{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:7TBF65D6DU7U46LJ26L6FAWWJS","short_pith_number":"pith:7TBF65D6","canonical_record":{"source":{"id":"math/0410374","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2004-10-17T18:25:10Z","cross_cats_sorted":[],"title_canon_sha256":"8513f1c0cb7940826b17a29c12672fabe37c2c25db028906a5459fff5574c69a","abstract_canon_sha256":"0ab65c8b39d9756ac13bcce334c91a7aeaf96136d8b2cb92fbb863d21539dce8"},"schema_version":"1.0"},"canonical_sha256":"fcc25f747e1d3f4e7969d797e282d64cbd9e243115910b445c1811bd9c098913","source":{"kind":"arxiv","id":"math/0410374","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0410374","created_at":"2026-05-18T02:41:32Z"},{"alias_kind":"arxiv_version","alias_value":"math/0410374v2","created_at":"2026-05-18T02:41:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410374","created_at":"2026-05-18T02:41:32Z"},{"alias_kind":"pith_short_12","alias_value":"7TBF65D6DU7U","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"7TBF65D6DU7U46LJ","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"7TBF65D6","created_at":"2026-05-18T12:25:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:7TBF65D6DU7U46LJ26L6FAWWJS","target":"record","payload":{"canonical_record":{"source":{"id":"math/0410374","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2004-10-17T18:25:10Z","cross_cats_sorted":[],"title_canon_sha256":"8513f1c0cb7940826b17a29c12672fabe37c2c25db028906a5459fff5574c69a","abstract_canon_sha256":"0ab65c8b39d9756ac13bcce334c91a7aeaf96136d8b2cb92fbb863d21539dce8"},"schema_version":"1.0"},"canonical_sha256":"fcc25f747e1d3f4e7969d797e282d64cbd9e243115910b445c1811bd9c098913","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:32.256919Z","signature_b64":"EFtJcWueEVKtTRuV8z2Hjl//6B58wcSEX7FjtvTnShbdgr5OKbm6Dzb/KBgsOaHIEI2Z9Gzr3OKAG5B5VSeoDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcc25f747e1d3f4e7969d797e282d64cbd9e243115910b445c1811bd9c098913","last_reissued_at":"2026-05-18T02:41:32.256579Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:32.256579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0410374","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":"2026-05-18T02:41:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MPmoRoDM3c9QcUzt8kjYO4CcKSXJU/M8FevZN/FgJNya0QNfTHq+PmPzi3VUxfTsjAun/exNGg6JYYrGRlXVDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:20:49.142823Z"},"content_sha256":"a20f49a6818f654862fe1afca7c20e154cd8c8df1655c5f6833826d4918436b0","schema_version":"1.0","event_id":"sha256:a20f49a6818f654862fe1afca7c20e154cd8c8df1655c5f6833826d4918436b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:7TBF65D6DU7U46LJ26L6FAWWJS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Implications of the Ganea Condition","license":"","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Donald Stanley, Jeffrey Strom, Norio Iwase","submitted_at":"2004-10-17T18:25:10Z","abstract_excerpt":"Suppose the spaces X and X cross A have the same Lusternik-Schnirelmann category: cat(X cross A)= cat(X). Then there is a strict inequality cat(X cross (A halfsmash B)) < cat (X) + cat(A halfsmash B) for every space B, provided the connectivity of A is large enough (depending only on X). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410374","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":"2026-05-18T02:41:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mv4f4iuRG3v4HSBkTqaju7BNgGLGdggaq5P0juDJpVPAuAVw6tP/gXYkoRwJSbNgMxb1vHO/qpr6sPzhNZArDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:20:49.143474Z"},"content_sha256":"2ec5162b31159b531f7969bd27f3ddb67dd6eade044483a463860e4d4d694c71","schema_version":"1.0","event_id":"sha256:2ec5162b31159b531f7969bd27f3ddb67dd6eade044483a463860e4d4d694c71"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7TBF65D6DU7U46LJ26L6FAWWJS/bundle.json","state_url":"https://pith.science/pith/7TBF65D6DU7U46LJ26