{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:XFXPNIUIVII3YTHWDETD7O4H7N","short_pith_number":"pith:XFXPNIUI","canonical_record":{"source":{"id":"1801.02957","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-01-09T14:43:41Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"e3785c15d7bd8f8cf555acfe17b73ca82b08734250fd295d29291e21452b9425","abstract_canon_sha256":"6db9e4bf18bdbdbec817092f299b15582d537c4620d4b01e9d05001dde43ede8"},"schema_version":"1.0"},"canonical_sha256":"b96ef6a288aa11bc4cf619263fbb87fb671563801c900df23d6281d9e978ed4b","source":{"kind":"arxiv","id":"1801.02957","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.02957","created_at":"2026-05-18T00:26:21Z"},{"alias_kind":"arxiv_version","alias_value":"1801.02957v1","created_at":"2026-05-18T00:26:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.02957","created_at":"2026-05-18T00:26:21Z"},{"alias_kind":"pith_short_12","alias_value":"XFXPNIUIVII3","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"XFXPNIUIVII3YTHW","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"XFXPNIUI","created_at":"2026-05-18T12:33:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:XFXPNIUIVII3YTHWDETD7O4H7N","target":"record","payload":{"canonical_record":{"source":{"id":"1801.02957","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-01-09T14:43:41Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"e3785c15d7bd8f8cf555acfe17b73ca82b08734250fd295d29291e21452b9425","abstract_canon_sha256":"6db9e4bf18bdbdbec817092f299b15582d537c4620d4b01e9d05001dde43ede8"},"schema_version":"1.0"},"canonical_sha256":"b96ef6a288aa11bc4cf619263fbb87fb671563801c900df23d6281d9e978ed4b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:21.921415Z","signature_b64":"SyKzhwC4L1/O5p+CkbFlLvY97OvQNdPSRBoNWftRR8DRrzuIb8nwVZ8L6RPeBGuAlYtvevKvSnwpF06z126UDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b96ef6a288aa11bc4cf619263fbb87fb671563801c900df23d6281d9e978ed4b","last_reissued_at":"2026-05-18T00:26:21.920801Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:21.920801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.02957","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":"2026-05-18T00:26:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fdLEQ/PaSxbglYfPRTkKIyyBYGKPdcHCHxpOCP5ozb7z0zUR5krMHYkfntAi5C00NGfYVWpELQnFmzhCcDzsCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T22:45:52.515861Z"},"content_sha256":"57582554d91dfdcee9116dde15bf2e3a7fbf849caace4e892546a99304d346c3","schema_version":"1.0","event_id":"sha256:57582554d91dfdcee9116dde15bf2e3a7fbf849caace4e892546a99304d346c3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:XFXPNIUIVII3YTHWDETD7O4H7N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Topology of planar self-affine tiles with collinear digit set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.GN","authors_text":"Beno\\^it Loridant, J\\\"org Thuswaldner, Shigeki Akiyama","submitted_at":"2018-01-09T14:43:41Z","abstract_excerpt":"We consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\\in\\mathbb{Z}$ satisfy $|A|\\leq B\\geq 2$ and $M\\in\\mathbb{Z}^{2\\times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\\mathcal{D}=\\{0,v,2v,\\ldots,(B-1)v\\}$ for some $v\\in\\mathbb{Z}^2$ such that $v,M v$ are linearly independent. We are interested in the topological properties of the self-affine tile $\\mathcal{T}$ defined by $M\\mathcal{T}=\\bigcup_{d\\in\\mathcal{D}}(\\mathcal{T}+d)$. Lau and Leung proved that $\\mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|\\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02957","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":"2026-05-18T00:26:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qIUvLci87fmGEmLEZnZT/PbC9+1aEwFPGU0tJGlxSm0lmId/JJXhZB3bS4xrMG4w+lDLrg6r4PNmIMxOGiVKCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T22:45:52.516628Z"},"content_sha256":"8265bf33265468b89a511200839294ab143bf002f842e23d1367b43a1782c213","schema_version":"1.0","event_id":"sha256:8265bf33265468b89a511200839294ab143bf002f842e23d1367b43a1782c213"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