{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:4RRZQBWW5YQDTA22PWGXUTGOKY","short_pith_number":"pith:4RRZQBWW","canonical_record":{"source":{"id":"2408.01559","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2024-08-02T20:16:43Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"8233e1bc7618c4ae79cd2db8b71a0575e7a74397c7a607cde6802d2705d5997e","abstract_canon_sha256":"a28eae42a3bd99e331e374e55db51fcc4b40de580c70ce15abd4c8f16dcb14c0"},"schema_version":"1.0"},"canonical_sha256":"e4639806d6ee2039835a7d8d7a4cce5606e618ac1a78670816037c2d33dae1c2","source":{"kind":"arxiv","id":"2408.01559","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.01559","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"arxiv_version","alias_value":"2408.01559v1","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.01559","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"pith_short_12","alias_value":"4RRZQBWW5YQD","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"pith_short_16","alias_value":"4RRZQBWW5YQDTA22","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"pith_short_8","alias_value":"4RRZQBWW","created_at":"2026-07-05T08:51:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:4RRZQBWW5YQDTA22PWGXUTGOKY","target":"record","payload":{"canonical_record":{"source":{"id":"2408.01559","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2024-08-02T20:16:43Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"8233e1bc7618c4ae79cd2db8b71a0575e7a74397c7a607cde6802d2705d5997e","abstract_canon_sha256":"a28eae42a3bd99e331e374e55db51fcc4b40de580c70ce15abd4c8f16dcb14c0"},"schema_version":"1.0"},"canonical_sha256":"e4639806d6ee2039835a7d8d7a4cce5606e618ac1a78670816037c2d33dae1c2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:51:59.632309Z","signature_b64":"0tbkK5Q3R2F8Wxsx+aBGos0f/CSlBF207PVYWXqdMJdlh7nmSZmqCci63KxZT55P3QxEs3Z7XR7G19swlIoRCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e4639806d6ee2039835a7d8d7a4cce5606e618ac1a78670816037c2d33dae1c2","last_reissued_at":"2026-07-05T08:51:59.631962Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:51:59.631962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2408.01559","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":"2026-07-05T08:51:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6QzGejQfzDb8IfNdnZPmQ8WdyvdckgazL9V1qkOHZtCEoi/XbzA77WacmmWA5S9Y6jMWUwOTJK82Wzen4jD1Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T20:46:07.993876Z"},"content_sha256":"4dfe67f3acb2be53a503a78123bff91113cf881d959350c32b60ca36f89da8c4","schema_version":"1.0","event_id":"sha256:4dfe67f3acb2be53a503a78123bff91113cf881d959350c32b60ca36f89da8c4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:4RRZQBWW5YQDTA22PWGXUTGOKY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Joseph H. Silverman","submitted_at":"2024-08-02T20:16:43Z","abstract_excerpt":"In this note we give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system f:X-->X, including the dynamical degree d(f), which gives a coarse measure of the geometric complexity of the iterates of f, the arithmetic degree a(f,P), which gives a coarse measure of the arithmetic complexity of the orbit of a an algebraic point P in X, and various versions of the canonical height h_f(P) that provide more refined measures of arithmetic complexity. Emphasis is placed on open problems and directions for further exploration."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.01559","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.01559/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:51:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gb91FN8iooS5N8uMvj62PGSVoQZ9txA3R7ueUmkl3TosM95UzlVI/jQDTNz5PoscvbyR4Tpvt1gRDJCEZUUlAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T20:46:07.994466Z"},"content_sha256":"87a9b331c5eb45982c402ca5bc31dd9a6bc6e0e053819c4d877b540002dadc0b","schema_version":"1.0","event_id":"sha256:87a9b331c5eb45982c402ca5bc31dd9a6bc6e0e053819c4d877b540002dadc0b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4RRZQBWW5YQDTA22PWGXUTGOKY/bundle.json","state_url":"https://pith.science/pith/4RRZQBWW5YQDTA22PWGXUTGOKY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