{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:ND4LFYQKLGTQJ6GZ3B4BMIABFB","short_pith_number":"pith:ND4LFYQK","canonical_record":{"source":{"id":"1208.4090","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-20T19:45:37Z","cross_cats_sorted":[],"title_canon_sha256":"678b6a591ed6f29f15924abe8072f9bcd93ef6212ad1ec68edce5f140841d987","abstract_canon_sha256":"5ef3aedbb03ec755525f48f6d41b434bc98e46191c5f6fb9e2ae2ce61f47c815"},"schema_version":"1.0"},"canonical_sha256":"68f8b2e20a59a704f8d9d8781620012872aaa8519f73315b5f0d69c01784f070","source":{"kind":"arxiv","id":"1208.4090","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4090","created_at":"2026-05-18T03:15:18Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4090v2","created_at":"2026-05-18T03:15:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4090","created_at":"2026-05-18T03:15:18Z"},{"alias_kind":"pith_short_12","alias_value":"ND4LFYQKLGTQ","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"ND4LFYQKLGTQJ6GZ","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"ND4LFYQK","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:ND4LFYQKLGTQJ6GZ3B4BMIABFB","target":"record","payload":{"canonical_record":{"source":{"id":"1208.4090","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-20T19:45:37Z","cross_cats_sorted":[],"title_canon_sha256":"678b6a591ed6f29f15924abe8072f9bcd93ef6212ad1ec68edce5f140841d987","abstract_canon_sha256":"5ef3aedbb03ec755525f48f6d41b434bc98e46191c5f6fb9e2ae2ce61f47c815"},"schema_version":"1.0"},"canonical_sha256":"68f8b2e20a59a704f8d9d8781620012872aaa8519f73315b5f0d69c01784f070","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:18.696871Z","signature_b64":"TA0DKzM3+LbR3Q7WgShVF5QrGPqCv1B4He83Teg3k894z3jrnhIjfz7sF2ZI3COkDVPDQqIEwXoh5XwDE5nPCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68f8b2e20a59a704f8d9d8781620012872aaa8519f73315b5f0d69c01784f070","last_reissued_at":"2026-05-18T03:15:18.696081Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:18.696081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.4090","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":"2026-05-18T03:15:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xWq7mkvZUHQa9UyUKbvpfhFalna6NtP9Esx/FTZr58Jy5/PgaMMjBVMRZvhfOWU8h+Xo1rvn9M2BW1dX853HBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T00:58:49.104223Z"},"content_sha256":"387e62a508f5cff94cb7d07cff8dda7cc5488c8e4231cbf423ac4b23b8511953","schema_version":"1.0","event_id":"sha256:387e62a508f5cff94cb7d07cff8dda7cc5488c8e4231cbf423ac4b23b8511953"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:ND4LFYQKLGTQJ6GZ3B4BMIABFB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonexpansive Z^2 subdynamics and Nivat's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bryna Kra, Van Cyr","submitted_at":"2012-08-20T19:45:37Z","abstract_excerpt":"For a finite alphabet $\\A$ and $\\eta\\colon \\Z\\to\\A$, the Morse-Hedlund Theorem states that $\\eta$ is periodic if and only if there exists $n\\in\\N$ such that the block complexity function $P_\\eta(n)$ satisfies $P_\\eta(n)\\leq n$, and this statement is naturally studied by analyzing the dynamics of a $\\Z$-action associated to $\\eta$. In dimension two, we analyze the subdynamics of a $\\ZZ$-action associated to $\\eta\\colon\\ZZ\\to\\A$ and show that if there exist $n,k\\in\\N$ such that the $n\\times k$ rectangular complexity $P_{\\eta}(n,k)$ satisfies $P_{\\eta}(n,k)\\leq nk$, then the periodicity of $\\eta$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4090","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":"2026-05-18T03:15:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v0iBG8s2baJn5DsGnibRlE0gez6BSCsabk4XgDmY9MLmYQfTFaQnXtYvZavUBAnlNWnhclmt2cjN3xIaO5/pDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T00:58:49.104587Z"},"content_sha256":"bd305db4aa53f8d5c4a5445c8698ef8bb1c4903c8ff50310b19c2142eee0ed94","schema_version":"1.0","event_id":"sha256:bd305db4aa53f8d5c4a5445c8698ef8bb1c4903c8ff50310b19c2142eee0ed94"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ND4LFYQKLGTQJ6GZ3B4BMIABFB/bundle.json","state_url":"https://pith.science/pith