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We obtain a concrete necessary and sufficient condition for the subnormality of $\\mathbf{T}\\equiv (T_{1},T_{2})\\in \\mathcal{TC}$ in terms of $c(\\mathbf{T})$, the marginal measures of $T_{1}$ and $T_{2}$, and the weight $\\alpha_{01}$."},"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":"0710.3977","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"2007-10-22T04:32:21Z","cross_cats_sorted":[],"title_canon_sha256":"382e191f9f3564d83e7c908884221aeaa912a8b49121ce8df46edd3d0568b659","abstract_canon_sha256":"48f3612dee70dc01927cf1c69a4f63a7899388dea419418cad5f851d94605817"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:05:05.250368Z","signature_b64":"jUQsqaCzKHaL8poNPbnmNhy0LpLuoXKTollPv9UDxkqWEVv0QnBXRIXUN2UvavP294ShkUZ381hRqmPiWmbkAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27ad6f3dd212fee3f2fc82d9a093a5a8fbae2a4aebb8964cbd8682dd080b02df","last_reissued_at":"2026-07-04T15:05:05.249988Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:05:05.249988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reconstruction of the Berger measure when the core is of tensor form","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jasang Yoon, Raul E. 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