{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:JY62K5YXDMDK7QF3RU4YS666FY","short_pith_number":"pith:JY62K5YX","canonical_record":{"source":{"id":"2411.15503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-11-23T09:46:22Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"2ab18622f91d939fe11c5fbd70c5b90c15666827be76aa7c6df3d01c106da9ae","abstract_canon_sha256":"18aca562317c56c6e1579a5f29c0018b22e672e75bdbce058f525a9f7211de0a"},"schema_version":"1.0"},"canonical_sha256":"4e3da577171b06afc0bb8d39897bde2e020eca0686c821845e77f1fcc41558c8","source":{"kind":"arxiv","id":"2411.15503","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.15503","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"arxiv_version","alias_value":"2411.15503v1","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.15503","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"pith_short_12","alias_value":"JY62K5YXDMDK","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"pith_short_16","alias_value":"JY62K5YXDMDK7QF3","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"pith_short_8","alias_value":"JY62K5YX","created_at":"2026-07-05T09:39:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:JY62K5YXDMDK7QF3RU4YS666FY","target":"record","payload":{"canonical_record":{"source":{"id":"2411.15503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-11-23T09:46:22Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"2ab18622f91d939fe11c5fbd70c5b90c15666827be76aa7c6df3d01c106da9ae","abstract_canon_sha256":"18aca562317c56c6e1579a5f29c0018b22e672e75bdbce058f525a9f7211de0a"},"schema_version":"1.0"},"canonical_sha256":"4e3da577171b06afc0bb8d39897bde2e020eca0686c821845e77f1fcc41558c8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:39:36.053170Z","signature_b64":"RHHzPETHxt19Ex60K2EMmGrBsFK6rBgOg/fjkkz3WPmR5INlc5wLPvi+UIjvNYx9UDFVThbLMd2FSYNN5oxrAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e3da577171b06afc0bb8d39897bde2e020eca0686c821845e77f1fcc41558c8","last_reissued_at":"2026-07-05T09:39:36.052639Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:39:36.052639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.15503","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:39:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ycFqPDQLO8le7lxFigJV+oJQANTJGv+k40udVBIZljkoU3AyH5+RF5QlZvcKFNm4J7+ljQPrZM1Z/GnGrVMjAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T19:54:25.642059Z"},"content_sha256":"62ac6ccb8780a04c58a32d1c922a2f75c359a07dc71b06b5db30e378894a568a","schema_version":"1.0","event_id":"sha256:62ac6ccb8780a04c58a32d1c922a2f75c359a07dc71b06b5db30e378894a568a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:JY62K5YXDMDK7QF3RU4YS666FY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the long-range order of the Spectre tilings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DS","authors_text":"Franz G\\\"ahler, Jan Maz\\'a\\v{c}, Lorenzo Sadun, Michael Baake","submitted_at":"2024-11-23T09:46:22Z","abstract_excerpt":"The Spectre is an aperiodic monotile for the Euclidean plane that is truly chiral in the sense that it tiles the plane without any need for a reflected tile. The topological and dynamical properties of the Spectre tilings are very similar to those of the Hat tilings. Specifically, the Spectre sits within a complex $2$-dimensional family of tilings, most of which involve two shapes rather than one. All tilings in the family give topologically conjugate dynamics, up to an overall rescaling and rotation. They all have pure point dynamical spectrum with continuous eigenfunctions and may be obtaine"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15503","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.15503/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T09:39:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PQZTULoKJl9rFvObY/13A6rwwy1H+rk+156DwUQwsdK1rTnRnZfc5LAfn+gRGtMBf+gQoKWHhM7OXUAlydHbCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T19:54:25.642623Z"},"content_sha256":"910435ce4ade3461a77181a726e06fd242040ea96e27fbd385bfe95343a54e91","schema_version":"1.0","event_id":"sha256:910435ce4ade3461a77181a726e06fd242040ea96e27fbd385bfe95343a54e91"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JY62K5YXDMDK7QF3RU4YS666FY/bundle.json","state_url":"https://pith.science/pith/JY62