eeb7cd5e2ffa2d3","sha256:8d7b67c59bc9085e896821b7ce151391afeb584c339e1f707187cd9264ec4ca7"],"state_sha256":"df8105295881e3ffa2400c5be69a029cda5b295cd5e1c581918ebf3625f30925"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kh3ZtKya6apvOY4IP/pPvrTAZ6p4VgF7P4Fy65kc5bsHszE6r0lbopZoCC0cP9hyI9zW0prepdAVsjNKpRHqAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T08:33:46.676205Z","bundle_sha256":"eaca780c1492e1a5bbb5468bdb59769608cb5dd1f416dc5e430674d44cc894fd"}}