{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:I45FSOZDXCVQB4USYZ6CJMTQ5R","short_pith_number":"pith:I45FSOZD","canonical_record":{"source":{"id":"1211.3248","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-14T09:31:09Z","cross_cats_sorted":[],"title_canon_sha256":"eef289cb85c5105d20b80090e1b4969c051e70aa1270738970593d11f470d582","abstract_canon_sha256":"338a84b2ebd8fa7cf64d07c4e408186fbabbf1f29f28a766bbe10a4dce3a8ea4"},"schema_version":"1.0"},"canonical_sha256":"473a593b23b8ab00f292c67c24b270ec7228545df1e8e5bf8babaf556d998ccb","source":{"kind":"arxiv","id":"1211.3248","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.3248","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"1211.3248v3","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.3248","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"I45FSOZDXCVQ","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"I45FSOZDXCVQB4US","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"I45FSOZD","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:I45FSOZDXCVQB4USYZ6CJMTQ5R","target":"record","payload":{"canonical_record":{"source":{"id":"1211.3248","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-14T09:31:09Z","cross_cats_sorted":[],"title_canon_sha256":"eef289cb85c5105d20b80090e1b4969c051e70aa1270738970593d11f470d582","abstract_canon_sha256":"338a84b2ebd8fa7cf64d07c4e408186fbabbf1f29f28a766bbe10a4dce3a8ea4"},"schema_version":"1.0"},"canonical_sha256":"473a593b23b8ab00f292c67c24b270ec7228545df1e8e5bf8babaf556d998ccb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:32.237398Z","signature_b64":"H7eJRlqirlx9Q8uQaUJy3OpjkLltmzE3GNavfKpaWGcBPGm12FNf/uCxYTRgQWcYMEuSK0121G/GR63MgsC6DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"473a593b23b8ab00f292c67c24b270ec7228545df1e8e5bf8babaf556d998ccb","last_reissued_at":"2026-05-18T03:26:32.236964Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:32.236964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.3248","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":"2026-05-18T03:26:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cEAVDM/rtDjL62SmNJfmmePDahv8g4rP7rlagwDCRg0VYfa4wUtx2mr+a4U86u7Uk0UznZ61D5QpP3T4GoVfCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T06:10:32.378613Z"},"content_sha256":"c3ba1e509fd0620d55c2a917c12874dce72185d4023f0f2863a2321f880afa98","schema_version":"1.0","event_id":"sha256:c3ba1e509fd0620d55c2a917c12874dce72185d4023f0f2863a2321f880afa98"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:I45FSOZDXCVQB4USYZ6CJMTQ5R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lower bounds on maximal determinants of +-1 matrices via the probabilistic method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Judy-anne H. Osborn, Richard P. Brent, Warren D. Smith","submitted_at":"2012-11-14T09:31:09Z","abstract_excerpt":"We show that the maximal determinant D(n) for $n \\times n$ ${\\pm 1}$-matrices satisfies $R(n) := D(n)/n^{n/2} \\ge \\kappa_d > 0$. Here $n^{n/2}$ is the Hadamard upper bound, and $\\kappa_d$ depends only on $d := n-h$, where $h$ is the maximal order of a Hadamard matrix with $h \\le n$. Previous lower bounds on R(n) depend on both $d$ and $n$. Our bounds are improvements, for all sufficiently large $n$, if $d > 1$.\n  We give various lower bounds on R(n) that depend only on $d$. For example, $R(n) \\ge 0.07 (0.352)^d > 3^{-(d+3)}$. For any fixed $d \\ge 0$ we have $R(n) \\ge (2/(\\pi e))^{d/2}$ for all"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3248","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":"2026-05-18T03:26:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wBAOmmNcQm/NXeYdIwdsrOYwx0ofIVjNcQM8zgqvRtZYAvZobwAitDk8zU1RCks6Hw1FsFPd1FN/osi8EpxWCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T06:10:32.379282Z"},"content_sha256":"a1f0c63a56ba039df4fe1966613b5791f7bd064be0c198b267b352c88f96f0e0","schema_version":"1.0","event_id":"sha256:a1f0c63a56ba039df4fe1966613b5791f7bd064be0c198b267b352c88f96f0e0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I45FSOZDXCVQB4USYZ6CJMTQ5R/bundle.json","state_url":"https://pith