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We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\\vec{\\mathcal{I}}$ is a tower of ideals which concentrates on the class $GIC_{\\omega_1}$ of $\\omega_1$-guessing, internally club sets, then $\\v"},"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":"1110.1584","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-10-07T16:52:41Z","cross_cats_sorted":[],"title_canon_sha256":"45f1fe6b0db14e386a5e68bfc0e9d491e8f014cda28392b8841053c241a50386","abstract_canon_sha256":"402e649d346f4b49e6193894021bba2e90a5f02381c3a73d2250f38c44e48e0b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:47.932649Z","signature_b64":"SqLdz7AWNQtC12K8d5ckD+YR5rCltFNIEuYBzMsozgVCjYl8a8qpv/8GC5DtRyMUePUZ7K0dneL/y/yo5fIdDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd81e88d8730dd9d1b333debcaf3f934d12c14133a4757d3cfcc3d33c998d43a","last_reissued_at":"2026-05-18T04:10:47.932238Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:47.932238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Martin's Maximum and tower forcing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Matteo Viale, Sean Cox","submitted_at":"2011-10-07T16:52:41Z","abstract_excerpt":"There are several examples in the literature showing that compactness-like properties of a cardinal $\\kappa$ cause poor behavior of some generic ultrapowers which have critical point $\\kappa$ (Burke \\cite{MR1472122} when $\\kappa$ is a supercompact cardinal; Foreman-Magidor \\cite{MR1359154} when $\\kappa = \\omega_2$ in the presence of strong forcing axioms). We prove more instances of this phenomenon. 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