N/bundle.json","state_url":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XFXPNIUIVII3YTHWDETD7O4H7N/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":"2026-06-11T22:45:52Z","links":{"resolver":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N","bundle":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N/bundle.json","state":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XFXPNIUIVII3YTHWDETD7O4H7N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:XFXPNIUIVII3YTHWDETD7O4H7N","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":"6db9e4bf18bdbdbec817092f299b15582d537c4620d4b01e9d05001dde43ede8","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-01-09T14:43:41Z","title_canon_sha256":"e3785c15d7bd8f8cf555acfe17b73ca82b08734250fd295d29291e21452b9425"},"schema_version":"1.0","source":{"id":"1801.02957","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.02957","created_at":"2026-05-18T00:26:21Z"},{"alias_kind":"arxiv_version","alias_value":"1801.02957v1","created_at":"2026-05-18T00:26:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.02957","created_at":"2026-05-18T00:26:21Z"},{"alias_kind":"pith_short_12","alias_value":"XFXPNIUIVII3","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"XFXPNIUIVII3YTHW","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"XFXPNIUI","created_at":"2026-05-18T12:33:01Z"}],"graph_snapshots":[{"event_id":"sha256:8265bf33265468b89a511200839294ab143bf002f842e23d1367b43a1782c213","target":"graph","created_at":"2026-05-18T00:26:21Z","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 consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\\in\\mathbb{Z}$ satisfy $|A|\\leq B\\geq 2$ and $M\\in\\mathbb{Z}^{2\\times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\\mathcal{D}=\\{0,v,2v,\\ldots,(B-1)v\\}$ for some $v\\in\\mathbb{Z}^2$ such that $v,M v$ are linearly independent. We are interested in the topological properties of the self-affine tile $\\mathcal{T}$ defined by $M\\mathcal{T}=\\bigcup_{d\\in\\mathcal{D}}(\\mathcal{T}+d)$. Lau and Leung proved that $\\mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|\\le","authors_text":"Beno\\^it Loridant, J\\\"org Thuswaldner, Shigeki Akiyama","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-01-09T14:43:41Z","title":"Topology of planar self-affine tiles with collinear digit set"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02957","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":"sha256:57582554d91dfdcee9116dde15bf2e3a7fbf849caace4e892546a99304d346c3","target":"record","created_at":"2026-05-18T00:26:21Z","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":"6db9e4bf18bdbdbec817092f299b15582d537c4620d4b01e9d05001dde43ede8","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-01-09T14:43:41Z","title_canon_sha256":"e3785c15d7bd8f8cf555acfe17b73ca82b08734250fd295d29291e21452b9425"},"schema_version":"1.0","source":{"id":"1801.02957","kind":"arxiv","version":1}},"canonical_sha256":"b96ef6a288aa11bc4cf619263fbb87fb671563801c900df23d6281d9e978ed4b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b96ef6a288aa11bc4cf619263fbb87fb671563801c900df23d6281d9e978ed4b","first_computed_at":"2026-05-18T00:26:21.920801Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:21.920801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SyKzhwC4L1/O5p+CkbFlLvY97OvQNdPSRBoNWftRR8DRrzuIb8nwVZ8L6RPeBGuAlYtvevKvSnwpF06z126UDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:21.921415Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.02957","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57582554d91dfdcee9116dde15bf2e3a7fbf849caace4e892546a99304d346c3","sha256:8265bf33265468b89a511200839294ab143bf002f842e23d1367b43a1782c213"],"state_sha256":"2b4efe41317777bf083ddc41f7383d6e4f2f63a30b5f919893f8b0c968d86d0a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ov+XZV2mz2mWroTLqI1nBOHfDHYmPCxfbxfXQ+/b0lvb60OWk2XhrZ1r2WTN9FkKKhpW2r4Gcwn/hUcLhpOaDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T22:45:52.526959Z","bundle_sha256":"894003dc525cafd8edc6dc7912fa898a1a758158d7299b94a06a45a8fd8a9363"}}