{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:AO5CC7GQE36MPJSY2JUWMULVM6","short_pith_number":"pith:AO5CC7GQ","canonical_record":{"source":{"id":"1107.2636","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-13T19:27:18Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a9ae6ebbafc78f5a3d87ba786063e10b83388dd83113beb0419bfcb57011a508","abstract_canon_sha256":"b1f2836a8622085a5bc6afbd0c8eb6a11cc6ca27b3b369a594c0aa9c9c9d2b53"},"schema_version":"1.0"},"canonical_sha256":"03ba217cd026fcc7a658d26966517567879a2498b520d537553cf60eaf5c1a54","source":{"kind":"arxiv","id":"1107.2636","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2636","created_at":"2026-05-18T03:50:25Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2636v3","created_at":"2026-05-18T03:50:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2636","created_at":"2026-05-18T03:50:25Z"},{"alias_kind":"pith_short_12","alias_value":"AO5CC7GQE36M","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AO5CC7GQE36MPJSY","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AO5CC7GQ","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:AO5CC7GQE36MPJSY2JUWMULVM6","target":"record","payload":{"canonical_record":{"source":{"id":"1107.2636","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-13T19:27:18Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a9ae6ebbafc78f5a3d87ba786063e10b83388dd83113beb0419bfcb57011a508","abstract_canon_sha256":"b1f2836a8622085a5bc6afbd0c8eb6a11cc6ca27b3b369a594c0aa9c9c9d2b53"},"schema_version":"1.0"},"canonical_sha256":"03ba217cd026fcc7a658d26966517567879a2498b520d537553cf60eaf5c1a54","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:25.824864Z","signature_b64":"GthWpG3pftTchnrsi7v/wZYoZtvEchWAyBuoGirMxfHhsoFWt+BBlRLvYgm4jF3A7KT3sLUzIPbITxPcaxvqCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03ba217cd026fcc7a658d26966517567879a2498b520d537553cf60eaf5c1a54","last_reissued_at":"2026-05-18T03:50:25.824165Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:25.824165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.2636","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:50:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"88p9sqPsnkYKflURx9717Rz9n9UEU/oHoN9brFpb62dSAxlKFZJpwIDZFU0Sn1p+hGH4n3dOBM7Dw1iejudoAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T14:15:50.421127Z"},"content_sha256":"60aaf7222cff096459387d22266ff1a9e397149b2221c71929521fef884e1622","schema_version":"1.0","event_id":"sha256:60aaf7222cff096459387d22266ff1a9e397149b2221c71929521fef884e1622"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:AO5CC7GQE36MPJSY2JUWMULVM6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Phase Transition for Dyadic Tilings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Alexander E. Holroyd, Gady Kozma, Johan W\\\"astlund, Omer Angel, Peter Winkler","submitted_at":"2011-07-13T19:27:18Z","abstract_excerpt":"A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that for p sufficiently close to 1, there exists a set of pairwise disjoint available tiles whose union is the unit square, with probability tending to 1 as n->infinity, as conjectured by Joel Spencer in 1999. In particular we prove that if p=7/8, such a tiling exists with probability at least 1-(3/4)^n. The proof involves a surprisingly delicate counting argument"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2636","kind":"arxiv","version":3},"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":"2026-05-18T03:50:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0JZKVkKVCtotyOpFylt4PCVj5ORVO6adoAWPKWlWwbJnRYI3MSEMMUgLdPvD1VgvRNrfWbI9HUmrbhp+DisbAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T14:15:50.421508Z"},"content_sha256":"3cce35e284c922453c894126b0b5b5d8432ee1c9d5780811fa266b8a6b758a43","schema_version":"1.0","event_id":"sha256:3cce35e284c922453c894126b0b5b5d8432ee1c9d5780811fa266b8a6b758a43"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AO5CC7GQE36MPJSY2JUWMULVM6/bundle.json","state_url":"https://pith.science/pith/