{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:TNG56QGRFFXGPFVVFVLN2GGXVJ","short_pith_number":"pith:TNG56QGR","canonical_record":{"source":{"id":"1510.02429","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-08T18:08:07Z","cross_cats_sorted":[],"title_canon_sha256":"facd997b1839210cdf4a434ba9833f16c1af314865c01eb3f46a655771d4eb7f","abstract_canon_sha256":"091760fd2ee5d847e4d416288a4de6b9bfdeeeda1be6fcd22fde55b7fc170db5"},"schema_version":"1.0"},"canonical_sha256":"9b4ddf40d1296e6796b52d56dd18d7aa5bb26441fe59e5cc07155a8384773021","source":{"kind":"arxiv","id":"1510.02429","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.02429","created_at":"2026-05-18T01:30:43Z"},{"alias_kind":"arxiv_version","alias_value":"1510.02429v1","created_at":"2026-05-18T01:30:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.02429","created_at":"2026-05-18T01:30:43Z"},{"alias_kind":"pith_short_12","alias_value":"TNG56QGRFFXG","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TNG56QGRFFXGPFVV","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TNG56QGR","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:TNG56QGRFFXGPFVVFVLN2GGXVJ","target":"record","payload":{"canonical_record":{"source":{"id":"1510.02429","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-08T18:08:07Z","cross_cats_sorted":[],"title_canon_sha256":"facd997b1839210cdf4a434ba9833f16c1af314865c01eb3f46a655771d4eb7f","abstract_canon_sha256":"091760fd2ee5d847e4d416288a4de6b9bfdeeeda1be6fcd22fde55b7fc170db5"},"schema_version":"1.0"},"canonical_sha256":"9b4ddf40d1296e6796b52d56dd18d7aa5bb26441fe59e5cc07155a8384773021","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:43.769472Z","signature_b64":"pVUqwP4pSVZbG+y9s2pSlN2In9HbPuLwV/3YBNLY6d5LFYRK9WQnmcIwWC+0hdNdrGKxeMIPPgbrsJWWOWMgDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b4ddf40d1296e6796b52d56dd18d7aa5bb26441fe59e5cc07155a8384773021","last_reissued_at":"2026-05-18T01:30:43.768821Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:43.768821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.02429","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":"2026-05-18T01:30:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oDsVLP5ULgsyochSPGRHA+TIzvzdnBV100Wm9wphRGjl36HWv9QIH1QosL6klZz7DpDJnrr0JZJFlVx/S8S9Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T05:46:58.810116Z"},"content_sha256":"c8f2f22167552537456161a88530bffe56d104e332a2b74f72e9377e2ed75940","schema_version":"1.0","event_id":"sha256:c8f2f22167552537456161a88530bffe56d104e332a2b74f72e9377e2ed75940"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:TNG56QGRFFXGPFVVFVLN2GGXVJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the arithmetic of density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Menachem Kojman","submitted_at":"2015-10-08T18:08:07Z","abstract_excerpt":"The $\\kappa$-density of a cardinal $\\mu\\ge\\kappa$ is the least cardinality of a dense collection of $\\kappa$-subsets of $\\mu$ and is denoted by $\\mathcal D(\\mu,\\kappa)$. The Singular Density Hypothesis (SDH) for a singular cardinal $\\mu$ of cofinality $cf\\mu=\\kappa$ is the equation $\\mathcal D(\\mu,\\kappa)=\\mu^+$. The Generalized Density Hypothesis (GDH) for $\\mu$ and $\\lambda$ such that $\\lambda\\le\\mu$ is: $\\mathcal D(\\mu,\\lambda)=\\mu$ if $cf\\mu\\not=cf\\lambda$ and $\\mathcal D(\\mu,\\lambda)=\\mu^+$ if $cf\\mu=cf\\lambda$.\n  Density is shown to satisfy Silver's theorem. The most important case