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For arbitrary non-finitely generated groups, we show that the cancellation property is $\\Pi^1_1$ $m$-hard; we know of no upper bound, but we conjecture that it is $\\Pi^1_2$ $m$-complete."},"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":"1809.07191","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-09-19T13:43:53Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"aa51177db8ee6e7ad7d00a1268a38d2967dbe2f038bb15b73dbf13119385f161","abstract_canon_sha256":"15fcb8338a16d5f155b9b197a627c245595c09b0954b7900d252968f90afa051"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:20.459570Z","signature_b64":"bsTaR/qRvkFrMAPKgLhpTdESboRFdcDqo1EnUVJoKWni5yJ/xAqqX8PdHePt0FMp//hWDy5Y5/8PgznsA+iyAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2939a892de52def19cc26206bcea98bd2b7c248b18a03d6867063e712bbd4b12","last_reissued_at":"2026-05-18T00:05:20.458978Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:20.458978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterizations of Cancellable Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.LO","authors_text":"Matthew Harrison-Trainor, Meng-Che \"Turbo\" Ho","submitted_at":"2018-09-19T13:43:53Z","abstract_excerpt":"An abelian group $A$ is said to be cancellable if whenever $A \\oplus G$ is isomorphic to $A \\oplus H$, $G$ is isomorphic to $H$. 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