4RRZQBWW5YQDTA22PWGXUTGOKY/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":"2026-08-19T20:46:07Z","links":{"resolver":"https://pith.science/pith/4RRZQBWW5YQDTA22PWGXUTGOKY","bundle":"https://pith.science/pith/4RRZQBWW5YQDTA22PWGXUTGOKY/bundle.json","state":"https://pith.science/pith/4RRZQBWW5YQDTA22PWGXUTGOKY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4RRZQBWW5YQDTA22PWGXUTGOKY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:4RRZQBWW5YQDTA22PWGXUTGOKY","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":"a28eae42a3bd99e331e374e55db51fcc4b40de580c70ce15abd4c8f16dcb14c0","cross_cats_sorted":["math.DS"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2024-08-02T20:16:43Z","title_canon_sha256":"8233e1bc7618c4ae79cd2db8b71a0575e7a74397c7a607cde6802d2705d5997e"},"schema_version":"1.0","source":{"id":"2408.01559","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.01559","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"arxiv_version","alias_value":"2408.01559v1","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.01559","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"pith_short_12","alias_value":"4RRZQBWW5YQD","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"pith_short_16","alias_value":"4RRZQBWW5YQDTA22","created_at":"2026-07-05T08:51:59Z"},{"alias_kind":"pith_short_8","alias_value":"4RRZQBWW","created_at":"2026-07-05T08:51:59Z"}],"graph_snapshots":[{"event_id":"sha256:87a9b331c5eb45982c402ca5bc31dd9a6bc6e0e053819c4d877b540002dadc0b","target":"graph","created_at":"2026-07-05T08:51:59Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2408.01559/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this note we give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system f:X-->X, including the dynamical degree d(f), which gives a coarse measure of the geometric complexity of the iterates of f, the arithmetic degree a(f,P), which gives a coarse measure of the arithmetic complexity of the orbit of a an algebraic point P in X, and various versions of the canonical height h_f(P) that provide more refined measures of arithmetic complexity. Emphasis is placed on open problems and directions for further exploration.","authors_text":"Joseph H. Silverman","cross_cats":["math.DS"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2024-08-02T20:16:43Z","title":"Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.01559","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":"sha256:4dfe67f3acb2be53a503a78123bff91113cf881d959350c32b60ca36f89da8c4","target":"record","created_at":"2026-07-05T08:51:59Z","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":"a28eae42a3bd99e331e374e55db51fcc4b40de580c70ce15abd4c8f16dcb14c0","cross_cats_sorted":["math.DS"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2024-08-02T20:16:43Z","title_canon_sha256":"8233e1bc7618c4ae79cd2db8b71a0575e7a74397c7a607cde6802d2705d5997e"},"schema_version":"1.0","source":{"id":"2408.01559","kind":"arxiv","version":1}},"canonical_sha256":"e4639806d6ee2039835a7d8d7a4cce5606e618ac1a78670816037c2d33dae1c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e4639806d6ee2039835a7d8d7a4cce5606e618ac1a78670816037c2d33dae1c2","first_computed_at":"2026-07-05T08:51:59.631962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:51:59.631962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0tbkK5Q3R2F8Wxsx+aBGos0f/CSlBF207PVYWXqdMJdlh7nmSZmqCci63KxZT55P3QxEs3Z7XR7G19swlIoRCg==","signature_status":"signed_v1","signed_at":"2026-07-05T08:51:59.632309Z","signed_message":"canonical_sha256_bytes"},"source_id":"2408.01559","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4dfe67f3acb2be53a503a78123bff91113cf881d959350c32b60ca36f89da8c4","sha256:87a9b331c5eb45982c402ca5bc31dd9a6bc6e0e053819c4d877b540002dadc0b"],"state_sha256":"89f80cfadf9a5afab7bd2950cb0bfae1c1e372205cdd563366f532094060d4f8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lDeUtb3VZzdHNpXc35Co/zVcigRMgihvTZ9pFLpLLQHj5HigUafV3DSqH9zKSbOEYyd7eNku/tZf/GgnKeKTBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T20:46:07.999496Z","bundle_sha256":"98fdb26064c6bb0d2b62971b8ca4d80a797cd799f40eafc0546080dc1af35f74"}}