{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:DRJGGHUECFADD7E4MJRKIOZREU","short_pith_number":"pith:DRJGGHUE","canonical_record":{"source":{"id":"2605.21873","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2026-05-21T01:34:47Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"050e279fcca48892559d24cdad72894791d0423f2fc4c9b6a377af9828c49976","abstract_canon_sha256":"7648a3dbd6f34f9de0c1905eb2ea3f31fdeef75b838b91fe36ebf14d9459c73f"},"schema_version":"1.0"},"canonical_sha256":"1c52631e84114031fc9c6262a43b31252049e58043ca82241c3d7cc69598e387","source":{"kind":"arxiv","id":"2605.21873","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.21873","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"arxiv_version","alias_value":"2605.21873v1","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21873","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"pith_short_12","alias_value":"DRJGGHUECFAD","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"pith_short_16","alias_value":"DRJGGHUECFADD7E4","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"pith_short_8","alias_value":"DRJGGHUE","created_at":"2026-05-22T01:04:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:DRJGGHUECFADD7E4MJRKIOZREU","target":"record","payload":{"canonical_record":{"source":{"id":"2605.21873","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2026-05-21T01:34:47Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"050e279fcca48892559d24cdad72894791d0423f2fc4c9b6a377af9828c49976","abstract_canon_sha256":"7648a3dbd6f34f9de0c1905eb2ea3f31fdeef75b838b91fe36ebf14d9459c73f"},"schema_version":"1.0"},"canonical_sha256":"1c52631e84114031fc9c6262a43b31252049e58043ca82241c3d7cc69598e387","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:12.438934Z","signature_b64":"GzZ0ETDwlkuV28b3+uT04/2lnXqbgpssmwXVc/fMxDy3RLX99IQgTB5iRYjeNehrgN4/iUIhJC9Ns85MZnYfDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c52631e84114031fc9c6262a43b31252049e58043ca82241c3d7cc69598e387","last_reissued_at":"2026-05-22T01:04:12.438195Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:12.438195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.21873","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":"2026-05-22T01:04:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OMWB09UG4w89ZUUGFdyv/7I0dbBUlKVetCntRw/MKKtTgizPfSKCgbV3Zul8ujsPsB2e8QAvpS0taxGB+ByKDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T05:36:41.714066Z"},"content_sha256":"cf3c6d768757e17372ad1a77acfd8ca5308a2e4cead56ffda480373ebc4759ea","schema_version":"1.0","event_id":"sha256:cf3c6d768757e17372ad1a77acfd8ca5308a2e4cead56ffda480373ebc4759ea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:DRJGGHUECFADD7E4MJRKIOZREU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ultrapowers of spectral subspaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.OA","authors_text":"Hiroshi Ando, Isaac Goldbring","submitted_at":"2026-05-21T01:34:47Z","abstract_excerpt":"We prove, for any W$^*$-probability space $(M,\\varphi)$ where $M$ is a type $\\mathrm{III}_1$ factor, any nontrivial, proper closed $F\\subseteq \\mathbb{R}$, and any nonprincipal ultrafilter $\\mathcal{U}$ on $\\mathbb{N}$, that the ultrapower $M(\\sigma^\\varphi,F)^{\\mathcal{U}}$ of the spectral subspace $M(\\sigma^\\varphi,F)$ is a proper subset of the spectral subspace $M^{\\mathcal{U}}(\\sigma^{\\varphi^{\\mathcal{U}}},F)$. We discuss the model-theoretic implications of this result."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21873","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/2605.21873/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":"2026-05-22T01:04:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eIVdsACvYKi84HuHgCaCzrYFkYfeHnYjYhbM+K9XbFsgtlMH5/L48bVnP6CxcO0scpmTrm3fS6C0OcSARx/uAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T05:36:41.714879Z"},"content_sha256":"62092768e9eb2de9b984023877a0131c15bf2135203e1efb0a025f296ed19b6a","schema_version":"1.0","event_id":"sha256:62092768e9eb2de9b984023877a0131c15bf2135203e1efb0a025f296ed19b6a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DRJGGHUECFADD7E4MJRKIOZREU/bundle.json","state_url":"https://pith.science/pith/DRJGGHUECFADD7