{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:V7WWUMM5WE5YHOVVVZIMHP4NIE","short_pith_number":"pith:V7WWUMM5","schema_version":"1.0","canonical_sha256":"afed6a319db13b83bab5ae50c3bf8d41001e0e63f6e77892a4b3ad588e9abcbb","source":{"kind":"arxiv","id":"2606.06122","version":1},"attestation_state":"computed","paper":{"title":"The stable Andrews-Curtis conjecture and thickenable presentations of the trivial group","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Marc Lackenby","submitted_at":"2026-06-04T13:09:26Z","abstract_excerpt":"We establish an explicit upper bound on the number of stable Andrews-Curtis moves that convert thickenable balanced presentations of the trivial group to the standard one-generator presentation. We also present a proof that thickenable balanced presentations of the trivial group satisfy the (unstable) Andrews-Curtis conjecture."},"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":"2606.06122","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-04T13:09:26Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"6a1624f8c1325cdcd606642759033d2bf72b5e7bd9af736b20ceed539b37f1ed","abstract_canon_sha256":"3ff6d5ea4d01975b1d3a24785f16d777d681f79adbff8d6c59d14bb0a3f3450d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T01:15:34.122418Z","signature_b64":"aptE6nkqxo5JlKCumTj0aC6BVJUZMQOjQFNRTbqY4xMiJmla2FOxAs+OZeqPJjOh166CGEKfYekIM8eKcsOxBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afed6a319db13b83bab5ae50c3bf8d41001e0e63f6e77892a4b3ad588e9abcbb","last_reissued_at":"2026-06-05T01:15:34.121752Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T01:15:34.121752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The stable Andrews-Curtis conjecture and thickenable presentations of the trivial group","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Marc Lackenby","submitted_at":"2026-06-04T13:09:26Z","abstract_excerpt":"We establish an explicit upper bound on the number of stable Andrews-Curtis moves that convert thickenable balanced presentations of the trivial group to the standard one-generator presentation. We also present a proof that thickenable balanced presentations of the trivial group satisfy the (unstable) Andrews-Curtis conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06122","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/2606.06122/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":"2606.06122","created_at":"2026-06-05T01:15:34.121883+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.06122v1","created_at":"2026-06-05T01:15:34.121883+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06122","created_at":"2026-06-05T01:15:34.121883+00:00"},{"alias_kind":"pith_short_12","alias_value":"V7WWUMM5WE5Y","created_at":"2026-06-05T01:15:34.121883+00:00"},{"alias_kind":"pith_short_16","alias_value":"V7WWUMM5WE5YHOVV","created_at":"2026-06-05T01:15:34.121883+00:00"},{"alias_kind":"pith_short_8","alias_value":"V7WWUMM5","created_at":"2026-06-05T01:15:34.121883+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/V7WWUMM5WE5YHOVVVZIMHP4NIE","json":"https://pith.science/pith/V7WWUMM5WE5YHOVVVZIMHP4NIE.json","graph_json":"https://pith.science/api/pith-number/V7WWUMM5WE5YHOVVVZIMHP4NIE/graph.json","events_json":"https://pith.science/api/pith-number/V7WWUMM5WE5YHOVVVZIMHP4NIE/events.json","paper":"https://pith.science/paper/V7WWUMM5"},"agent_actions":{"view_html":"https://pith.science/pith/V7WWUMM5WE5YHOVVVZIMHP4NIE","download_json":"https://pith.science/pith/V7WWUMM5WE5YHOVVVZIMHP4NIE.json","view_paper":"https://pith.science/paper/V7WWUMM5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.06122&json=true","fetch_graph":"https://pith.science/api/pith-number/V7WWUMM5WE5YHOVVVZIMHP4NIE/graph.json","fetch_events":"https://pith.science/api/pith-number/V7WWUMM5WE5YHOVVVZIMHP4NIE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V7WWUMM5WE5YHOVVVZIMHP4NIE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V7WWUMM5WE5YHOVVVZIMHP4NIE/action/storage_attestation","attest_author":"https://pith.science/pith/V7WWUMM5WE5YHOVVVZIMHP4NIE/action/author_attestation","sign_citation":"https://pith.science/pith/V7WWUMM5WE5YHOVVVZIMHP4NIE/action/citation_signature","submit_replication":"https://pith.science/pith/V7WWUMM5WE5YHOVVVZIMHP4NIE/action/replication_record"}},"created_at":"2026-06-05T01:15:34.121883+00:00","updated_at":"2026-06-05T01:15:34.121883+00:00"}