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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1510.02429","created_at":"2026-05-18T01:30:43.768923+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.02429v1","created_at":"2026-05-18T01:30:43.768923+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.02429","created_at":"2026-05-18T01:30:43.768923+00:00"},{"alias_kind":"pith_short_12","alias_value":"TNG56QGRFFXG","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TNG56QGRFFXGPFVV","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TNG56QGR","created_at":"2026-05-18T12:29:42.218222+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ","json":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ.json","graph_json":"https://pith.science/api/pith-number/TNG56QGRFFXGPFVVFVLN2GGXVJ/graph.json","events_json":"https://pith.science/api/pith-number/TNG56QGRFFXGPFVVFVLN2GGXVJ/events.json","paper":"https://pith.science/paper/TNG56QGR"},"agent_actions":{"view_html":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ","download_json":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ.json","view_paper":"https://pith.science/paper/TNG56QGR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.02429&json=true","fetch_graph":"https://pith.science/api/pith-number/TNG56QGRFFXGPFVVFVLN2GGXVJ/graph.json","fetch_events":"https://pith.science/api/pith-number/TNG56QGRFFXGPFVVFVLN2GGXVJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/action/storage_attestation","attest_author":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/action/author_attestation","sign_citation":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/action/citation_signature","submit_replication":"https://pith.science/pith/TNG56QGRFFXGPFVVFVLN2GGXVJ/action/replication_record"}},"created_at":"2026-05-18T01:30:43.768923+00:00","updated_at":"2026-05-18T01:30:43.768923+00:00"}