{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:4CZZMLQTXHXWJQDVHXRQM5JKEY","short_pith_number":"pith:4CZZMLQT","canonical_record":{"source":{"id":"1406.0766","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-06-03T16:09:09Z","cross_cats_sorted":[],"title_canon_sha256":"0e5605f34e878329da618d16883167475e7bfeb933482665e1422cfc436f9bc0","abstract_canon_sha256":"4a9f35b36aa333ea17f5ef6fe1e5a9c7f413ef64a5eb8a389a2c66dd94f6c6f5"},"schema_version":"1.0"},"canonical_sha256":"e0b3962e13b9ef64c0753de306752a262ad75da2e837f0e55d92d5e77ce70c75","source":{"kind":"arxiv","id":"1406.0766","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.0766","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1406.0766v4","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.0766","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"4CZZMLQTXHXW","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"4CZZMLQTXHXWJQDV","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"4CZZMLQT","created_at":"2026-05-18T12:28:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:4CZZMLQTXHXWJQDVHXRQM5JKEY","target":"record","payload":{"canonical_record":{"source":{"id":"1406.0766","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-06-03T16:09:09Z","cross_cats_sorted":[],"title_canon_sha256":"0e5605f34e878329da618d16883167475e7bfeb933482665e1422cfc436f9bc0","abstract_canon_sha256":"4a9f35b36aa333ea17f5ef6fe1e5a9c7f413ef64a5eb8a389a2c66dd94f6c6f5"},"schema_version":"1.0"},"canonical_sha256":"e0b3962e13b9ef64c0753de306752a262ad75da2e837f0e55d92d5e77ce70c75","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:32.286984Z","signature_b64":"/Cy2qWIRNuJ+BovOopAOXV/UsPRLIcPYOjWFrkCtIz4Jn+exKXT0t63M+0HS+59JIJ3FyglXG50MLs+pDtEtAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0b3962e13b9ef64c0753de306752a262ad75da2e837f0e55d92d5e77ce70c75","last_reissued_at":"2026-05-18T00:46:32.286291Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:32.286291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1406.0766","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":"2026-05-18T00:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c+RDiuHY88RzApOJSU1DzenKWsGwXuBEsiluIBGYP6aFxWpzUsPt4eGZGqCsJMIeJTAUvLEz2Zy11HstfjA5Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T09:54:00.179242Z"},"content_sha256":"3e5776379366009a61e3cebd3499023eda2a28077e40a05abc889d1b8d4b0320","schema_version":"1.0","event_id":"sha256:3e5776379366009a61e3cebd3499023eda2a28077e40a05abc889d1b8d4b0320"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:4CZZMLQTXHXWJQDVHXRQM5JKEY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"P\\'eter Csikv\\'ari","submitted_at":"2014-06-03T16:09:09Z","abstract_excerpt":"Friedland's Lower Matching Conjecture asserts that if $G$ is a $d$--regular bipartite graph on $v(G)=2n$ vertices, and $m_k(G)$ denotes the number of matchings of size $k$, then $$m_k(G)\\geq {n \\choose k}^2\\left(\\frac{d-p}{d}\\right)^{n(d-p)}(dp)^{np},$$ where $p=\\frac{k}{n}$. When $p=1$, this conjecture reduces to a theorem of Schrijver which says that a $d$--regular bipartite graph on $v(G)=2n$ vertices has at least $$\\left(\\frac{(d-1)^{d-1}}{d^{d-2}}\\right)^n$$ perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that $$\\frac{\\ln m_k(G"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0766","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":"2026-05-18T00:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WtSDEd6ghkiGgmJVqi2u9WVTVPfBfz66u2JVeqHlGrRQHv1QBoqWfbNL7ITZC7N1yf+hG6V1QiF32lr3zHbpBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T09:54:00.179603Z"},"content_sha256":"0e6b827b08ba330af590097713b2032675b63d74e192ed2870c501ce8f8ba1e1","schema_version":"1.0","event_id":"sha256:0e6b827b08ba330af590097713b2032675b63d74e192ed2870c501ce8f8ba1e1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4CZZMLQTXHXWJQDVHXRQM5JKEY/bundle.json","state_url":"https://pith.science/pith/4CZZMLQTXHXWJQDVHXRQM5JKEY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4CZZMLQTXHXWJQDVHXRQM5JKEY/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":"2026-06-02T09:54:00Z","links":{"resolver":"https://pith.science/pith/4CZZMLQTXHXWJQDVHXRQM5JKEY","bundle":"https://pith.science/pith/4CZZMLQTXHXWJQDVHXRQM5JKEY/bundle.json","state":"https://pith.science/pith/4CZZMLQTXHXWJQDVHXRQM5JKEY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4CZZMLQTXHXWJQDVHXRQM5JKEY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:4CZZMLQTXHXWJQDVHXRQM5JKEY","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":"4a9f35b36aa333ea17f5ef6fe1e5a9c7f413ef64a5eb8a389a2c66dd94f6c6f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-06-03T16:09:09Z","title_canon_sha256":"0e5605f34e878329da618d16883167475e7bfeb933482665e1422cfc436f9bc0"},"schema_version":"1.0","source":{"id":"1406.0766","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.0766","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1406.0766v4","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.0766","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"4CZZMLQTXHXW","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"4CZZMLQTXHXWJQDV","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"4CZZMLQT","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:0e6b827b08ba330af590097713b2032675b63d74e192ed2870c501ce8f8ba1e1","target":"graph","created_at":"2026-05-18T00:46:32Z","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":"Friedland's Lower Matching Conjecture asserts that if $G$ is a $d$--regular bipartite graph on $v(G)=2n$ vertices, and $m_k(G)$ denotes the number of matchings of size $k$, then $$m_k(G)\\geq {n \\choose k}^2\\left(\\frac{d-p}{d}\\right)^{n(d-p)}(dp)^{np},$$ where $p=\\frac{k}{n}$. When $p=1$, this conjecture reduces to a theorem of Schrijver which says that a $d$--regular bipartite graph on $v(G)=2n$ vertices has at least $$\\left(\\frac{(d-1)^{d-1}}{d^{d-2}}\\right)^n$$ perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that $$\\frac{\\ln m_k(G","authors_text":"P\\'eter Csikv\\'ari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-06-03T16:09:09Z","title":"Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0766","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":"sha256:3e5776379366009a61e3cebd3499023eda2a28077e40a05abc889d1b8d4b0320","target":"record","created_at":"2026-05-18T00:46:32Z","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":"4a9f35b36aa333ea17f5ef6fe1e5a9c7f413ef64a5eb8a389a2c66dd94f6c6f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-06-03T16:09:09Z","title_canon_sha256":"0e5605f34e878329da618d16883167475e7bfeb933482665e1422cfc436f9bc0"},"schema_version":"1.0","source":{"id":"1406.0766","kind":"arxiv","version":4}},"canonical_sha256":"e0b3962e13b9ef64c0753de306752a262ad75da2e837f0e55d92d5e77ce70c75","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e0b3962e13b9ef64c0753de306752a262ad75da2e837f0e55d92d5e77ce70c75","first_computed_at":"2026-05-18T00:46:32.286291Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:32.286291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/Cy2qWIRNuJ+BovOopAOXV/UsPRLIcPYOjWFrkCtIz4Jn+exKXT0t63M+0HS+59JIJ3FyglXG50MLs+pDtEtAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:32.286984Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.0766","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e5776379366009a61e3cebd3499023eda2a28077e40a05abc889d1b8d4b0320","sha256:0e6b827b08ba330af590097713b2032675b63d74e192ed2870c501ce8f8ba1e1"],"state_sha256":"37a16a6b672c4d2509192390b48e5a09a026d5d97015c81512de5ec826e1a335"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VrDc2zQRtfklxAxY3q1IVijOS3TT13gc9I+njHcAn50IlRkYW7EVOST1LcnHRKPfTgva/OcW4ml3HtzswnTLCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T09:54:00.181518Z","bundle_sha256":"099e2b3a046879ff4e97f77d2c3a51d8b66f510d12760ec2246f551db389f3a7"}}