{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ISGH6VU7IOB3JGKRJHS3BEYN3O","short_pith_number":"pith:ISGH6VU7","canonical_record":{"source":{"id":"2606.20181","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2026-06-18T12:51:58Z","cross_cats_sorted":[],"title_canon_sha256":"d6e058183dba09dc4e2b7a7e66c5f5f2edc5fd6625a8d08d7a5d1737b2d2c5c9","abstract_canon_sha256":"0b7af09695b75863cc4d8d4463a65392cf3ae20bc3971f9632b37f163a64eb06"},"schema_version":"1.0"},"canonical_sha256":"448c7f569f4383b4995149e5b0930ddbaf4bad60fe4d50a665505af294e37e30","source":{"kind":"arxiv","id":"2606.20181","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.20181","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"arxiv_version","alias_value":"2606.20181v1","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.20181","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"pith_short_12","alias_value":"ISGH6VU7IOB3","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"pith_short_16","alias_value":"ISGH6VU7IOB3JGKR","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"pith_short_8","alias_value":"ISGH6VU7","created_at":"2026-06-19T16:13:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ISGH6VU7IOB3JGKRJHS3BEYN3O","target":"record","payload":{"canonical_record":{"source":{"id":"2606.20181","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2026-06-18T12:51:58Z","cross_cats_sorted":[],"title_canon_sha256":"d6e058183dba09dc4e2b7a7e66c5f5f2edc5fd6625a8d08d7a5d1737b2d2c5c9","abstract_canon_sha256":"0b7af09695b75863cc4d8d4463a65392cf3ae20bc3971f9632b37f163a64eb06"},"schema_version":"1.0"},"canonical_sha256":"448c7f569f4383b4995149e5b0930ddbaf4bad60fe4d50a665505af294e37e30","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:13:04.992152Z","signature_b64":"d0hWc7RW/HXw9T2wCk29cSMDmWfWeOnBCiW0abTEgvXDRyb+UR1tpTFZ3RmcbR7CQfyAUcWNrRLXQcs+9kBzAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"448c7f569f4383b4995149e5b0930ddbaf4bad60fe4d50a665505af294e37e30","last_reissued_at":"2026-06-19T16:13:04.991668Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:13:04.991668Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.20181","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-19T16:13:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k4VrtoowNGoefvUVeuKmbO/WpDu3lOzboRT37KkSykYCQIfeQNdCyFiY6638AD8qDMOhW08ayLPGX0iqdMlyDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:06:25.417247Z"},"content_sha256":"c55511b3d10d218be56c5c3fdeef191e96d3481b1f93f7dc8e57ea13e53a5686","schema_version":"1.0","event_id":"sha256:c55511b3d10d218be56c5c3fdeef191e96d3481b1f93f7dc8e57ea13e53a5686"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ISGH6VU7IOB3JGKRJHS3BEYN3O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the relation between the product of KK-groups and the KK-group of the product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Diego Mart\\'inez","submitted_at":"2026-06-18T12:51:58Z","abstract_excerpt":"We observe that the canonical map \\(KK(A, \\prod_{n \\in \\mathbb{N}} B_n) \\to \\prod_{n \\in \\mathbb{N}} KK(A,B_n)\\) is an isomorphism of abelian groups whenever \\(A\\) enjoys the Universal Coefficient Theorem and \\(B_n\\) are unital, simple and purely infinite C*-algebras. This clarifies an aspect of previous work of Dadarlat--Eilers and Tikuisis--White--Winter."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.20181","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/2606.20181/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-19T16:13:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zybCntuNk8/O8+XjI9ZwWaPz8jJdN1spmq3jjhuvdWYBatwGAz+RQ9AG29JoYz0b4QlHj1XZBnJWDZECAbGyAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:06:25.417716Z"},"content_sha256":"0bc570eed48b0a2fd18f8a929a0f5d69957ac0a9d592f6852f04fbd07d8181ac","schema_version":"1.0","event_id":"sha256:0bc570eed48b0a2fd18f8a929a0f5d69957ac0a9d592f6852f04fbd07d8181ac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ISGH6VU7IOB3JGKRJHS3BEYN3O/bundle.json","state_url":"https://pith.science/pith/ISGH6VU7IOB3JGKRJHS3BEYN3O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ISGH6VU7IOB3JGKRJHS3BEYN3O/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":"2026-06-21T21:06:25Z","links":{"resolver":"https://pith.science/pith/ISGH6VU7IOB3JGKRJHS3BEYN3O","bundle":"https://pith.science/pith/ISGH6VU7IOB3JGKRJHS3BEYN3O/bundle.json","state":"https://pith.science/pith/ISGH6VU7IOB3JGKRJHS3BEYN3O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ISGH6VU7IOB3JGKRJHS3BEYN3O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ISGH6VU7IOB3JGKRJHS3BEYN3O","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":"0b7af09695b75863cc4d8d4463a65392cf3ae20bc3971f9632b37f163a64eb06","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2026-06-18T12:51:58Z","title_canon_sha256":"d6e058183dba09dc4e2b7a7e66c5f5f2edc5fd6625a8d08d7a5d1737b2d2c5c9"},"schema_version":"1.0","source":{"id":"2606.20181","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.20181","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"arxiv_version","alias_value":"2606.20181v1","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.20181","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"pith_short_12","alias_value":"ISGH6VU7IOB3","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"pith_short_16","alias_value":"ISGH6VU7IOB3JGKR","created_at":"2026-06-19T16:13:04Z"},{"alias_kind":"pith_short_8","alias_value":"ISGH6VU7","created_at":"2026-06-19T16:13:04Z"}],"graph_snapshots":[{"event_id":"sha256:0bc570eed48b0a2fd18f8a929a0f5d69957ac0a9d592f6852f04fbd07d8181ac","target":"graph","created_at":"2026-06-19T16:13:04Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.20181/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We observe that the canonical map \\(KK(A, \\prod_{n \\in \\mathbb{N}} B_n) \\to \\prod_{n \\in \\mathbb{N}} KK(A,B_n)\\) is an isomorphism of abelian groups whenever \\(A\\) enjoys the Universal Coefficient Theorem and \\(B_n\\) are unital, simple and purely infinite C*-algebras. This clarifies an aspect of previous work of Dadarlat--Eilers and Tikuisis--White--Winter.","authors_text":"Diego Mart\\'inez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2026-06-18T12:51:58Z","title":"On the relation between the product of KK-groups and the KK-group of the product"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.20181","kind":"arxiv","version":1},"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:c55511b3d10d218be56c5c3fdeef191e96d3481b1f93f7dc8e57ea13e53a5686","target":"record","created_at":"2026-06-19T16:13:04Z","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":"0b7af09695b75863cc4d8d4463a65392cf3ae20bc3971f9632b37f163a64eb06","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2026-06-18T12:51:58Z","title_canon_sha256":"d6e058183dba09dc4e2b7a7e66c5f5f2edc5fd6625a8d08d7a5d1737b2d2c5c9"},"schema_version":"1.0","source":{"id":"2606.20181","kind":"arxiv","version":1}},"canonical_sha256":"448c7f569f4383b4995149e5b0930ddbaf4bad60fe4d50a665505af294e37e30","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"448c7f569f4383b4995149e5b0930ddbaf4bad60fe4d50a665505af294e37e30","first_computed_at":"2026-06-19T16:13:04.991668Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:13:04.991668Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d0hWc7RW/HXw9T2wCk29cSMDmWfWeOnBCiW0abTEgvXDRyb+UR1tpTFZ3RmcbR7CQfyAUcWNrRLXQcs+9kBzAw==","signature_status":"signed_v1","signed_at":"2026-06-19T16:13:04.992152Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.20181","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c55511b3d10d218be56c5c3fdeef191e96d3481b1f93f7dc8e57ea13e53a5686","sha256:0bc570eed48b0a2fd18f8a929a0f5d69957ac0a9d592f6852f04fbd07d8181ac"],"state_sha256":"aa5b0802b055ec81e7e772054aa0b7188c4fe6ae868e82602884d18b88130cf1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QSzUMLKLBR/hpzotgYp7j0ybfWrulPDxroYieYJRicHZH8n07A3bEPSZn/inOx8ChLv/W645F6QHoe4SX/BRAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T21:06:25.419668Z","bundle_sha256":"82d26722f01b5fd659394476566448ce6bdc0090ad968f245dd125e060771f29"}}