/ND4LFYQKLGTQJ6GZ3B4BMIABFB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ND4LFYQKLGTQJ6GZ3B4BMIABFB/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":"2026-06-23T00:58:49Z","links":{"resolver":"https://pith.science/pith/ND4LFYQKLGTQJ6GZ3B4BMIABFB","bundle":"https://pith.science/pith/ND4LFYQKLGTQJ6GZ3B4BMIABFB/bundle.json","state":"https://pith.science/pith/ND4LFYQKLGTQJ6GZ3B4BMIABFB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ND4LFYQKLGTQJ6GZ3B4BMIABFB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ND4LFYQKLGTQJ6GZ3B4BMIABFB","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":"5ef3aedbb03ec755525f48f6d41b434bc98e46191c5f6fb9e2ae2ce61f47c815","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-20T19:45:37Z","title_canon_sha256":"678b6a591ed6f29f15924abe8072f9bcd93ef6212ad1ec68edce5f140841d987"},"schema_version":"1.0","source":{"id":"1208.4090","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4090","created_at":"2026-05-18T03:15:18Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4090v2","created_at":"2026-05-18T03:15:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4090","created_at":"2026-05-18T03:15:18Z"},{"alias_kind":"pith_short_12","alias_value":"ND4LFYQKLGTQ","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"ND4LFYQKLGTQJ6GZ","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"ND4LFYQK","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:bd305db4aa53f8d5c4a5445c8698ef8bb1c4903c8ff50310b19c2142eee0ed94","target":"graph","created_at":"2026-05-18T03:15:18Z","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":"For a finite alphabet $\\A$ and $\\eta\\colon \\Z\\to\\A$, the Morse-Hedlund Theorem states that $\\eta$ is periodic if and only if there exists $n\\in\\N$ such that the block complexity function $P_\\eta(n)$ satisfies $P_\\eta(n)\\leq n$, and this statement is naturally studied by analyzing the dynamics of a $\\Z$-action associated to $\\eta$. In dimension two, we analyze the subdynamics of a $\\ZZ$-action associated to $\\eta\\colon\\ZZ\\to\\A$ and show that if there exist $n,k\\in\\N$ such that the $n\\times k$ rectangular complexity $P_{\\eta}(n,k)$ satisfies $P_{\\eta}(n,k)\\leq nk$, then the periodicity of $\\eta$","authors_text":"Bryna Kra, Van Cyr","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-20T19:45:37Z","title":"Nonexpansive Z^2 subdynamics and Nivat's conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4090","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":"sha256:387e62a508f5cff94cb7d07cff8dda7cc5488c8e4231cbf423ac4b23b8511953","target":"record","created_at":"2026-05-18T03:15:18Z","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":"5ef3aedbb03ec755525f48f6d41b434bc98e46191c5f6fb9e2ae2ce61f47c815","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-20T19:45:37Z","title_canon_sha256":"678b6a591ed6f29f15924abe8072f9bcd93ef6212ad1ec68edce5f140841d987"},"schema_version":"1.0","source":{"id":"1208.4090","kind":"arxiv","version":2}},"canonical_sha256":"68f8b2e20a59a704f8d9d8781620012872aaa8519f73315b5f0d69c01784f070","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68f8b2e20a59a704f8d9d8781620012872aaa8519f73315b5f0d69c01784f070","first_computed_at":"2026-05-18T03:15:18.696081Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:18.696081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TA0DKzM3+LbR3Q7WgShVF5QrGPqCv1B4He83Teg3k894z3jrnhIjfz7sF2ZI3COkDVPDQqIEwXoh5XwDE5nPCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:18.696871Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.4090","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:387e62a508f5cff94cb7d07cff8dda7cc5488c8e4231cbf423ac4b23b8511953","sha256:bd305db4aa53f8d5c4a5445c8698ef8bb1c4903c8ff50310b19c2142eee0ed94"],"state_sha256":"df49525f5707fdcd2fda2d8137a90e1cd0d151aeb129df6a6b866f4b4462cd58"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FbFtLsIGhIy33A8Ur/LGqadSxB0uZOZhMQJNKHtOyAisoi0TkuXYXclNteZIW9CGUzLWeXjImx0TwTa6EkAGDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T00:58:49.106518Z","bundle_sha256":"b65540d0b850cb49f3be483c7486889983fd639ff39c89f329adcb21eabfde9b"}}