{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:M7XRH4KAV7AHD3JYNXPACVEHC7","short_pith_number":"pith:M7XRH4KA","canonical_record":{"source":{"id":"1908.05920","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-08-16T10:17:23Z","cross_cats_sorted":[],"title_canon_sha256":"bbb3fa8b96c9350df869c98d97478efbe069a1cec0fb788c2db84a1d4b8e0cff","abstract_canon_sha256":"4b094eff24a1eda2404301ef4b913172bd8197b56fe9cebb89b01b2d4518d28a"},"schema_version":"1.0"},"canonical_sha256":"67ef13f140afc071ed386dde01548717caf44f88b0d06a1cfbd49aacac3ff68f","source":{"kind":"arxiv","id":"1908.05920","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05920","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05920v1","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05920","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"pith_short_12","alias_value":"M7XRH4KAV7AH","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"pith_short_16","alias_value":"M7XRH4KAV7AHD3JY","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"pith_short_8","alias_value":"M7XRH4KA","created_at":"2026-07-04T23:57:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:M7XRH4KAV7AHD3JYNXPACVEHC7","target":"record","payload":{"canonical_record":{"source":{"id":"1908.05920","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-08-16T10:17:23Z","cross_cats_sorted":[],"title_canon_sha256":"bbb3fa8b96c9350df869c98d97478efbe069a1cec0fb788c2db84a1d4b8e0cff","abstract_canon_sha256":"4b094eff24a1eda2404301ef4b913172bd8197b56fe9cebb89b01b2d4518d28a"},"schema_version":"1.0"},"canonical_sha256":"67ef13f140afc071ed386dde01548717caf44f88b0d06a1cfbd49aacac3ff68f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:57:54.428370Z","signature_b64":"rWGyYQZxU5uqFKouzJYRAkETKoQwzf0V9BSQlrqFTyM773Qnp0f+hIy44vnLrFwML8+tZ60r6Y8Hs7gV4BnECA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67ef13f140afc071ed386dde01548717caf44f88b0d06a1cfbd49aacac3ff68f","last_reissued_at":"2026-07-04T23:57:54.428020Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:57:54.428020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.05920","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:57:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LvGaQgrZZZcAGKqWFvzrBOmDExp8zIhsXoRkmZoKjpXaRgdlMu9iSuRcxfZo11mvEVEOluUGblAJ32uOjYt9AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T09:05:43.342786Z"},"content_sha256":"5402bd6185480ddbd78b8aa00b4a9f5569a1afed55fb4c67fdedaf211589f710","schema_version":"1.0","event_id":"sha256:5402bd6185480ddbd78b8aa00b4a9f5569a1afed55fb4c67fdedaf211589f710"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:M7XRH4KAV7AHD3JYNXPACVEHC7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalisations of Stationarity, Closed and Unboundedness, and of Jensen's $\\Box$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"H. Brickhill, P. D. Welch","submitted_at":"2019-08-16T10:17:23Z","abstract_excerpt":"The concepts of closed unbounded (club) and stationary sets are generalised to $\\gamma$-club and $\\gamma$-stationary sets, which are closely related to stationary reflection. We use these notions to define generalisations of Jensen's combinatorial principles {$\\Box$} and $\\diamondsuit$.\n  We define $\\Pi^1_\\gamma$-indescribability and use the new $\\Box^{\\gamma}$-sequences to extend the result of Jensen that in the constructible universe a regular cardinal is stationary reflecting if and only if it is $\\Pi^1_1$-indescribable: we show that in $L$ a cardinal is $\\Pi^{1}_{\\gamma} $-indescribable if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05920","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.05920/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:57:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iHdIqr4A4oN4oJXfjdpM6cpn8mSUHjo3azkc+e9mgmKiWsxdYHinZvTPnrGEti/spoyZNggNWbKq3B4PADqEDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T09:05:43.343306Z"},"content_sha256":"f53666143b1df25581ba77d3ff0b490b19a07cbc7f31119293e4acdf4378a9ff","schema_version":"1.0","event_id":"sha256:f53666143b1df25581ba77d3ff0b490b19a07cbc7f31119293e4acdf4378a9ff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M7XRH4KAV7AHD3JYNXPACVEHC7/bundle.json","state_url":"https://pith.science/pith/M7XRH4KAV7AHD3JYNXPACVEHC7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M7XRH4KAV7AHD3JYNXPACVEHC7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-16T09:05:43Z","links":{"resolver":"https://pith.science/pith/M7XRH4KAV7AHD3JYNXPACVEHC7","bundle":"https://pith.science/pith/M7XRH4KAV7AHD3JYNXPACVEHC7/bundle.json","state":"https://pith.science/pith/M7XRH4KAV7AHD3JYNXPACVEHC7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M7XRH4KAV7AHD3JYNXPACVEHC7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:M7XRH4KAV7AHD3JYNXPACVEHC7","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":"4b094eff24a1eda2404301ef4b913172bd8197b56fe9cebb89b01b2d4518d28a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-08-16T10:17:23Z","title_canon_sha256":"bbb3fa8b96c9350df869c98d97478efbe069a1cec0fb788c2db84a1d4b8e0cff"},"schema_version":"1.0","source":{"id":"1908.05920","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05920","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05920v1","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05920","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"pith_short_12","alias_value":"M7XRH4KAV7AH","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"pith_short_16","alias_value":"M7XRH4KAV7AHD3JY","created_at":"2026-07-04T23:57:54Z"},{"alias_kind":"pith_short_8","alias_value":"M7XRH4KA","created_at":"2026-07-04T23:57:54Z"}],"graph_snapshots":[{"event_id":"sha256:f53666143b1df25581ba77d3ff0b490b19a07cbc7f31119293e4acdf4378a9ff","target":"graph","created_at":"2026-07-04T23:57:54Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.05920/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The concepts of closed unbounded (club) and stationary sets are generalised to $\\gamma$-club and $\\gamma$-stationary sets, which are closely related to stationary reflection. We use these notions to define generalisations of Jensen's combinatorial principles {$\\Box$} and $\\diamondsuit$.\n  We define $\\Pi^1_\\gamma$-indescribability and use the new $\\Box^{\\gamma}$-sequences to extend the result of Jensen that in the constructible universe a regular cardinal is stationary reflecting if and only if it is $\\Pi^1_1$-indescribable: we show that in $L$ a cardinal is $\\Pi^{1}_{\\gamma} $-indescribable if","authors_text":"H. Brickhill, P. D. Welch","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-08-16T10:17:23Z","title":"Generalisations of Stationarity, Closed and Unboundedness, and of Jensen's $\\Box$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05920","kind":"arxiv","version":1},"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":"sha256:5402bd6185480ddbd78b8aa00b4a9f5569a1afed55fb4c67fdedaf211589f710","target":"record","created_at":"2026-07-04T23:57:54Z","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":"4b094eff24a1eda2404301ef4b913172bd8197b56fe9cebb89b01b2d4518d28a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-08-16T10:17:23Z","title_canon_sha256":"bbb3fa8b96c9350df869c98d97478efbe069a1cec0fb788c2db84a1d4b8e0cff"},"schema_version":"1.0","source":{"id":"1908.05920","kind":"arxiv","version":1}},"canonical_sha256":"67ef13f140afc071ed386dde01548717caf44f88b0d06a1cfbd49aacac3ff68f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"67ef13f140afc071ed386dde01548717caf44f88b0d06a1cfbd49aacac3ff68f","first_computed_at":"2026-07-04T23:57:54.428020Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:57:54.428020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rWGyYQZxU5uqFKouzJYRAkETKoQwzf0V9BSQlrqFTyM773Qnp0f+hIy44vnLrFwML8+tZ60r6Y8Hs7gV4BnECA==","signature_status":"signed_v1","signed_at":"2026-07-04T23:57:54.428370Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.05920","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5402bd6185480ddbd78b8aa00b4a9f5569a1afed55fb4c67fdedaf211589f710","sha256:f53666143b1df25581ba77d3ff0b490b19a07cbc7f31119293e4acdf4378a9ff"],"state_sha256":"29bd79b3d02f58ddc9e35def8c3d2cbbf9fbce5a68e13c66927473a01079582d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kCeNRim402qwibXTvKQVjpN/i5IR/cReTXftGMY/iyJn/J4Rud8akigNdPLmh16Ea41eGeD535erJY/sVJSzCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T09:05:43.348449Z","bundle_sha256":"7c803d9e69a0a42fd6b9ca61fa846768357bd29abce3efc71621566c11b5cdc7"}}