{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:IAM4THYGN2Z6FG3QOYHOIIOGDT","short_pith_number":"pith:IAM4THYG","canonical_record":{"source":{"id":"1904.11537","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-25T18:55:11Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"d29bd11f5cd452479427c4e69ba936f74b05a5272e24f5ab6ccd3ef258061eeb","abstract_canon_sha256":"2663b3fce2ad19e8c52dc52431285e500cd4627e287498d501d6ab59d7646ca2"},"schema_version":"1.0"},"canonical_sha256":"4019c99f066eb3e29b70760ee421c61cec42f3dcb5f30e0f49f70bb2e5cacb46","source":{"kind":"arxiv","id":"1904.11537","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.11537","created_at":"2026-05-17T23:47:35Z"},{"alias_kind":"arxiv_version","alias_value":"1904.11537v2","created_at":"2026-05-17T23:47:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.11537","created_at":"2026-05-17T23:47:35Z"},{"alias_kind":"pith_short_12","alias_value":"IAM4THYGN2Z6","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"IAM4THYGN2Z6FG3Q","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"IAM4THYG","created_at":"2026-05-18T12:33:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:IAM4THYGN2Z6FG3QOYHOIIOGDT","target":"record","payload":{"canonical_record":{"source":{"id":"1904.11537","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-25T18:55:11Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"d29bd11f5cd452479427c4e69ba936f74b05a5272e24f5ab6ccd3ef258061eeb","abstract_canon_sha256":"2663b3fce2ad19e8c52dc52431285e500cd4627e287498d501d6ab59d7646ca2"},"schema_version":"1.0"},"canonical_sha256":"4019c99f066eb3e29b70760ee421c61cec42f3dcb5f30e0f49f70bb2e5cacb46","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:35.513939Z","signature_b64":"6+9KrSyXXH8OzUmI6k9eGBtjTD+PSUcft89nB2BHCw8HhSyA7NX+077qyuBzUX9kL44umNtgbFjZwduYKJJTAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4019c99f066eb3e29b70760ee421c61cec42f3dcb5f30e0f49f70bb2e5cacb46","last_reissued_at":"2026-05-17T23:47:35.513559Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:35.513559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.11537","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:47:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n3LizRCQVVLFoDG/CCAd7Sm/Oey9gAo4a1yIzm1cCASZ9iwt0JWmUSfXmcr33I0HdjzS0oJdnKAhPp+qRJFXDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T12:31:04.962179Z"},"content_sha256":"675b19fa6e173e9383e697e906059d848f7a35094ea2049dbc6de5901fd9e33b","schema_version":"1.0","event_id":"sha256:675b19fa6e173e9383e697e906059d848f7a35094ea2049dbc6de5901fd9e33b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:IAM4THYGN2Z6FG3QOYHOIIOGDT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A counterexample to Fuglede's conjecture in $(\\mathbb{Z}/p\\mathbb{Z})^4$ for all odd primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CO","authors_text":"Sam Mattheus","submitted_at":"2019-04-25T18:55:11Z","abstract_excerpt":"In this short note we construct a spectral, non-tiling set of size $2p$ in $(\\mathbb{Z}/p\\mathbb{Z})^4$, $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \\equiv 3 \\pmod{4}$. On the contrary we show that the conjecture does hold in $(\\mathbb{Z}/2\\mathbb{Z})^4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.11537","kind":"arxiv","version":2},"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":"2026-05-17T23:47:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PPHoQi/tgQvfE8q0ANPO2hozUpaH0VOCmDfUsuI9tGdZELXqUitIJQsYRdGgSLKsiLJK6t4csu/upsZyhOfKBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T12:31:04.962694Z"},"content_sha256":"37491e62c92267f7246145d6e2866a037b11058d62334599b7ba3ed51b9beb69","schema_version":"1.0","event_id":"sha256:37491e62c92267f7246145d6e2866a037b11058d62334599b7ba3ed51b9beb69"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IAM4THYGN2Z6FG3QOYHOIIOGDT/bundle.json","state_url":"https://pith.