{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7WXI43C2UU2JHU5VMCVX3UIK3J","short_pith_number":"pith:7WXI43C2","canonical_record":{"source":{"id":"2606.27366","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-06-25T17:59:24Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"3b640a1fcaa377a8b9672f09c0b4a1f40dc1db888f71fde4172e1d4c5c6d1432","abstract_canon_sha256":"f81759ee8ecba0e74c6ef3811591e5d291fb12bdb7e1ba7a1ff8e6a9c9af72df"},"schema_version":"1.0"},"canonical_sha256":"fdae8e6c5aa53493d3b560ab7dd10ada4d870efb793da85faddab473a15dabeb","source":{"kind":"arxiv","id":"2606.27366","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27366","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27366v1","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27366","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"pith_short_12","alias_value":"7WXI43C2UU2J","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"pith_short_16","alias_value":"7WXI43C2UU2JHU5V","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"pith_short_8","alias_value":"7WXI43C2","created_at":"2026-06-26T01:16:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7WXI43C2UU2JHU5VMCVX3UIK3J","target":"record","payload":{"canonical_record":{"source":{"id":"2606.27366","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-06-25T17:59:24Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"3b640a1fcaa377a8b9672f09c0b4a1f40dc1db888f71fde4172e1d4c5c6d1432","abstract_canon_sha256":"f81759ee8ecba0e74c6ef3811591e5d291fb12bdb7e1ba7a1ff8e6a9c9af72df"},"schema_version":"1.0"},"canonical_sha256":"fdae8e6c5aa53493d3b560ab7dd10ada4d870efb793da85faddab473a15dabeb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-26T01:16:19.972248Z","signature_b64":"TeYCsI+TFtlvXoFgy1rzY1y3o+zKXX0fk50jr+N2vmIZOk42eVMlxImnRUlPQgiRhbGPEJXv6B81ALKCO3nXCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fdae8e6c5aa53493d3b560ab7dd10ada4d870efb793da85faddab473a15dabeb","last_reissued_at":"2026-06-26T01:16:19.971827Z","signature_status":"signed_v1","first_computed_at":"2026-06-26T01:16:19.971827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.27366","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":"2026-06-26T01:16:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qrXFaTjV4v8RMNk6qGXbqN4kLJzVl9UjnmiQlpBNOtY6dPQ3mROGbWbhIQt5K7/JtezEiJqtBhyRYbJPYs9EBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T12:36:49.950407Z"},"content_sha256":"878cb3f9e4d3846c20f8f87f3d51ca8564233f836374d7bae144b8153ecf7355","schema_version":"1.0","event_id":"sha256:878cb3f9e4d3846c20f8f87f3d51ca8564233f836374d7bae144b8153ecf7355"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7WXI43C2UU2JHU5VMCVX3UIK3J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\mathrm{W}^*$-algebraic Integration Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jan G{\\l}owacki, Yui Kuramochi","submitted_at":"2026-06-25T17:59:24Z","abstract_excerpt":"Given a pair of $\\mathrm{W}^*$-algebras $(\\mathcal{M}_\\mathcal{S},\\mathcal{M}_\\mathcal{R})$ with $(\\mathcal{M}_\\mathcal{S})_*$ separable, a measurable space $(\\Sigma, \\mathcal{F})$ and a POVM $\\mathsf{E}: \\mathcal{F} \\to \\mathcal{E}(\\mathcal{M}_\\mathcal{R})$, the integral of a function $f: \\Sigma \\to \\mathcal{M}_\\mathcal{S}$ is defined as an element of the spatial tensor product $\\int f \\otimes d\\mathsf{E} \\in \\mathcal{M}_\\mathcal{S} \\bar{\\otimes} \\mathcal{M}_\\mathcal{R}$. The space $B_b(\\Sigma,\\mathcal{F},\\mathcal{M}_\\mathcal{S})$ of uniformly bounded ultraweakly measurable functions is the u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27366","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/2606.27366/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":"2026-06-26T01:16:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ACs52qIoZ97cAakFHRXsQ7he8SwjHmUFBBXIJ5SHukJ1FyQ3b5S/Lu1cOdO8OotAIuGCPbWawT2pscvKPlZOBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T12:36:49.950794Z"},"content_sha256":"533894aec3091d7220ddc0e1f6e984526a9267bfb60712bf96c579426aecb47f","schema_version":"1.0","event_id":"sha256:533894aec3091d7220ddc0e1f6e984526a9267bfb60712bf96c579426aecb47f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7WXI43C2UU2JHU5VMCVX3UIK3J/bundle.json","state_url":"https