{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:XGV5ICUT3VEKUM3KKGMGKIMRWM","short_pith_number":"pith:XGV5ICUT","canonical_record":{"source":{"id":"0807.4715","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DS","submitted_at":"2008-07-29T17:43:59Z","cross_cats_sorted":[],"title_canon_sha256":"5c9c75409bda2b1649f48e75b25adc57dd91365a0477039666ccd25241e95fab","abstract_canon_sha256":"4623bcf20babff724fec1ab03304b9561f2398a801f3bcef81f500ee650b2190"},"schema_version":"1.0"},"canonical_sha256":"b9abd40a93dd48aa336a5198652191b322d9436df78eac462762dcf74117a0b1","source":{"kind":"arxiv","id":"0807.4715","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.4715","created_at":"2026-05-18T04:15:19Z"},{"alias_kind":"arxiv_version","alias_value":"0807.4715v2","created_at":"2026-05-18T04:15:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.4715","created_at":"2026-05-18T04:15:19Z"},{"alias_kind":"pith_short_12","alias_value":"XGV5ICUT3VEK","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"XGV5ICUT3VEKUM3K","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"XGV5ICUT","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:XGV5ICUT3VEKUM3KKGMGKIMRWM","target":"record","payload":{"canonical_record":{"source":{"id":"0807.4715","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DS","submitted_at":"2008-07-29T17:43:59Z","cross_cats_sorted":[],"title_canon_sha256":"5c9c75409bda2b1649f48e75b25adc57dd91365a0477039666ccd25241e95fab","abstract_canon_sha256":"4623bcf20babff724fec1ab03304b9561f2398a801f3bcef81f500ee650b2190"},"schema_version":"1.0"},"canonical_sha256":"b9abd40a93dd48aa336a5198652191b322d9436df78eac462762dcf74117a0b1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:19.076884Z","signature_b64":"r2Vh12F00NaOj+GnNroKciR09SeDwVJ2QyBVTnSnO/9Fs87LOEbQMfmYahdp1a/JNo9BA4Cr3FWKUUSvGHXiAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9abd40a93dd48aa336a5198652191b322d9436df78eac462762dcf74117a0b1","last_reissued_at":"2026-05-18T04:15:19.076013Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:19.076013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0807.4715","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:15:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oUpKDra/tnygLzbDnQ9/c+mTfXwnF/fR3PDUCa0A8NH/2s3K8opAKS0LcRGx1DSd6R5QZPL/2q3I59kSbeykDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T15:24:50.679062Z"},"content_sha256":"c27418efc079aedd99e7d53c878de205e1d6815670b974e49d0d67e23bdb2fab","schema_version":"1.0","event_id":"sha256:c27418efc079aedd99e7d53c878de205e1d6815670b974e49d0d67e23bdb2fab"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:XGV5ICUT3VEKUM3KKGMGKIMRWM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Eventually Expanding Maps","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Pawel Gora, Peyman Eslami","submitted_at":"2008-07-29T17:43:59Z","abstract_excerpt":"In this paper we show that the piecewise linear map f(x) = px for x in [0,1/p], and sx-s/p for x in (1/p,1], p > 1, 0 < s < 1 which has an expanding, onto branch and a contracting branch is eventually piecewise expanding and exact."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.4715","kind":"arxiv","version":2},"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":"2026-05-18T04:15:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/TH5BynwqHEPbptch3W5daZ2YX+vVFZfyhsv6UtXfwzUTz8eoHLqyqmlcn9i1O3g0KI0JU+xcmUDeVZUIAlzCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T15:24:50.679426Z"},"content_sha256":"90a888b42b1c343b2daf4aa66b27adbc0b6d5e32949308185ef897da2d34b5c3","schema_version":"1.0","event_id":"sha256:90a888b42b1c343b2daf4aa66b27adbc0b6d5e32949308185ef897da2d34b5c3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XGV5ICUT3VEKUM3KKGMGKIMRWM/bundle.json","state_url":"https://pith.science/pith/XGV5ICUT3VEKUM3