{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:CDVBPT4JQXE7LXMCNVBSVBQS7U","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":"cb9b123a3b297f7980d5dd168831c4f13a5d0134fd984868e323198fdb4e6867","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2025-02-01T23:53:52Z","title_canon_sha256":"affffe2416f32aeb3b1e3d5c6f89167b92ca263a4d02e7301a5000330aa61dcb"},"schema_version":"1.0","source":{"id":"2502.00598","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.00598","created_at":"2026-07-05T10:08:35Z"},{"alias_kind":"arxiv_version","alias_value":"2502.00598v1","created_at":"2026-07-05T10:08:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.00598","created_at":"2026-07-05T10:08:35Z"},{"alias_kind":"pith_short_12","alias_value":"CDVBPT4JQXE7","created_at":"2026-07-05T10:08:35Z"},{"alias_kind":"pith_short_16","alias_value":"CDVBPT4JQXE7LXMC","created_at":"2026-07-05T10:08:35Z"},{"alias_kind":"pith_short_8","alias_value":"CDVBPT4J","created_at":"2026-07-05T10:08:35Z"}],"graph_snapshots":[{"event_id":"sha256:9acfc1bac35c9639d85fe7846c4c7c4fbe2ad0acbfca9d4328d0efef28a69100","target":"graph","created_at":"2026-07-05T10:08:35Z","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/2502.00598/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove the existence of clopen marker sets with some strong regularity property. For each $n\\geq 1$ and any integer $d\\geq 1$, we show that there are a positive integer $D$ and a clopen marker set $M$ in $F(2^{\\mathbb{Z}^n})$ such that (1) for any distinct $x,y\\in M$ in the same orbit, $\\rho(x,y)\\geq d$; (2) for any $1\\leq i\\leq n$ and any $x\\in F(2^{\\mathbb{Z}^n})$, there are non-negative integers $a, b\\leq D$ such that $a\\cdot x\\in M$ and $-b\\cdot x\\in M$. As an application, we obtain a clopen tree section for $F(2^{\\mathbb{Z}^n})$. Based on the strong marker sets, we get a quick proof tha","authors_text":"Su Gao, Tianhao Wang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2025-02-01T23:53:52Z","title":"Strong marker sets and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.00598","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":"sha256:70176067a8e1effd06cdac708691904ab0f865f6b26c721132667168d7182f09","target":"record","created_at":"2026-07-05T10:08:35Z","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":"cb9b123a3b297f7980d5dd168831c4f13a5d0134fd984868e323198fdb4e6867","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2025-02-01T23:53:52Z","title_canon_sha256":"affffe2416f32aeb3b1e3d5c6f89167b92ca263a4d02e7301a5000330aa61dcb"},"schema_version":"1.0","source":{"id":"2502.00598","kind":"arxiv","version":1}},"canonical_sha256":"10ea17cf8985c9f5dd826d432a8612fd17417d94e89c8d0cda7f4f604dd466c4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"10ea17cf8985c9f5dd826d432a8612fd17417d94e89c8d0cda7f4f604dd466c4","first_computed_at":"2026-07-05T10:08:35.974108Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:08:35.974108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xINhBYbPa5z9hE1hmj+ZCjc3mb8/AR2Tbc0GwQYYDVJUPkPBJAkXw22Cl4xk0bdISx6mecziDN37LBuhJ3beCw==","signature_status":"signed_v1","signed_at":"2026-07-05T10:08:35.974526Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.00598","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:70176067a8e1effd06cdac708691904ab0f865f6b26c721132667168d7182f09","sha256:9acfc1bac35c9639d85fe7846c4c7c4fbe2ad0acbfca9d4328d0efef28a69100"],"state_sha256":"f51b02297937b80aac66ab3da65444a1a32b8d400f770021dffb38d3c88d5b48"}