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We say that a subgroup $H$ of $G$ is: (i) \\emph{uniformly controllable} in $G$ provided that for every finite set $J\\subseteq I$ there exists a finite set $K\\subseteq I$ such that $p_{J}(H)=p_{J}(H\\cap\\bigoplus_{i\\in K} G_i)$; (ii) \\emph{controllable} in $G$ provided that $p_{J}(H)=p_{J}(H\\cap\\bigoplus_{i\\in I} G_i)$ for every finite set $J\\subseteq I$; (iii) \\emph{weakly controllable} in "},"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":"1306.3954","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2013-06-17T19:10:43Z","cross_cats_sorted":["cs.IT","math.GR","math.IT"],"title_canon_sha256":"d58f5a584ffc21c251caae39ac0239690b8dc79afb243e5b402d98ffe1d89d52","abstract_canon_sha256":"acf030eefeba89a87f2bad19a4c7a68c5da8f36fb90c58d14bdae4afeca303c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:32.836221Z","signature_b64":"edGYA21fPFq8vSuV4izlD+cSAJxwGhS3UNDcGpDCKsmNhlfmZ6nL9GuiZzr3jyQQKbP4K7+pzDZbbWJvKEU9Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46d7223fe9407e3cec51b83b36214f55265deb4dfba92306cfc5dc41814f9445","last_reissued_at":"2026-05-18T00:32:32.835714Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:32.835714Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Subgroups of direct products closely approximated by direct sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.GR","math.IT"],"primary_cat":"math.GN","authors_text":"Dmitri Shakhmatov, Maria V. 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We say that a subgroup $H$ of $G$ is: (i) \\emph{uniformly controllable} in $G$ provided that for every finite set $J\\subseteq I$ there exists a finite set $K\\subseteq I$ such that $p_{J}(H)=p_{J}(H\\cap\\bigoplus_{i\\in K} G_i)$; (ii) \\emph{controllable} in $G$ provided that $p_{J}(H)=p_{J}(H\\cap\\bigoplus_{i\\in I} G_i)$ for every finite set $J\\subseteq I$; (iii) \\emph{weakly controllable} in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3954","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1306.3954","created_at":"2026-05-18T00:32:32.835788+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.3954v1","created_at":"2026-05-18T00:32:32.835788+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.3954","created_at":"2026-05-18T00:32:32.835788+00:00"},{"alias_kind":"pith_short_12","alias_value":"I3LSEP7JIB7D","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_16","alias_value":"I3LSEP7JIB7DZ3CR","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_8","alias_value":"I3LSEP7J","created_at":"2026-05-18T12:27:46.883200+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/I3LSEP7JIB7DZ3CRXA5TMIKPKU","json":"https://pith.science/pith/I3LSEP7JIB7DZ3CRXA5TMIKPKU.json","graph_json":"https://pith.science/api/pith-number/I3LSEP7JIB7DZ3CRXA5TMIKPKU/graph.json","events_json":"https://pith.science/api/pith-number/I3LSEP7JIB7DZ3CRXA5TMIKPKU/events.json","paper":"https://pith.science/paper/I3LSEP7J"},"agent_actions":{"view_html":"https://pith.science/pith/I3LSEP7JIB7DZ3CRXA5TMIKPKU","download_json":"https://pith.science/pith/I3LSEP7JIB7DZ3CRXA5TMIKPKU.json","view_paper":"https://pith.science/paper/I3LSEP7J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.3954&json=true","fetch_graph":"https://pith.science/api/pith-number/I3LSEP7JIB7DZ3CRXA5TMIKPKU/graph.json","fetch_events":"https://pith.science/api/pith-number/I3LSEP7JIB7DZ3CRXA5TMIKPKU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I3LSEP7JIB7DZ3CRXA5TMIKPKU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I3LSEP7JIB7DZ3CRXA5TMIKPKU/action/storage_attestation","attest_author":"https://pith.science/pith/I3LSEP7JIB7DZ3CRXA5TMIKPKU/action/author_attestation","sign_citation":"https://pith.science/pith/I3LSEP7JIB7DZ3CRXA5TMIKPKU/action/citation_signature","submit_replication":"https://pith.science/pith/I3LSEP7JIB7DZ3CRXA5TMIKPKU/action/replication_record"}},"created_at":"2026-05-18T00:32:32.835788+00:00","updated_at":"2026-05-18T00:32:32.835788+00:00"}