KKGMGKIMRWM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XGV5ICUT3VEKUM3KKGMGKIMRWM/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":"2026-06-23T15:24:50Z","links":{"resolver":"https://pith.science/pith/XGV5ICUT3VEKUM3KKGMGKIMRWM","bundle":"https://pith.science/pith/XGV5ICUT3VEKUM3KKGMGKIMRWM/bundle.json","state":"https://pith.science/pith/XGV5ICUT3VEKUM3KKGMGKIMRWM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XGV5ICUT3VEKUM3KKGMGKIMRWM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:XGV5ICUT3VEKUM3KKGMGKIMRWM","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":"4623bcf20babff724fec1ab03304b9561f2398a801f3bcef81f500ee650b2190","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DS","submitted_at":"2008-07-29T17:43:59Z","title_canon_sha256":"5c9c75409bda2b1649f48e75b25adc57dd91365a0477039666ccd25241e95fab"},"schema_version":"1.0","source":{"id":"0807.4715","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.4715","created_at":"2026-05-18T04:15:19Z"},{"alias_kind":"arxiv_version","alias_value":"0807.4715v2","created_at":"2026-05-18T04:15:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.4715","created_at":"2026-05-18T04:15:19Z"},{"alias_kind":"pith_short_12","alias_value":"XGV5ICUT3VEK","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"XGV5ICUT3VEKUM3K","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"XGV5ICUT","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:90a888b42b1c343b2daf4aa66b27adbc0b6d5e32949308185ef897da2d34b5c3","target":"graph","created_at":"2026-05-18T04:15:19Z","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":"In this paper we show that the piecewise linear map f(x) = px for x in [0,1/p], and sx-s/p for x in (1/p,1], p > 1, 0 < s < 1 which has an expanding, onto branch and a contracting branch is eventually piecewise expanding and exact.","authors_text":"Pawel Gora, Peyman Eslami","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DS","submitted_at":"2008-07-29T17:43:59Z","title":"Eventually Expanding Maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.4715","kind":"arxiv","version":2},"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:c27418efc079aedd99e7d53c878de205e1d6815670b974e49d0d67e23bdb2fab","target":"record","created_at":"2026-05-18T04:15:19Z","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":"4623bcf20babff724fec1ab03304b9561f2398a801f3bcef81f500ee650b2190","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DS","submitted_at":"2008-07-29T17:43:59Z","title_canon_sha256":"5c9c75409bda2b1649f48e75b25adc57dd91365a0477039666ccd25241e95fab"},"schema_version":"1.0","source":{"id":"0807.4715","kind":"arxiv","version":2}},"canonical_sha256":"b9abd40a93dd48aa336a5198652191b322d9436df78eac462762dcf74117a0b1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b9abd40a93dd48aa336a5198652191b322d9436df78eac462762dcf74117a0b1","first_computed_at":"2026-05-18T04:15:19.076013Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:15:19.076013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r2Vh12F00NaOj+GnNroKciR09SeDwVJ2QyBVTnSnO/9Fs87LOEbQMfmYahdp1a/JNo9BA4Cr3FWKUUSvGHXiAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:15:19.076884Z","signed_message":"canonical_sha256_bytes"},"source_id":"0807.4715","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c27418efc079aedd99e7d53c878de205e1d6815670b974e49d0d67e23bdb2fab","sha256:90a888b42b1c343b2daf4aa66b27adbc0b6d5e32949308185ef897da2d34b5c3"],"state_sha256":"6e82189e3fbcdd3b54a2dcf2ca1cb3cac31e145f0bf9df27d6625d21772ea42c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RkCGopoR1PqpkiwoGs9MtO5RFnwlQnICfw5XFjFVEQFtdeoMOCRYrvrayL1bP7BCDfLw81VQvmjkF5THKZWRBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T15:24:50.681368Z","bundle_sha256":"86ffa8105629f7a47da3391ac9dd1d8f16cd9f9450ba597210a4fc1dea70e604"}}