://pith.science/pith/7WXI43C2UU2JHU5VMCVX3UIK3J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7WXI43C2UU2JHU5VMCVX3UIK3J/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":"2026-07-01T12:36:49Z","links":{"resolver":"https://pith.science/pith/7WXI43C2UU2JHU5VMCVX3UIK3J","bundle":"https://pith.science/pith/7WXI43C2UU2JHU5VMCVX3UIK3J/bundle.json","state":"https://pith.science/pith/7WXI43C2UU2JHU5VMCVX3UIK3J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7WXI43C2UU2JHU5VMCVX3UIK3J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7WXI43C2UU2JHU5VMCVX3UIK3J","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":"f81759ee8ecba0e74c6ef3811591e5d291fb12bdb7e1ba7a1ff8e6a9c9af72df","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-06-25T17:59:24Z","title_canon_sha256":"3b640a1fcaa377a8b9672f09c0b4a1f40dc1db888f71fde4172e1d4c5c6d1432"},"schema_version":"1.0","source":{"id":"2606.27366","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27366","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27366v1","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27366","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"pith_short_12","alias_value":"7WXI43C2UU2J","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"pith_short_16","alias_value":"7WXI43C2UU2JHU5V","created_at":"2026-06-26T01:16:19Z"},{"alias_kind":"pith_short_8","alias_value":"7WXI43C2","created_at":"2026-06-26T01:16:19Z"}],"graph_snapshots":[{"event_id":"sha256:533894aec3091d7220ddc0e1f6e984526a9267bfb60712bf96c579426aecb47f","target":"graph","created_at":"2026-06-26T01:16:19Z","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/2606.27366/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a pair of $\\mathrm{W}^*$-algebras $(\\mathcal{M}_\\mathcal{S},\\mathcal{M}_\\mathcal{R})$ with $(\\mathcal{M}_\\mathcal{S})_*$ separable, a measurable space $(\\Sigma, \\mathcal{F})$ and a POVM $\\mathsf{E}: \\mathcal{F} \\to \\mathcal{E}(\\mathcal{M}_\\mathcal{R})$, the integral of a function $f: \\Sigma \\to \\mathcal{M}_\\mathcal{S}$ is defined as an element of the spatial tensor product $\\int f \\otimes d\\mathsf{E} \\in \\mathcal{M}_\\mathcal{S} \\bar{\\otimes} \\mathcal{M}_\\mathcal{R}$. The space $B_b(\\Sigma,\\mathcal{F},\\mathcal{M}_\\mathcal{S})$ of uniformly bounded ultraweakly measurable functions is the u","authors_text":"Jan G{\\l}owacki, Yui Kuramochi","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-06-25T17:59:24Z","title":"$\\mathrm{W}^*$-algebraic Integration Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27366","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":"sha256:878cb3f9e4d3846c20f8f87f3d51ca8564233f836374d7bae144b8153ecf7355","target":"record","created_at":"2026-06-26T01:16:19Z","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":"f81759ee8ecba0e74c6ef3811591e5d291fb12bdb7e1ba7a1ff8e6a9c9af72df","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-06-25T17:59:24Z","title_canon_sha256":"3b640a1fcaa377a8b9672f09c0b4a1f40dc1db888f71fde4172e1d4c5c6d1432"},"schema_version":"1.0","source":{"id":"2606.27366","kind":"arxiv","version":1}},"canonical_sha256":"fdae8e6c5aa53493d3b560ab7dd10ada4d870efb793da85faddab473a15dabeb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fdae8e6c5aa53493d3b560ab7dd10ada4d870efb793da85faddab473a15dabeb","first_computed_at":"2026-06-26T01:16:19.971827Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-26T01:16:19.971827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TeYCsI+TFtlvXoFgy1rzY1y3o+zKXX0fk50jr+N2vmIZOk42eVMlxImnRUlPQgiRhbGPEJXv6B81ALKCO3nXCg==","signature_status":"signed_v1","signed_at":"2026-06-26T01:16:19.972248Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.27366","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:878cb3f9e4d3846c20f8f87f3d51ca8564233f836374d7bae144b8153ecf7355","sha256:533894aec3091d7220ddc0e1f6e984526a9267bfb60712bf96c579426aecb47f"],"state_sha256":"3865ed859c9179a85c7dbe1b80767c898b458fa734f48b7868680bc66dbd3b95"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TsM7LUajrrxJzP3hKwPt/hnorK/mLRUIrzUJIKMfj14FdsbbIQM79DcuuCJ31RTGtUF5FKjNDtk/1ExnX+S3Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T12:36:49.952976Z","bundle_sha256":"acc16364bd4a806a73f7f1e73a8ea89e86dc97c209de25def2b79e4e91cd49d2"}}