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We prove that I(T,\\kappa), as a function of \\kappa > \\aleph_0, is the minimum of 2^{\\kappa} and one of the following functions:\n  1. 2^{\\kappa};\n  2. the constant function 1;\n  3. |\\hat\\mu^n/{\\sim_G}|-|(\\hat\\mu - 1)^n/{\\sim_G}| if \\hat\\mu<\\omega for some 1<n<\\omega and \\hat\\mu if \\hat\\"},"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":"math/0007199","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"2000-07-01T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"3c1a8bb675747db231e262fa72b250526d70ac8c442b4d01328eecd3be03f4cd","abstract_canon_sha256":"ba8f7def2fb185005b9f3edd9b5247131ea07b2745de2b88dbc56dcbfd5c7b31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:38.449982Z","signature_b64":"Mpi9D7tsixjbvO1d4+F3tbxHTOP5+fB/VQhc9zJC77zQWLURtYgrBs8kdBAUOnjBfD00eNMWX/lYc/R9fZbQCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd6264762731658225e4c1ae1ccf1dea01204dea4916008202142df3d8142edf","last_reissued_at":"2026-05-18T01:05:38.449489Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:38.449489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The uncountable spectra of countable theories","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Bradd Hart, Ehud Hrushovski, Michael C. 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