E4MJRKIOZREU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DRJGGHUECFADD7E4MJRKIOZREU/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":"2026-06-09T05:36:41Z","links":{"resolver":"https://pith.science/pith/DRJGGHUECFADD7E4MJRKIOZREU","bundle":"https://pith.science/pith/DRJGGHUECFADD7E4MJRKIOZREU/bundle.json","state":"https://pith.science/pith/DRJGGHUECFADD7E4MJRKIOZREU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DRJGGHUECFADD7E4MJRKIOZREU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:DRJGGHUECFADD7E4MJRKIOZREU","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":"7648a3dbd6f34f9de0c1905eb2ea3f31fdeef75b838b91fe36ebf14d9459c73f","cross_cats_sorted":["math.LO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2026-05-21T01:34:47Z","title_canon_sha256":"050e279fcca48892559d24cdad72894791d0423f2fc4c9b6a377af9828c49976"},"schema_version":"1.0","source":{"id":"2605.21873","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.21873","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"arxiv_version","alias_value":"2605.21873v1","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21873","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"pith_short_12","alias_value":"DRJGGHUECFAD","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"pith_short_16","alias_value":"DRJGGHUECFADD7E4","created_at":"2026-05-22T01:04:12Z"},{"alias_kind":"pith_short_8","alias_value":"DRJGGHUE","created_at":"2026-05-22T01:04:12Z"}],"graph_snapshots":[{"event_id":"sha256:62092768e9eb2de9b984023877a0131c15bf2135203e1efb0a025f296ed19b6a","target":"graph","created_at":"2026-05-22T01:04:12Z","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/2605.21873/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove, for any W$^*$-probability space $(M,\\varphi)$ where $M$ is a type $\\mathrm{III}_1$ factor, any nontrivial, proper closed $F\\subseteq \\mathbb{R}$, and any nonprincipal ultrafilter $\\mathcal{U}$ on $\\mathbb{N}$, that the ultrapower $M(\\sigma^\\varphi,F)^{\\mathcal{U}}$ of the spectral subspace $M(\\sigma^\\varphi,F)$ is a proper subset of the spectral subspace $M^{\\mathcal{U}}(\\sigma^{\\varphi^{\\mathcal{U}}},F)$. We discuss the model-theoretic implications of this result.","authors_text":"Hiroshi Ando, Isaac Goldbring","cross_cats":["math.LO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2026-05-21T01:34:47Z","title":"Ultrapowers of spectral subspaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21873","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:cf3c6d768757e17372ad1a77acfd8ca5308a2e4cead56ffda480373ebc4759ea","target":"record","created_at":"2026-05-22T01:04:12Z","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":"7648a3dbd6f34f9de0c1905eb2ea3f31fdeef75b838b91fe36ebf14d9459c73f","cross_cats_sorted":["math.LO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2026-05-21T01:34:47Z","title_canon_sha256":"050e279fcca48892559d24cdad72894791d0423f2fc4c9b6a377af9828c49976"},"schema_version":"1.0","source":{"id":"2605.21873","kind":"arxiv","version":1}},"canonical_sha256":"1c52631e84114031fc9c6262a43b31252049e58043ca82241c3d7cc69598e387","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1c52631e84114031fc9c6262a43b31252049e58043ca82241c3d7cc69598e387","first_computed_at":"2026-05-22T01:04:12.438195Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:04:12.438195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GzZ0ETDwlkuV28b3+uT04/2lnXqbgpssmwXVc/fMxDy3RLX99IQgTB5iRYjeNehrgN4/iUIhJC9Ns85MZnYfDA==","signature_status":"signed_v1","signed_at":"2026-05-22T01:04:12.438934Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.21873","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cf3c6d768757e17372ad1a77acfd8ca5308a2e4cead56ffda480373ebc4759ea","sha256:62092768e9eb2de9b984023877a0131c15bf2135203e1efb0a025f296ed19b6a"],"state_sha256":"b9f89a928866d0a4c1a86730d26854cd6765c03c9a0bef24eeec8237ae4d2a91"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PjA4zaUc3L4tPTjJ4iskQrjBnorBLDhGRtyo8LE6+mWoQ42rT0dfJjsjntx9m2uvmsoiycQ/lqm/Xyuq3/k2Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T05:36:41.718584Z","bundle_sha256":"e9c89d4ab644b1f1d7c37845bca37b02a5ca17dcd01c04e922b4dc0e698085b9"}}