K5YXDMDK7QF3RU4YS666FY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JY62K5YXDMDK7QF3RU4YS666FY/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":"2026-08-13T19:54:25Z","links":{"resolver":"https://pith.science/pith/JY62K5YXDMDK7QF3RU4YS666FY","bundle":"https://pith.science/pith/JY62K5YXDMDK7QF3RU4YS666FY/bundle.json","state":"https://pith.science/pith/JY62K5YXDMDK7QF3RU4YS666FY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JY62K5YXDMDK7QF3RU4YS666FY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:JY62K5YXDMDK7QF3RU4YS666FY","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":"18aca562317c56c6e1579a5f29c0018b22e672e75bdbce058f525a9f7211de0a","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-11-23T09:46:22Z","title_canon_sha256":"2ab18622f91d939fe11c5fbd70c5b90c15666827be76aa7c6df3d01c106da9ae"},"schema_version":"1.0","source":{"id":"2411.15503","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.15503","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"arxiv_version","alias_value":"2411.15503v1","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.15503","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"pith_short_12","alias_value":"JY62K5YXDMDK","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"pith_short_16","alias_value":"JY62K5YXDMDK7QF3","created_at":"2026-07-05T09:39:36Z"},{"alias_kind":"pith_short_8","alias_value":"JY62K5YX","created_at":"2026-07-05T09:39:36Z"}],"graph_snapshots":[{"event_id":"sha256:910435ce4ade3461a77181a726e06fd242040ea96e27fbd385bfe95343a54e91","target":"graph","created_at":"2026-07-05T09:39:36Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2411.15503/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Spectre is an aperiodic monotile for the Euclidean plane that is truly chiral in the sense that it tiles the plane without any need for a reflected tile. The topological and dynamical properties of the Spectre tilings are very similar to those of the Hat tilings. Specifically, the Spectre sits within a complex $2$-dimensional family of tilings, most of which involve two shapes rather than one. All tilings in the family give topologically conjugate dynamics, up to an overall rescaling and rotation. They all have pure point dynamical spectrum with continuous eigenfunctions and may be obtaine","authors_text":"Franz G\\\"ahler, Jan Maz\\'a\\v{c}, Lorenzo Sadun, Michael Baake","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-11-23T09:46:22Z","title":"On the long-range order of the Spectre tilings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15503","kind":"arxiv","version":1},"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":"sha256:62ac6ccb8780a04c58a32d1c922a2f75c359a07dc71b06b5db30e378894a568a","target":"record","created_at":"2026-07-05T09:39:36Z","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":"18aca562317c56c6e1579a5f29c0018b22e672e75bdbce058f525a9f7211de0a","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-11-23T09:46:22Z","title_canon_sha256":"2ab18622f91d939fe11c5fbd70c5b90c15666827be76aa7c6df3d01c106da9ae"},"schema_version":"1.0","source":{"id":"2411.15503","kind":"arxiv","version":1}},"canonical_sha256":"4e3da577171b06afc0bb8d39897bde2e020eca0686c821845e77f1fcc41558c8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e3da577171b06afc0bb8d39897bde2e020eca0686c821845e77f1fcc41558c8","first_computed_at":"2026-07-05T09:39:36.052639Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:39:36.052639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RHHzPETHxt19Ex60K2EMmGrBsFK6rBgOg/fjkkz3WPmR5INlc5wLPvi+UIjvNYx9UDFVThbLMd2FSYNN5oxrAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:39:36.053170Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.15503","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:62ac6ccb8780a04c58a32d1c922a2f75c359a07dc71b06b5db30e378894a568a","sha256:910435ce4ade3461a77181a726e06fd242040ea96e27fbd385bfe95343a54e91"],"state_sha256":"ab27b65cc065fd1d8fc431ec27e8b4e19b46e7717920fd1341791a2185c44f60"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PB9vPJ/vlE2YK5mlMFwZ4vCS3iuNPDeLTvg1LmDKZHc5qOLzRaj73Thafo+J/aA0HyGtjbXcF9JOkdPmrGDgAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T19:54:25.648488Z","bundle_sha256":"e96f010493cfe18d33f68f7947c793d2492287fe096f623b8a85bf0726204ba6"}}