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The main result is the following theorem:\n  Theorem: Suppose $\\kappa$ is a singular strong limit cardinal and $2^\\kappa >= \\lambda$ where $\\lambda$ is not the successor of a cardinal of cofinality at most $\\kappa$.\n (i) If $\\cofinality(\\kappa)>\\gw$ then $o(\\kappa)\\ge\\lambda$.\n (ii) If $\\cofinality(\\kappa)=\\gw$ then either $o(\\kappa)\\ge\\lambda$ or $\\set{\\ga:K\\sat o(\\ga)\\ge\\ga^{+n}}$ is cofinal in $\\kappa$ for each $n\\in\\gw$.\n  In order to prove this theorem we give a d"},"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/9507214","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1995-07-27T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"70409033dfccaebfc6834b120bf7927f4e8eb98405162103ca00fa3a3c43497f","abstract_canon_sha256":"0783708ee19a33d023c007ef0d455da4ab57d2d1e478791ab68efa714f25f760"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:48.497602Z","signature_b64":"zrvc4oomj8XodVM+OX3QsYopkda5U2Wn1L8XfeTyFawWNsapK/U35GUI726G5RyOWu4Df077eO8EmKv+qWi1AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2659b2b6e3bf8058d268e329cbe18a8b281dacbdce7c1c3bd796309008d76271","last_reissued_at":"2026-05-18T01:05:48.497133Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:48.497133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Indiscernible Sequences for Extenders, and the Singular Cardinal Hypothesis","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Moti Gitik, William Mitchell","submitted_at":"1995-07-27T00:00:00Z","abstract_excerpt":"We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis.  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