L6FAWWJS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7TBF65D6DU7U46LJ26L6FAWWJS/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":"2026-05-27T23:20:49Z","links":{"resolver":"https://pith.science/pith/7TBF65D6DU7U46LJ26L6FAWWJS","bundle":"https://pith.science/pith/7TBF65D6DU7U46LJ26L6FAWWJS/bundle.json","state":"https://pith.science/pith/7TBF65D6DU7U46LJ26L6FAWWJS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7TBF65D6DU7U46LJ26L6FAWWJS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:7TBF65D6DU7U46LJ26L6FAWWJS","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":"0ab65c8b39d9756ac13bcce334c91a7aeaf96136d8b2cb92fbb863d21539dce8","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2004-10-17T18:25:10Z","title_canon_sha256":"8513f1c0cb7940826b17a29c12672fabe37c2c25db028906a5459fff5574c69a"},"schema_version":"1.0","source":{"id":"math/0410374","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0410374","created_at":"2026-05-18T02:41:32Z"},{"alias_kind":"arxiv_version","alias_value":"math/0410374v2","created_at":"2026-05-18T02:41:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410374","created_at":"2026-05-18T02:41:32Z"},{"alias_kind":"pith_short_12","alias_value":"7TBF65D6DU7U","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"7TBF65D6DU7U46LJ","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"7TBF65D6","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:2ec5162b31159b531f7969bd27f3ddb67dd6eade044483a463860e4d4d694c71","target":"graph","created_at":"2026-05-18T02:41:32Z","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":"Suppose the spaces X and X cross A have the same Lusternik-Schnirelmann category: cat(X cross A)= cat(X). Then there is a strict inequality cat(X cross (A halfsmash B)) < cat (X) + cat(A halfsmash B) for every space B, provided the connectivity of A is large enough (depending only on X). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres.","authors_text":"Donald Stanley, Jeffrey Strom, Norio Iwase","cross_cats":[],"headline":"","license":"","primary_cat":"math.AT","submitted_at":"2004-10-17T18:25:10Z","title":"Implications of the Ganea Condition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410374","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:a20f49a6818f654862fe1afca7c20e154cd8c8df1655c5f6833826d4918436b0","target":"record","created_at":"2026-05-18T02:41:32Z","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":"0ab65c8b39d9756ac13bcce334c91a7aeaf96136d8b2cb92fbb863d21539dce8","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2004-10-17T18:25:10Z","title_canon_sha256":"8513f1c0cb7940826b17a29c12672fabe37c2c25db028906a5459fff5574c69a"},"schema_version":"1.0","source":{"id":"math/0410374","kind":"arxiv","version":2}},"canonical_sha256":"fcc25f747e1d3f4e7969d797e282d64cbd9e243115910b445c1811bd9c098913","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fcc25f747e1d3f4e7969d797e282d64cbd9e243115910b445c1811bd9c098913","first_computed_at":"2026-05-18T02:41:32.256579Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:32.256579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EFtJcWueEVKtTRuV8z2Hjl//6B58wcSEX7FjtvTnShbdgr5OKbm6Dzb/KBgsOaHIEI2Z9Gzr3OKAG5B5VSeoDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:32.256919Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0410374","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a20f49a6818f654862fe1afca7c20e154cd8c8df1655c5f6833826d4918436b0","sha256:2ec5162b31159b531f7969bd27f3ddb67dd6eade044483a463860e4d4d694c71"],"state_sha256":"281582fbd71b4fb564096c902a357da5e7d4cb58974e6833cb34dc77bf86e851"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rd6G8yAOpyEwwa1RJRy+gWijjOf/U9swaWzY9h7ewzZ8A7/8O62lrWJkAi64i0reF0934vukjA7aBmnmKHe6AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T23:20:49.146506Z","bundle_sha256":"8ce42ef28e3902b65dc811a962ba2b98a098b9b4c31c9f69e7257ac26d9b64db"}}