.science/pith/I45FSOZDXCVQB4USYZ6CJMTQ5R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I45FSOZDXCVQB4USYZ6CJMTQ5R/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":"2026-05-31T06:10:32Z","links":{"resolver":"https://pith.science/pith/I45FSOZDXCVQB4USYZ6CJMTQ5R","bundle":"https://pith.science/pith/I45FSOZDXCVQB4USYZ6CJMTQ5R/bundle.json","state":"https://pith.science/pith/I45FSOZDXCVQB4USYZ6CJMTQ5R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I45FSOZDXCVQB4USYZ6CJMTQ5R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:I45FSOZDXCVQB4USYZ6CJMTQ5R","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":"338a84b2ebd8fa7cf64d07c4e408186fbabbf1f29f28a766bbe10a4dce3a8ea4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-14T09:31:09Z","title_canon_sha256":"eef289cb85c5105d20b80090e1b4969c051e70aa1270738970593d11f470d582"},"schema_version":"1.0","source":{"id":"1211.3248","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.3248","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"1211.3248v3","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.3248","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"I45FSOZDXCVQ","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"I45FSOZDXCVQB4US","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"I45FSOZD","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:a1f0c63a56ba039df4fe1966613b5791f7bd064be0c198b267b352c88f96f0e0","target":"graph","created_at":"2026-05-18T03:26:32Z","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":"We show that the maximal determinant D(n) for $n \\times n$ ${\\pm 1}$-matrices satisfies $R(n) := D(n)/n^{n/2} \\ge \\kappa_d > 0$. Here $n^{n/2}$ is the Hadamard upper bound, and $\\kappa_d$ depends only on $d := n-h$, where $h$ is the maximal order of a Hadamard matrix with $h \\le n$. Previous lower bounds on R(n) depend on both $d$ and $n$. Our bounds are improvements, for all sufficiently large $n$, if $d > 1$.\n  We give various lower bounds on R(n) that depend only on $d$. For example, $R(n) \\ge 0.07 (0.352)^d > 3^{-(d+3)}$. For any fixed $d \\ge 0$ we have $R(n) \\ge (2/(\\pi e))^{d/2}$ for all","authors_text":"Judy-anne H. Osborn, Richard P. Brent, Warren D. Smith","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-14T09:31:09Z","title":"Lower bounds on maximal determinants of +-1 matrices via the probabilistic method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3248","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:c3ba1e509fd0620d55c2a917c12874dce72185d4023f0f2863a2321f880afa98","target":"record","created_at":"2026-05-18T03:26:32Z","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":"338a84b2ebd8fa7cf64d07c4e408186fbabbf1f29f28a766bbe10a4dce3a8ea4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-14T09:31:09Z","title_canon_sha256":"eef289cb85c5105d20b80090e1b4969c051e70aa1270738970593d11f470d582"},"schema_version":"1.0","source":{"id":"1211.3248","kind":"arxiv","version":3}},"canonical_sha256":"473a593b23b8ab00f292c67c24b270ec7228545df1e8e5bf8babaf556d998ccb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"473a593b23b8ab00f292c67c24b270ec7228545df1e8e5bf8babaf556d998ccb","first_computed_at":"2026-05-18T03:26:32.236964Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:32.236964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H7eJRlqirlx9Q8uQaUJy3OpjkLltmzE3GNavfKpaWGcBPGm12FNf/uCxYTRgQWcYMEuSK0121G/GR63MgsC6DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:32.237398Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.3248","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3ba1e509fd0620d55c2a917c12874dce72185d4023f0f2863a2321f880afa98","sha256:a1f0c63a56ba039df4fe1966613b5791f7bd064be0c198b267b352c88f96f0e0"],"state_sha256":"6d1c8648c6cf89f3003fd0702141f26f14c5b53f5d04907823ce21a0cdb5428c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RK53jAsRjZjeSAnEbLivhjmjel9iez8/DmI0zi9KnOV7ZK4l1t2MSdbuzsbJzPAhz7jeWwAkCku8wYSRduoWCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T06:10:32.382697Z","bundle_sha256":"4ebb10a60de29e228a5f4c87fae496bdd31b686b299a2599e84ad9a7d8196b77"}}