{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:6H7XRIJRG63Z3RYSD2GXQSPVMX","short_pith_number":"pith:6H7XRIJR","canonical_record":{"source":{"id":"0903.4289","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-03-25T10:34:07Z","cross_cats_sorted":[],"title_canon_sha256":"86c24f8b8b228571bff93aeccf35461543a0ffa8ddcb3fefe3fecfd58dbc6f8f","abstract_canon_sha256":"be878778adba830f1059d84597d0daa65edd31cd03dd387df2ffa9de0490b10e"},"schema_version":"1.0"},"canonical_sha256":"f1ff78a13137b79dc7121e8d7849f565c4c5406f3dfccc922db7c07409563ec9","source":{"kind":"arxiv","id":"0903.4289","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4289","created_at":"2026-05-18T00:14:35Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4289v4","created_at":"2026-05-18T00:14:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4289","created_at":"2026-05-18T00:14:35Z"},{"alias_kind":"pith_short_12","alias_value":"6H7XRIJRG63Z","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6H7XRIJRG63Z3RYS","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6H7XRIJR","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:6H7XRIJRG63Z3RYSD2GXQSPVMX","target":"record","payload":{"canonical_record":{"source":{"id":"0903.4289","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-03-25T10:34:07Z","cross_cats_sorted":[],"title_canon_sha256":"86c24f8b8b228571bff93aeccf35461543a0ffa8ddcb3fefe3fecfd58dbc6f8f","abstract_canon_sha256":"be878778adba830f1059d84597d0daa65edd31cd03dd387df2ffa9de0490b10e"},"schema_version":"1.0"},"canonical_sha256":"f1ff78a13137b79dc7121e8d7849f565c4c5406f3dfccc922db7c07409563ec9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:35.118368Z","signature_b64":"Nioyl5Q3wRweBk0EKBviayhTdWhmu3+lbOkKO8uQJSAO36Nz/9TqFJs5paL7jSdRzMkfKI3Nxv4YqshBaO8CDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f1ff78a13137b79dc7121e8d7849f565c4c5406f3dfccc922db7c07409563ec9","last_reissued_at":"2026-05-18T00:14:35.117679Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:35.117679Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0903.4289","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:14:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7K6ntcS2jWSqynSQVt4gUzk+iEl/JSfIhHJd0ZpuPqI/LB0kNgDvnB7HUxAUjSF4sO415SkuS7bffQgAHyv2Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T06:17:20.665371Z"},"content_sha256":"de414bcbeefc4df3c25fe8612bc32f4b2b5d8606c3d055baff521774b475ac84","schema_version":"1.0","event_id":"sha256:de414bcbeefc4df3c25fe8612bc32f4b2b5d8606c3d055baff521774b475ac84"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:6H7XRIJRG63Z3RYSD2GXQSPVMX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Combinatorics and topology of straightening maps II: Discontinuity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Hiroyuki Inou","submitted_at":"2009-03-25T10:34:07Z","abstract_excerpt":"We continue the study of straightening maps for the family of polynomials of degree $d \\ge 3$. The notion of straightening map is originally introduced by Douady and Hubbard to study relationship between polynomial-like renormalizations and self-similarity of the Mandelbrot set. In the quadratic case, straightening maps are always continuous, and this is one of the critical steps to prove the Mandelbrot set has small copies in itself.\n  On the other hand, for higher degree case, we do not have such a nice self-similar property: As expected from an example of a cubic-like family with discontinu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4289","kind":"arxiv","version":4},"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":"2026-05-18T00:14:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t/XsAaoUO/4QaMTCF5B9eZUMDKXAogLpBVlAQmCY8u2rkfAIQ0PN7vipksla2rowlrFlch9D1HCsYPRXDksbCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T06:17:20.665930Z"},"content_sha256":"c66e1181bdca8082aed9bd7a6c08e24ca5aa1acaadb2ee7b307a7923c3a0e9d4","schema_version":"1.0","event_id":"sha256:c66e1181bdca8082aed9bd7a6c08e24ca5aa1acaadb2ee7b307a7923c3a0e9d4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6H7XRIJRG63Z3RYSD2GXQSPVMX/bundle.json","state_url":"https://pith.science