science/pith/IAM4THYGN2Z6FG3QOYHOIIOGDT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IAM4THYGN2Z6FG3QOYHOIIOGDT/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":"2026-06-28T12:31:04Z","links":{"resolver":"https://pith.science/pith/IAM4THYGN2Z6FG3QOYHOIIOGDT","bundle":"https://pith.science/pith/IAM4THYGN2Z6FG3QOYHOIIOGDT/bundle.json","state":"https://pith.science/pith/IAM4THYGN2Z6FG3QOYHOIIOGDT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IAM4THYGN2Z6FG3QOYHOIIOGDT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:IAM4THYGN2Z6FG3QOYHOIIOGDT","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":"2663b3fce2ad19e8c52dc52431285e500cd4627e287498d501d6ab59d7646ca2","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-25T18:55:11Z","title_canon_sha256":"d29bd11f5cd452479427c4e69ba936f74b05a5272e24f5ab6ccd3ef258061eeb"},"schema_version":"1.0","source":{"id":"1904.11537","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.11537","created_at":"2026-05-17T23:47:35Z"},{"alias_kind":"arxiv_version","alias_value":"1904.11537v2","created_at":"2026-05-17T23:47:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.11537","created_at":"2026-05-17T23:47:35Z"},{"alias_kind":"pith_short_12","alias_value":"IAM4THYGN2Z6","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"IAM4THYGN2Z6FG3Q","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"IAM4THYG","created_at":"2026-05-18T12:33:18Z"}],"graph_snapshots":[{"event_id":"sha256:37491e62c92267f7246145d6e2866a037b11058d62334599b7ba3ed51b9beb69","target":"graph","created_at":"2026-05-17T23:47:35Z","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":"In this short note we construct a spectral, non-tiling set of size $2p$ in $(\\mathbb{Z}/p\\mathbb{Z})^4$, $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \\equiv 3 \\pmod{4}$. On the contrary we show that the conjecture does hold in $(\\mathbb{Z}/2\\mathbb{Z})^4$.","authors_text":"Sam Mattheus","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-25T18:55:11Z","title":"A counterexample to Fuglede's conjecture in $(\\mathbb{Z}/p\\mathbb{Z})^4$ for all odd primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.11537","kind":"arxiv","version":2},"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":"sha256:675b19fa6e173e9383e697e906059d848f7a35094ea2049dbc6de5901fd9e33b","target":"record","created_at":"2026-05-17T23:47:35Z","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":"2663b3fce2ad19e8c52dc52431285e500cd4627e287498d501d6ab59d7646ca2","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-25T18:55:11Z","title_canon_sha256":"d29bd11f5cd452479427c4e69ba936f74b05a5272e24f5ab6ccd3ef258061eeb"},"schema_version":"1.0","source":{"id":"1904.11537","kind":"arxiv","version":2}},"canonical_sha256":"4019c99f066eb3e29b70760ee421c61cec42f3dcb5f30e0f49f70bb2e5cacb46","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4019c99f066eb3e29b70760ee421c61cec42f3dcb5f30e0f49f70bb2e5cacb46","first_computed_at":"2026-05-17T23:47:35.513559Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:47:35.513559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6+9KrSyXXH8OzUmI6k9eGBtjTD+PSUcft89nB2BHCw8HhSyA7NX+077qyuBzUX9kL44umNtgbFjZwduYKJJTAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:47:35.513939Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.11537","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:675b19fa6e173e9383e697e906059d848f7a35094ea2049dbc6de5901fd9e33b","sha256:37491e62c92267f7246145d6e2866a037b11058d62334599b7ba3ed51b9beb69"],"state_sha256":"e765af98794f2a1116e470f9ef597cb6f1a23503bc976f30bdb3389ffd3b9a0b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8AtQF8kY2GnV3mTGug5kesxXSUR8DU8GWz7jAm6TPNlOj3epWAjI7cE3KVY9+qBncUnM68abkQpMnV4hJsXFDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T12:31:04.965354Z","bundle_sha256":"9ce8ba51144cd82bce81bc383ef7c1ec093cfd94816c46d1d8101c5905269c69"}}