AO5CC7GQE36MPJSY2JUWMULVM6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AO5CC7GQE36MPJSY2JUWMULVM6/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":"2026-05-30T14:15:50Z","links":{"resolver":"https://pith.science/pith/AO5CC7GQE36MPJSY2JUWMULVM6","bundle":"https://pith.science/pith/AO5CC7GQE36MPJSY2JUWMULVM6/bundle.json","state":"https://pith.science/pith/AO5CC7GQE36MPJSY2JUWMULVM6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AO5CC7GQE36MPJSY2JUWMULVM6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:AO5CC7GQE36MPJSY2JUWMULVM6","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":"b1f2836a8622085a5bc6afbd0c8eb6a11cc6ca27b3b369a594c0aa9c9c9d2b53","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-13T19:27:18Z","title_canon_sha256":"a9ae6ebbafc78f5a3d87ba786063e10b83388dd83113beb0419bfcb57011a508"},"schema_version":"1.0","source":{"id":"1107.2636","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2636","created_at":"2026-05-18T03:50:25Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2636v3","created_at":"2026-05-18T03:50:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2636","created_at":"2026-05-18T03:50:25Z"},{"alias_kind":"pith_short_12","alias_value":"AO5CC7GQE36M","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AO5CC7GQE36MPJSY","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AO5CC7GQ","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:3cce35e284c922453c894126b0b5b5d8432ee1c9d5780811fa266b8a6b758a43","target":"graph","created_at":"2026-05-18T03:50:25Z","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":"A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that for p sufficiently close to 1, there exists a set of pairwise disjoint available tiles whose union is the unit square, with probability tending to 1 as n->infinity, as conjectured by Joel Spencer in 1999. In particular we prove that if p=7/8, such a tiling exists with probability at least 1-(3/4)^n. The proof involves a surprisingly delicate counting argument","authors_text":"Alexander E. Holroyd, Gady Kozma, Johan W\\\"astlund, Omer Angel, Peter Winkler","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-13T19:27:18Z","title":"The Phase Transition for Dyadic Tilings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2636","kind":"arxiv","version":3},"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":"sha256:60aaf7222cff096459387d22266ff1a9e397149b2221c71929521fef884e1622","target":"record","created_at":"2026-05-18T03:50:25Z","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":"b1f2836a8622085a5bc6afbd0c8eb6a11cc6ca27b3b369a594c0aa9c9c9d2b53","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-13T19:27:18Z","title_canon_sha256":"a9ae6ebbafc78f5a3d87ba786063e10b83388dd83113beb0419bfcb57011a508"},"schema_version":"1.0","source":{"id":"1107.2636","kind":"arxiv","version":3}},"canonical_sha256":"03ba217cd026fcc7a658d26966517567879a2498b520d537553cf60eaf5c1a54","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"03ba217cd026fcc7a658d26966517567879a2498b520d537553cf60eaf5c1a54","first_computed_at":"2026-05-18T03:50:25.824165Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:25.824165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GthWpG3pftTchnrsi7v/wZYoZtvEchWAyBuoGirMxfHhsoFWt+BBlRLvYgm4jF3A7KT3sLUzIPbITxPcaxvqCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:25.824864Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.2636","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:60aaf7222cff096459387d22266ff1a9e397149b2221c71929521fef884e1622","sha256:3cce35e284c922453c894126b0b5b5d8432ee1c9d5780811fa266b8a6b758a43"],"state_sha256":"5d1421daf5d9db8d9f2e2923e9ae0817bba962e2d20dc95220ae246141320044"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6jtLhGr0a5jVg8c9sVhHnAJd143ANzf1iS7c1mEfWVOGU+g6ZSNJoEmUWAmpSdQ65zB231ISNDarpj/zknEIBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T14:15:50.423573Z","bundle_sha256":"7d9d45d91ec9bb9542eb820281d976047e3b55c586c9f58aa7679386ee464930"}}