/pith/6H7XRIJRG63Z3RYSD2GXQSPVMX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6H7XRIJRG63Z3RYSD2GXQSPVMX/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":"2026-07-01T06:17:20Z","links":{"resolver":"https://pith.science/pith/6H7XRIJRG63Z3RYSD2GXQSPVMX","bundle":"https://pith.science/pith/6H7XRIJRG63Z3RYSD2GXQSPVMX/bundle.json","state":"https://pith.science/pith/6H7XRIJRG63Z3RYSD2GXQSPVMX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6H7XRIJRG63Z3RYSD2GXQSPVMX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:6H7XRIJRG63Z3RYSD2GXQSPVMX","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":"be878778adba830f1059d84597d0daa65edd31cd03dd387df2ffa9de0490b10e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-03-25T10:34:07Z","title_canon_sha256":"86c24f8b8b228571bff93aeccf35461543a0ffa8ddcb3fefe3fecfd58dbc6f8f"},"schema_version":"1.0","source":{"id":"0903.4289","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4289","created_at":"2026-05-18T00:14:35Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4289v4","created_at":"2026-05-18T00:14:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4289","created_at":"2026-05-18T00:14:35Z"},{"alias_kind":"pith_short_12","alias_value":"6H7XRIJRG63Z","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6H7XRIJRG63Z3RYS","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6H7XRIJR","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:c66e1181bdca8082aed9bd7a6c08e24ca5aa1acaadb2ee7b307a7923c3a0e9d4","target":"graph","created_at":"2026-05-18T00:14:35Z","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 continue the study of straightening maps for the family of polynomials of degree $d \\ge 3$. The notion of straightening map is originally introduced by Douady and Hubbard to study relationship between polynomial-like renormalizations and self-similarity of the Mandelbrot set. In the quadratic case, straightening maps are always continuous, and this is one of the critical steps to prove the Mandelbrot set has small copies in itself.\n  On the other hand, for higher degree case, we do not have such a nice self-similar property: As expected from an example of a cubic-like family with discontinu","authors_text":"Hiroyuki Inou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-03-25T10:34:07Z","title":"Combinatorics and topology of straightening maps II: Discontinuity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4289","kind":"arxiv","version":4},"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:de414bcbeefc4df3c25fe8612bc32f4b2b5d8606c3d055baff521774b475ac84","target":"record","created_at":"2026-05-18T00:14:35Z","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":"be878778adba830f1059d84597d0daa65edd31cd03dd387df2ffa9de0490b10e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-03-25T10:34:07Z","title_canon_sha256":"86c24f8b8b228571bff93aeccf35461543a0ffa8ddcb3fefe3fecfd58dbc6f8f"},"schema_version":"1.0","source":{"id":"0903.4289","kind":"arxiv","version":4}},"canonical_sha256":"f1ff78a13137b79dc7121e8d7849f565c4c5406f3dfccc922db7c07409563ec9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f1ff78a13137b79dc7121e8d7849f565c4c5406f3dfccc922db7c07409563ec9","first_computed_at":"2026-05-18T00:14:35.117679Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:35.117679Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Nioyl5Q3wRweBk0EKBviayhTdWhmu3+lbOkKO8uQJSAO36Nz/9TqFJs5paL7jSdRzMkfKI3Nxv4YqshBaO8CDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:35.118368Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.4289","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de414bcbeefc4df3c25fe8612bc32f4b2b5d8606c3d055baff521774b475ac84","sha256:c66e1181bdca8082aed9bd7a6c08e24ca5aa1acaadb2ee7b307a7923c3a0e9d4"],"state_sha256":"261217a49bca3654c2cc0dff1d8a51b886606cfc39436eed0712c69e8b96ed2a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2cgnItGJ2uitTECtOzyluWM+dcwQS3jQ4pPZxeU7Z9I1TG+v6ydLNc3VScQWDkfLJPLqrEKK7TnAxaAX1pQtDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T06:17:20.669199Z","bundle_sha256":"2c21975fe1dbcc1dab09bc65a099b7cac8ee9a11544e5f55e86ebec8311edd83"}}