{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:A2SEDGWESL5Y6LESKV677LVNKE","short_pith_number":"pith:A2SEDGWE","canonical_record":{"source":{"id":"1601.06629","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-25T14:55:58Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"7b41affee4c7b8116a9dbdbbf7a1edecc869c39378db57687ab8ee80a08f0efe","abstract_canon_sha256":"c28a3d8fb0c7d2597684d618d4152d699fcd15408607b9c432cc83df508cdf98"},"schema_version":"1.0"},"canonical_sha256":"06a4419ac492fb8f2c92557dffaead5113485e847048d0b289018dd92851aa6f","source":{"kind":"arxiv","id":"1601.06629","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.06629","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"arxiv_version","alias_value":"1601.06629v1","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06629","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"pith_short_12","alias_value":"A2SEDGWESL5Y","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"A2SEDGWESL5Y6LES","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"A2SEDGWE","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:A2SEDGWESL5Y6LESKV677LVNKE","target":"record","payload":{"canonical_record":{"source":{"id":"1601.06629","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-25T14:55:58Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"7b41affee4c7b8116a9dbdbbf7a1edecc869c39378db57687ab8ee80a08f0efe","abstract_canon_sha256":"c28a3d8fb0c7d2597684d618d4152d699fcd15408607b9c432cc83df508cdf98"},"schema_version":"1.0"},"canonical_sha256":"06a4419ac492fb8f2c92557dffaead5113485e847048d0b289018dd92851aa6f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:29.735241Z","signature_b64":"WucJ1zKzqZhLV5FX+16hdtieVPVWOjpetj5X52v6QJpDzCcMcd4s4iYgkcjNd5zd6iyjkmbnYMoZgtO3gCw9Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06a4419ac492fb8f2c92557dffaead5113485e847048d0b289018dd92851aa6f","last_reissued_at":"2026-05-18T00:50:29.734557Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:29.734557Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.06629","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":"2026-05-18T00:50:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hrCzXRrnzHu8ez3V9DA7r8gBTh71DXr0ZW24W9FKwr1KgMvQC5d1w0kIxaIwzWBVmFhJa+d1DzmYBGnUR8yIBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T22:01:03.507825Z"},"content_sha256":"6e7d36d0f42ff76191d4478676c8b473cb925e2284a049d7bcd36618e9c649d3","schema_version":"1.0","event_id":"sha256:6e7d36d0f42ff76191d4478676c8b473cb925e2284a049d7bcd36618e9c649d3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:A2SEDGWESL5Y6LESKV677LVNKE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral notions of aperiodic order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.DS","authors_text":"Daniel Lenz (Jena), Michael Baake (Bielefeld)","submitted_at":"2016-01-25T14:55:58Z","abstract_excerpt":"Various spectral notions have been employed to grasp the structure of point sets, in particular non-periodic ones. In this article, we present them in a unified setting and explain the relations between them. For the sake of readability, we use Delone sets in Euclidean space as our main object class, and give generalisations in the form of further examples and remarks."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06629","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":""},"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":"2026-05-18T00:50:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oR1whXH94ha0G7zX2We5jJgtD5tiQ/dWmt86/pT4wyJh7TDRFOBHS3SBHA6mnnQqKPac9BfPO2PvfOcz3TVtAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T22:01:03.508155Z"},"content_sha256":"67ac2c4b66c0a1d40b01c1e3eb95bb832221119c0ba8d1f43833386efa5d8f4c","schema_version":"1.0","event_id":"sha256:67ac2c4b66c0a1d40b01c1e3eb95bb832221119c0ba8d1f43833386efa5d8f4c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A2SEDGWESL5Y6LESKV677LVNKE/bundle.json","state_url":"https://pith.science/pith/A2SEDGWESL5Y6LESKV677LVNKE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A2SEDGWESL5Y6LESKV677LVNKE/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":"2026-06-29T22:01:03Z","links":{"resolver":"https://pith.science/pith/A2SEDGWESL5Y6LESKV677LVNKE","bundle":"https://pith.science/pith/A2SEDGWESL5Y6LESKV677LVNKE/bundle.json","state":"https://pith.science/pith/A2SEDGWESL5Y6LESKV677LVNKE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A2SEDGWESL5Y6LESKV677LVNKE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:A2SEDGWESL5Y6LESKV677LVNKE","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":"c28a3d8fb0c7d2597684d618d4152d699fcd15408607b9c432cc83df508cdf98","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-25T14:55:58Z","title_canon_sha256":"7b41affee4c7b8116a9dbdbbf7a1edecc869c39378db57687ab8ee80a08f0efe"},"schema_version":"1.0","source":{"id":"1601.06629","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.06629","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"arxiv_version","alias_value":"1601.06629v1","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06629","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"pith_short_12","alias_value":"A2SEDGWESL5Y","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"A2SEDGWESL5Y6LES","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"A2SEDGWE","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:67ac2c4b66c0a1d40b01c1e3eb95bb832221119c0ba8d1f43833386efa5d8f4c","target":"graph","created_at":"2026-05-18T00:50:29Z","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":"Various spectral notions have been employed to grasp the structure of point sets, in particular non-periodic ones. In this article, we present them in a unified setting and explain the relations between them. For the sake of readability, we use Delone sets in Euclidean space as our main object class, and give generalisations in the form of further examples and remarks.","authors_text":"Daniel Lenz (Jena), Michael Baake (Bielefeld)","cross_cats":["math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-25T14:55:58Z","title":"Spectral notions of aperiodic order"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06629","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":"sha256:6e7d36d0f42ff76191d4478676c8b473cb925e2284a049d7bcd36618e9c649d3","target":"record","created_at":"2026-05-18T00:50:29Z","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":"c28a3d8fb0c7d2597684d618d4152d699fcd15408607b9c432cc83df508cdf98","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-25T14:55:58Z","title_canon_sha256":"7b41affee4c7b8116a9dbdbbf7a1edecc869c39378db57687ab8ee80a08f0efe"},"schema_version":"1.0","source":{"id":"1601.06629","kind":"arxiv","version":1}},"canonical_sha256":"06a4419ac492fb8f2c92557dffaead5113485e847048d0b289018dd92851aa6f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06a4419ac492fb8f2c92557dffaead5113485e847048d0b289018dd92851aa6f","first_computed_at":"2026-05-18T00:50:29.734557Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:29.734557Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WucJ1zKzqZhLV5FX+16hdtieVPVWOjpetj5X52v6QJpDzCcMcd4s4iYgkcjNd5zd6iyjkmbnYMoZgtO3gCw9Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:29.735241Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.06629","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e7d36d0f42ff76191d4478676c8b473cb925e2284a049d7bcd36618e9c649d3","sha256:67ac2c4b66c0a1d40b01c1e3eb95bb832221119c0ba8d1f43833386efa5d8f4c"],"state_sha256":"71d996394a0326df685e5cb0386fa7b3ad28d4aa913f906c95fe16ab9e08da37"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2jmP+SLQFxEYmJfBmliNX8IhPncJTdkD2Z5mjCdQ6NyHs9deNkoMlqL7mSOTG9DVa71zoe9+j1WCeVVVbIzNBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T22:01:03.509982Z","bundle_sha256":"d3bb2061e64f5cbbf909a4cf6cacdd64f750cf7b1ebaf12542e952dda8e6ef80"}}