is:\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02429","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":"2026-05-18T01:30:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UR5J8DrjFUzMEl1qvShFNJ9nx3XZHpxkGnstIS16BbFxymZXLF5eBzwgfzqfgqC/zjIF+PzhOC7lFgS2Bl5nDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T05:46:58.810483Z"},"content_sha256":"091374b85e15a69e9f8f70245551b7aeddfce229fca7c3b1541053c7a0d19427","schema_version":"1.0","event_id":"sha256:091374b85e15a69e9f8f70245551b7aeddfce229fca7c3b1541053c7a0d19427"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/bundle.json","state_url":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/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":"2026-06-22T05:46:58Z","links":{"resolver":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ","bundle":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/bundle.json","state":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:TNG56QGRFFXGPFVVFVLN2GGXVJ","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":"091760fd2ee5d847e4d416288a4de6b9bfdeeeda1be6fcd22fde55b7fc170db5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-08T18:08:07Z","title_canon_sha256":"facd997b1839210cdf4a434ba9833f16c1af314865c01eb3f46a655771d4eb7f"},"schema_version":"1.0","source":{"id":"1510.02429","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.02429","created_at":"2026-05-18T01:30:43Z"},{"alias_kind":"arxiv_version","alias_value":"1510.02429v1","created_at":"2026-05-18T01:30:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.02429","created_at":"2026-05-18T01:30:43Z"},{"alias_kind":"pith_short_12","alias_value":"TNG56QGRFFXG","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TNG56QGRFFXGPFVV","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TNG56QGR","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:091374b85e15a69e9f8f70245551b7aeddfce229fca7c3b1541053c7a0d19427","target":"graph","created_at":"2026-05-18T01:30:43Z","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":"The $\\kappa$-density of a cardinal $\\mu\\ge\\kappa$ is the least cardinality of a dense collection of $\\kappa$-subsets of $\\mu$ and is denoted by $\\mathcal D(\\mu,\\kappa)$. The Singular Density Hypothesis (SDH) for a singular cardinal $\\mu$ of cofinality $cf\\mu=\\kappa$ is the equation $\\mathcal D(\\mu,\\kappa)=\\mu^+$. The Generalized Density Hypothesis (GDH) for $\\mu$ and $\\lambda$ such that $\\lambda\\le\\mu$ is: $\\mathcal D(\\mu,\\lambda)=\\mu$ if $cf\\mu\\not=cf\\lambda$ and $\\mathcal D(\\mu,\\lambda)=\\mu^+$ if $cf\\mu=cf\\lambda$.\n  Density is shown to satisfy Silver's theorem. The most important case is:\n ","authors_text":"Menachem Kojman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-08T18:08:07Z","title":"On the arithmetic of density"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02429","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:c8f2f22167552537456161a88530bffe56d104e332a2b74f72e9377e2ed75940","target":"record","created_at":"2026-05-18T01:30:43Z","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":"091760fd2ee5d847e4d416288a4de6b9bfdeeeda1be6fcd22fde55b7fc170db5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-08T18:08:07Z","title_canon_sha256":"facd997b1839210cdf4a434ba9833f16c1af314865c01eb3f46a655771d4eb7f"},"schema_version":"1.0","source":{"id":"1510.02429","kind":"arxiv","version":1}},"canonical_sha256":"9b4ddf40d1296e6796b52d56dd18d7aa5bb26441fe59e5cc07155a8384773021","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b4ddf40d1296e6796b52d56dd18d7aa5bb26441fe59e5cc07155a8384773021","first_computed_at":"2026-05-18T01:30:43.768821Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:43.768821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pVUqwP4pSVZbG+y9s2pSlN2In9HbPuLwV/3YBNLY6d5LFYRK9WQnmcIwWC+0hdNdrGKxeMIPPgbrsJWWOWMgDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:43.769472Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.02429","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c8f2f22167552537456161a88530bffe56d104e332a2b74f72e9377e2ed75940","sha256:091374b85e15a69e9f8f70245551b7aeddfce229fca7c3b1541053c7a0d19427"],"state_sha256":"1c9230d3d2b27d143691b3034651ebdf0b6bd97bab71f7acb1601046784342bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qnKnihggJA7OwY1HPnJc04aUNmNz9Ah2+9t6kiX7uqOnPcNqBR12ESXAf6ZBO+tp3JrsLn8MgbZ6GyeB/lj4DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T05:46:58.812433Z","bundle_sha256":"59a23ef57de83f5149973e59cf5729d3bdf70258ac52d5655e386387b8123036"}}