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The two hypotheses are independent and play disjoint roles: bi-reversibility is exactly what makes $Y_A$ a complete square complex, so that its universal cover splits as a product of two trees and anti-tori can be discussed at all; and, within that setting, an anti-torus is precisely a period-fr"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.11899","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-06-10T10:26:19Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"4c1ea90925ad9b08e67e36487b6a541997f8287f68449f0fa02cc184a577c8a4","abstract_canon_sha256":"868bbe8ef5e92ade0e3ae508df7c02059b5392f1d036eccded441f175c66e2cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:10:14.462233Z","signature_b64":"vsYti0kjdB/+DrbLgqkYg7hOCl+Stm6dKGr0gE7cFz/Vcy47lIrQY3rtWx9yX0tIMpjAUhgKGBT7qeMp/xdCAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b460ecd7900e7b8fd353bc69c0670a8c3ac70b52b018ec2a104054ac33200ac4","last_reissued_at":"2026-06-11T01:10:14.461288Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:10:14.461288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Full Mealy automata, complete square complexes, and anti-tori","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.GT","authors_text":"David Pask","submitted_at":"2026-06-10T10:26:19Z","abstract_excerpt":"To a full $m\\times n$ Mealy automaton $A$ we associate a bijection $\\theta_A$, a one-vertex rank-two graph $F_{\\theta_A}$, and a one-vertex $VH$-square complex $Y_A$ tiled by $mn$ Wang tiles. We prove that $Y_A$ contains an anti-torus if and only if $A$ is bi-reversible and $F_{\\theta_A}$ is aperiodic. The two hypotheses are independent and play disjoint roles: bi-reversibility is exactly what makes $Y_A$ a complete square complex, so that its universal cover splits as a product of two trees and anti-tori can be discussed at all; and, within that setting, an anti-torus is precisely a period-fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11899","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/2606.11899/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.11899","created_at":"2026-06-11T01:10:14.461474+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.11899v1","created_at":"2026-06-11T01:10:14.461474+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11899","created_at":"2026-06-11T01:10:14.461474+00:00"},{"alias_kind":"pith_short_12","alias_value":"WRQOZV4QBZ5Y","created_at":"2026-06-11T01:10:14.461474+00:00"},{"alias_kind":"pith_short_16","alias_value":"WRQOZV4QBZ5Y7U2T","created_at":"2026-06-11T01:10:14.461474+00:00"},{"alias_kind":"pith_short_8","alias_value":"WRQOZV4Q","created_at":"2026-06-11T01:10:14.461474+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ","json":"https://pith.science/pith/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ.json","graph_json":"https://pith.science/api/pith-number/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ/graph.json","events_json":"https://pith.science/api/pith-number/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ/events.json","paper":"https://pith.science/paper/WRQOZV4Q"},"agent_actions":{"view_html":"https://pith.science/pith/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ","download_json":"https://pith.science/pith/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ.json","view_paper":"https://pith.science/paper/WRQOZV4Q","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.11899&json=true","fetch_graph":"https://pith.science/api/pith-number/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ/graph.json","fetch_events":"https://pith.science/api/pith-number/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ/action/storage_attestation","attest_author":"https://pith.science/pith/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ/action/author_attestation","sign_citation":"https://pith.science/pith/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ/action/citation_signature","submit_replication":"https://pith.science/pith/WRQOZV4QBZ5Y7U2TXRU4AZYKRQ/action/replication_record"}},"created_at":"2026-06-11T01:10:14.461474+00:00","updated_at":"2026-06-11T01:10:14.461474+00:00"}