{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VVRJRKZIKLJFI7QYGI4JORDC6H","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":"fb384306adc9395aec8fefb629596db09861f7a5f00c486b87a68cbbf9f2d1ac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-01-14T15:13:51Z","title_canon_sha256":"0e3b3b4f9318ded152326a2aeedbe11465f1301f1c720b101fc9014e5ed0e5d9"},"schema_version":"1.0","source":{"id":"2601.09553","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.09553","created_at":"2026-05-28T01:04:35Z"},{"alias_kind":"arxiv_version","alias_value":"2601.09553v2","created_at":"2026-05-28T01:04:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.09553","created_at":"2026-05-28T01:04:35Z"},{"alias_kind":"pith_short_12","alias_value":"VVRJRKZIKLJF","created_at":"2026-05-28T01:04:35Z"},{"alias_kind":"pith_short_16","alias_value":"VVRJRKZIKLJFI7QY","created_at":"2026-05-28T01:04:35Z"},{"alias_kind":"pith_short_8","alias_value":"VVRJRKZI","created_at":"2026-05-28T01:04:35Z"}],"graph_snapshots":[{"event_id":"sha256:3d44d3607bf2a869c900c3faf1d8910e830205a82be758fa743e4340162597a8","target":"graph","created_at":"2026-05-28T01:04:35Z","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/2601.09553/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish a non-ergodic version of the Host-Kra-Ziegler structure theorem for measure-preserving $\\mathbb{Z}^d$-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for $d\\ge 2$ due to Candela and Szegedy) via a measurable selection procedure. We also establish a non-ergodic vertical nilcharacter version of our main result. The non-ergodic version of the Host-Kra-Ziegler structure theorem is a key input in the companion paper by the second author and Fraczyk classifying point processes (i.e. random subsets) of $\\mathbb{Z}^d$ whose law is invariant under the group $\\mat","authors_text":"Asgar Jamneshan, Simon Machado","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-01-14T15:13:51Z","title":"The non-ergodic Host-Kra-Ziegler structure theorem for $\\mathbb{Z}^d$-actions via measurable selections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.09553","kind":"arxiv","version":2},"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":"sha256:60f8a46bc5c24c5709d3d9cadbeb73d54e98d8cd6ed7d36c20a8cfa9df86d353","target":"record","created_at":"2026-05-28T01:04:35Z","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":"fb384306adc9395aec8fefb629596db09861f7a5f00c486b87a68cbbf9f2d1ac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-01-14T15:13:51Z","title_canon_sha256":"0e3b3b4f9318ded152326a2aeedbe11465f1301f1c720b101fc9014e5ed0e5d9"},"schema_version":"1.0","source":{"id":"2601.09553","kind":"arxiv","version":2}},"canonical_sha256":"ad6298ab2852d2547e183238974462f1c61b3a8e2f026a6fe0e1bc8ddd6474f8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad6298ab2852d2547e183238974462f1c61b3a8e2f026a6fe0e1bc8ddd6474f8","first_computed_at":"2026-05-28T01:04:35.302910Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-28T01:04:35.302910Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U22y05u6ryF/ZzbiRKMtxu0hO9PA+2NEAyxYlHQwwN0wXjD2zR163yB0YlSYNFobEQE/KFb5KYqd9Xj0bo+MBQ==","signature_status":"signed_v1","signed_at":"2026-05-28T01:04:35.303343Z","signed_message":"canonical_sha256_bytes"},"source_id":"2601.09553","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:60f8a46bc5c24c5709d3d9cadbeb73d54e98d8cd6ed7d36c20a8cfa9df86d353","sha256:3d44d3607bf2a869c900c3faf1d8910e830205a82be758fa743e4340162597a8"],"state_sha256":"c601adcd246de7f8ac272e0f89ae11e09b8dc051c92d289abe0eb8725ff0617d"}