{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:546WCDT6KCZMREGGASHZ3CHGD2","short_pith_number":"pith:546WCDT6","canonical_record":{"source":{"id":"1712.09551","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-12-27T11:23:15Z","cross_cats_sorted":["math.DS","math.KT"],"title_canon_sha256":"8358d4b18df0c0837e257687b850a3390bea331c06568358d7c8286b8e3655e3","abstract_canon_sha256":"7697da8c2fa402d2b3cf2ca2b3d9e3d84509e2a8ab71b7a00de1bf805367ffdd"},"schema_version":"1.0"},"canonical_sha256":"ef3d610e7e50b2c890c6048f9d88e61e92d9853efa861b842619efa4848db36e","source":{"kind":"arxiv","id":"1712.09551","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.09551","created_at":"2026-05-18T00:05:05Z"},{"alias_kind":"arxiv_version","alias_value":"1712.09551v2","created_at":"2026-05-18T00:05:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.09551","created_at":"2026-05-18T00:05:05Z"},{"alias_kind":"pith_short_12","alias_value":"546WCDT6KCZM","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"546WCDT6KCZMREGG","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"546WCDT6","created_at":"2026-05-18T12:31:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:546WCDT6KCZMREGGASHZ3CHGD2","target":"record","payload":{"canonical_record":{"source":{"id":"1712.09551","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-12-27T11:23:15Z","cross_cats_sorted":["math.DS","math.KT"],"title_canon_sha256":"8358d4b18df0c0837e257687b850a3390bea331c06568358d7c8286b8e3655e3","abstract_canon_sha256":"7697da8c2fa402d2b3cf2ca2b3d9e3d84509e2a8ab71b7a00de1bf805367ffdd"},"schema_version":"1.0"},"canonical_sha256":"ef3d610e7e50b2c890c6048f9d88e61e92d9853efa861b842619efa4848db36e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:05.374850Z","signature_b64":"BinFGy+YTcxqGi5UG8LSlE3E+un73R8ecLD5idqxpNldOUWC0Sg54u1quk3C19HI89Q4lQQXLcFURtc/FG8dAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef3d610e7e50b2c890c6048f9d88e61e92d9853efa861b842619efa4848db36e","last_reissued_at":"2026-05-18T00:05:05.374284Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:05.374284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.09551","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":"2026-05-18T00:05:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Lgsgzs3Ppp6IWf5hICN5vTp9JhUx9EchpMre53hAnUqIF72zmTjhda9KkzG1LcN+YijG1/V17hgZZYcsc8LqAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T13:23:58.684904Z"},"content_sha256":"dec19f8e7e0b925c14e012b34938aaa5395671231a62099628cc3a77e6cce91f","schema_version":"1.0","event_id":"sha256:dec19f8e7e0b925c14e012b34938aaa5395671231a62099628cc3a77e6cce91f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:546WCDT6KCZMREGGASHZ3CHGD2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the K-theory of C*-algebras for substitution tilings (a pedestrian version)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.KT"],"primary_cat":"math.OA","authors_text":"Daniel Gon\\c{c}alves, Maria Ramirez-Solano","submitted_at":"2017-12-27T11:23:15Z","abstract_excerpt":"Under suitable conditions, a substitution tiling gives rise to a Smale space, from which three equivalence relations can be constructed, namely the stable, unstable, and asymptotic equivalence relations. We denote with $S$, $U$, and $A$ their corresponding $C^*$-algebras in the sense of Renault. In this article we show that the $K$-theories of $S$ and $U$ can be computed from the cohomology and homology of a single cochain complex with connecting maps for tilings of the line and of the plane. Moreover, we provide formulas to compute the $K$-theory for these three $C^*$-algebras. Furthermore, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.09551","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":"2026-05-18T00:05:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IqWJ1phKnhQ2othA6VllWC/s0NHizypUKVLZvEhF6xrmWaLmk/7ZthWZp0vovFgh0k1v52UWYUvh3yugPe21DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T13:23:58.685258Z"},"content_sha256":"4c7a8ca16438bec3e98695f7133833d00d628c5dd4fc419ff0c3f10ea535b327","schema_version":"1.0","event_id":"sha256:4c7a8ca16438bec3e98695f7133833d00d628c5dd4fc419ff0c3f10ea535b327"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/546WCDT6KCZMREGGASHZ3CHGD2/bundle.json","state_url":"https://pith.science/pith/546