{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:EK5Z2NWP6OAJI2TN2TN3HB6DVC","short_pith_number":"pith:EK5Z2NWP","canonical_record":{"source":{"id":"1109.5438","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-09-26T03:50:30Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4fbb8856e5fc31bc1ec9fc655dde6ab59d7b9e16930f2960955916f16121112e","abstract_canon_sha256":"7dd3d9594f6ea9c3c644b31d0e1159a310bef8b8af2e4d41d1d871957dcd37b4"},"schema_version":"1.0"},"canonical_sha256":"22bb9d36cff380946a6dd4dbb387c3a8813743a3cd9acd684a7e1144481b53c8","source":{"kind":"arxiv","id":"1109.5438","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5438","created_at":"2026-05-18T04:12:19Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5438v1","created_at":"2026-05-18T04:12:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5438","created_at":"2026-05-18T04:12:19Z"},{"alias_kind":"pith_short_12","alias_value":"EK5Z2NWP6OAJ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"EK5Z2NWP6OAJI2TN","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"EK5Z2NWP","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:EK5Z2NWP6OAJI2TN2TN3HB6DVC","target":"record","payload":{"canonical_record":{"source":{"id":"1109.5438","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-09-26T03:50:30Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4fbb8856e5fc31bc1ec9fc655dde6ab59d7b9e16930f2960955916f16121112e","abstract_canon_sha256":"7dd3d9594f6ea9c3c644b31d0e1159a310bef8b8af2e4d41d1d871957dcd37b4"},"schema_version":"1.0"},"canonical_sha256":"22bb9d36cff380946a6dd4dbb387c3a8813743a3cd9acd684a7e1144481b53c8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:19.803494Z","signature_b64":"neVBGJXi3lW1+i+V5zXyS1Yuqg9pB3fjpAIgw2hy/9DR1vjsZBpECYmNC8EF3QJ0ZAxCgvvV3oytEpzhd/TyAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22bb9d36cff380946a6dd4dbb387c3a8813743a3cd9acd684a7e1144481b53c8","last_reissued_at":"2026-05-18T04:12:19.803015Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:19.803015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.5438","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":"2026-05-18T04:12:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EsftbJ71M1yfSuJ5AUW/cAIY0JcDvzmp3vgkXRgwaHAidkoeH6cabdNR2eNrKuYveAtiqVVWc3kZyOCJFTv5Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:15:02.566061Z"},"content_sha256":"dd84979bbb2c5a6a13bb6db2273b2cbbf1f2b6cfbe750987dbc96f040bdb2599","schema_version":"1.0","event_id":"sha256:dd84979bbb2c5a6a13bb6db2273b2cbbf1f2b6cfbe750987dbc96f040bdb2599"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:EK5Z2NWP6OAJI2TN2TN3HB6DVC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Vapnik-Chervonenkis density in some theories without the independence property, I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.LO","authors_text":"A. Dolich, D. Haskell, D. Macpherson, M. Aschenbrenner, S. Starchenko","submitted_at":"2011-09-26T03:50:30Z","abstract_excerpt":"We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and $P$-minimal theories."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5438","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":"2026-05-18T04:12:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mUGg/d42XDXdihHf3IT7U4CVSYJmwL+8S8V0ioSbp0rOzHicOOWqEXNevpV0Btgf7HXI3oO/EPYcm+O8NkPHBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:15:02.566737Z"},"content_sha256":"d046b905fedb8c694e36ca6d3f30812d4571b3ea75d61a59adbdea584fad8ef5","schema_version":"1.0","event_id":"sha256:d046b905fedb8c694e36ca6d3f30812d4571b3ea75d61a59adbdea584fad8ef5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EK5Z2NWP6OAJI2TN2TN3HB6DVC/bundle.json","state_url":"https://pith.science/pith/EK5Z2NWP6OAJI2TN2TN3HB6DVC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EK5Z2NWP6OAJI2TN2TN3HB6DVC/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":"2026-05-27T23:15:02Z","links":{"resolver":"https://pith.science/pith/EK5Z2NWP6OAJI2TN2TN3HB6DVC","bundle":"https://pith.science/pith/EK5Z2NWP6OAJI2TN2TN3HB6DVC/bundle.json","state":"https://pith.science/pith/EK5Z2NWP6OAJI2TN2TN3HB6DVC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EK5Z2NWP6OAJI2TN2TN3HB6DVC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:EK5Z2NWP6OAJI2TN2TN3HB6DVC","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":"7dd3d9594f6ea9c3c644b31d0e1159a310bef8b8af2e4d41d1d871957dcd37b4","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-09-26T03:50:30Z","title_canon_sha256":"4fbb8856e5fc31bc1ec9fc655dde6ab59d7b9e16930f2960955916f16121112e"},"schema_version":"1.0","source":{"id":"1109.5438","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5438","created_at":"2026-05-18T04:12:19Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5438v1","created_at":"2026-05-18T04:12:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5438","created_at":"2026-05-18T04:12:19Z"},{"alias_kind":"pith_short_12","alias_value":"EK5Z2NWP6OAJ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"EK5Z2NWP6OAJI2TN","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"EK5Z2NWP","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:d046b905fedb8c694e36ca6d3f30812d4571b3ea75d61a59adbdea584fad8ef5","target":"graph","created_at":"2026-05-18T04:12:19Z","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":"We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and $P$-minimal theories.","authors_text":"A. Dolich, D. Haskell, D. Macpherson, M. Aschenbrenner, S. Starchenko","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-09-26T03:50:30Z","title":"Vapnik-Chervonenkis density in some theories without the independence property, I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5438","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:dd84979bbb2c5a6a13bb6db2273b2cbbf1f2b6cfbe750987dbc96f040bdb2599","target":"record","created_at":"2026-05-18T04:12:19Z","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":"7dd3d9594f6ea9c3c644b31d0e1159a310bef8b8af2e4d41d1d871957dcd37b4","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-09-26T03:50:30Z","title_canon_sha256":"4fbb8856e5fc31bc1ec9fc655dde6ab59d7b9e16930f2960955916f16121112e"},"schema_version":"1.0","source":{"id":"1109.5438","kind":"arxiv","version":1}},"canonical_sha256":"22bb9d36cff380946a6dd4dbb387c3a8813743a3cd9acd684a7e1144481b53c8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"22bb9d36cff380946a6dd4dbb387c3a8813743a3cd9acd684a7e1144481b53c8","first_computed_at":"2026-05-18T04:12:19.803015Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:19.803015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"neVBGJXi3lW1+i+V5zXyS1Yuqg9pB3fjpAIgw2hy/9DR1vjsZBpECYmNC8EF3QJ0ZAxCgvvV3oytEpzhd/TyAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:19.803494Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.5438","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd84979bbb2c5a6a13bb6db2273b2cbbf1f2b6cfbe750987dbc96f040bdb2599","sha256:d046b905fedb8c694e36ca6d3f30812d4571b3ea75d61a59adbdea584fad8ef5"],"state_sha256":"2d015be6a08d453e928d13f69d47f517d2ed75ef50aeeac092760885427e682a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"07aH0KY/zQsXVgtgaMLPYZQjHdPSfNlhIw7nZf8nGuTbGYPYLQUdrQsRX93AV/Zk0azfVfG7+EnXKZdEMF5GCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T23:15:02.569904Z","bundle_sha256":"535a2b406e4e3c9f34c584781113d1444828d97b81cf6be1aa7fca0bf07a1239"}}