{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7WA6RDMHGDOZ2GZTHXV4V47ZGT","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"402e649d346f4b49e6193894021bba2e90a5f02381c3a73d2250f38c44e48e0b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-10-07T16:52:41Z","title_canon_sha256":"45f1fe6b0db14e386a5e68bfc0e9d491e8f014cda28392b8841053c241a50386"},"schema_version":"1.0","source":{"id":"1110.1584","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1584","created_at":"2026-05-18T04:10:47Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1584v3","created_at":"2026-05-18T04:10:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1584","created_at":"2026-05-18T04:10:47Z"},{"alias_kind":"pith_short_12","alias_value":"7WA6RDMHGDOZ","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7WA6RDMHGDOZ2GZT","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7WA6RDMH","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:1f38b20c779d13a9b39fe341b2d570be764135ddf82c165b164cbcb2a4c9b3c1","target":"graph","created_at":"2026-05-18T04:10:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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. 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","authors_text":"Matteo Viale, Sean Cox","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-10-07T16:52:41Z","title":"Martin's Maximum and tower forcing"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1584","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:73917abb515a852d83ca3f84670f1a9b14d1a9cf9150f2aa99141f4b6f719580","target":"record","created_at":"2026-05-18T04:10:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"402e649d346f4b49e6193894021bba2e90a5f02381c3a73d2250f38c44e48e0b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-10-07T16:52:41Z","title_canon_sha256":"45f1fe6b0db14e386a5e68bfc0e9d491e8f014cda28392b8841053c241a50386"},"schema_version":"1.0","source":{"id":"1110.1584","kind":"arxiv","version":3}},"canonical_sha256":"fd81e88d8730dd9d1b333debcaf3f934d12c14133a4757d3cfcc3d33c998d43a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd81e88d8730dd9d1b333debcaf3f934d12c14133a4757d3cfcc3d33c998d43a","first_computed_at":"2026-05-18T04:10:47.932238Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:10:47.932238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SqLdz7AWNQtC12K8d5ckD+YR5rCltFNIEuYBzMsozgVCjYl8a8qpv/8GC5DtRyMUePUZ7K0dneL/y/yo5fIdDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:10:47.932649Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.1584","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73917abb515a852d83ca3f84670f1a9b14d1a9cf9150f2aa99141f4b6f719580","sha256:1f38b20c779d13a9b39fe341b2d570be764135ddf82c165b164cbcb2a4c9b3c1"],"state_sha256":"7ae803c544148fc82f4cbe6867bdb7cb97f370f929e1d8d3adedcacb00dcdf57"}