WCDT6KCZMREGGASHZ3CHGD2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/546WCDT6KCZMREGGASHZ3CHGD2/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":"2026-06-01T13:23:58Z","links":{"resolver":"https://pith.science/pith/546WCDT6KCZMREGGASHZ3CHGD2","bundle":"https://pith.science/pith/546WCDT6KCZMREGGASHZ3CHGD2/bundle.json","state":"https://pith.science/pith/546WCDT6KCZMREGGASHZ3CHGD2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/546WCDT6KCZMREGGASHZ3CHGD2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:546WCDT6KCZMREGGASHZ3CHGD2","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":"7697da8c2fa402d2b3cf2ca2b3d9e3d84509e2a8ab71b7a00de1bf805367ffdd","cross_cats_sorted":["math.DS","math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-12-27T11:23:15Z","title_canon_sha256":"8358d4b18df0c0837e257687b850a3390bea331c06568358d7c8286b8e3655e3"},"schema_version":"1.0","source":{"id":"1712.09551","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.09551","created_at":"2026-05-18T00:05:05Z"},{"alias_kind":"arxiv_version","alias_value":"1712.09551v2","created_at":"2026-05-18T00:05:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.09551","created_at":"2026-05-18T00:05:05Z"},{"alias_kind":"pith_short_12","alias_value":"546WCDT6KCZM","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"546WCDT6KCZMREGG","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"546WCDT6","created_at":"2026-05-18T12:31:00Z"}],"graph_snapshots":[{"event_id":"sha256:4c7a8ca16438bec3e98695f7133833d00d628c5dd4fc419ff0c3f10ea535b327","target":"graph","created_at":"2026-05-18T00:05:05Z","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":"Under suitable conditions, a substitution tiling gives rise to a Smale space, from which three equivalence relations can be constructed, namely the stable, unstable, and asymptotic equivalence relations. We denote with $S$, $U$, and $A$ their corresponding $C^*$-algebras in the sense of Renault. In this article we show that the $K$-theories of $S$ and $U$ can be computed from the cohomology and homology of a single cochain complex with connecting maps for tilings of the line and of the plane. Moreover, we provide formulas to compute the $K$-theory for these three $C^*$-algebras. Furthermore, w","authors_text":"Daniel Gon\\c{c}alves, Maria Ramirez-Solano","cross_cats":["math.DS","math.KT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-12-27T11:23:15Z","title":"On the K-theory of C*-algebras for substitution tilings (a pedestrian version)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.09551","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:dec19f8e7e0b925c14e012b34938aaa5395671231a62099628cc3a77e6cce91f","target":"record","created_at":"2026-05-18T00:05:05Z","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":"7697da8c2fa402d2b3cf2ca2b3d9e3d84509e2a8ab71b7a00de1bf805367ffdd","cross_cats_sorted":["math.DS","math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-12-27T11:23:15Z","title_canon_sha256":"8358d4b18df0c0837e257687b850a3390bea331c06568358d7c8286b8e3655e3"},"schema_version":"1.0","source":{"id":"1712.09551","kind":"arxiv","version":2}},"canonical_sha256":"ef3d610e7e50b2c890c6048f9d88e61e92d9853efa861b842619efa4848db36e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ef3d610e7e50b2c890c6048f9d88e61e92d9853efa861b842619efa4848db36e","first_computed_at":"2026-05-18T00:05:05.374284Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:05:05.374284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BinFGy+YTcxqGi5UG8LSlE3E+un73R8ecLD5idqxpNldOUWC0Sg54u1quk3C19HI89Q4lQQXLcFURtc/FG8dAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:05:05.374850Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.09551","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dec19f8e7e0b925c14e012b34938aaa5395671231a62099628cc3a77e6cce91f","sha256:4c7a8ca16438bec3e98695f7133833d00d628c5dd4fc419ff0c3f10ea535b327"],"state_sha256":"16bfc5a1d5b688d12d4211e7a8b7096a860b52153588203bfb0d8fc42127930f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"quQ0/e7ogrUtFKL/H4g36r0GWbDig0p94QmLnYcTum79rdjHXM3OVWdY6D3RjX+npIpvwWGz/yVnriz+GD/2AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T13:23:58.687545Z","bundle_sha256":"052f5a33bed5d7b0b2ba8f8e758b7dfd36447c8f5c1f4d5af5202f73c1b689ca"}}