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We call a function $f$ on $R_\\infty$ positive definite if for all $r_1, r_2, \\ldots, r"},"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":"2508.13760","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2025-08-19T11:57:29Z","cross_cats_sorted":[],"title_canon_sha256":"00d09e7e026305335775c24657b7ae33f5c5e3a64faf7f9ce91dbba1d15bafb2","abstract_canon_sha256":"3152e90a34313c68cbb7fff94bbd041b3a2e72393079ac602c8f6fb229de3dfa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:07:06.670587Z","signature_b64":"wU/zWEfbgKhytuS6HHXzj+PSvMhybRO61tnBDqh/6AkGR4UvDUOmbQdAJJaKLDIb9rJ3QCAoVUIQCzgnqkhQBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10fdc778d4347f579356aa5c2a08c38f04c650444e7a95d47227dd6da4fc2241","last_reissued_at":"2026-06-09T02:07:06.669408Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:07:06.669408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classification of indecomposable states on the infinite symmetric inverse semigroup invariant under the infinite symmetric group. 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