{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:MSQ2HB2SVIRLZGXYGEZQIYLUCH","short_pith_number":"pith:MSQ2HB2S","canonical_record":{"source":{"id":"2607.26446","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-07-29T03:51:18Z","cross_cats_sorted":[],"title_canon_sha256":"b8bfe64262ffdd0cd512b460895e8d62d7c1a7a6b5f8145dab134402fca1bece","abstract_canon_sha256":"168aae18813e01cee4e5ad4b83b582eb8f5fa675558382fcae6098714a035244"},"schema_version":"1.0"},"canonical_sha256":"64a1a38752aa22bc9af8313304617411ce74159e313d8d2e5e029cedf9016d37","source":{"kind":"arxiv","id":"2607.26446","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26446","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26446v1","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26446","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"pith_short_12","alias_value":"MSQ2HB2SVIRL","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"pith_short_16","alias_value":"MSQ2HB2SVIRLZGXY","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"pith_short_8","alias_value":"MSQ2HB2S","created_at":"2026-07-30T01:20:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:MSQ2HB2SVIRLZGXYGEZQIYLUCH","target":"record","payload":{"canonical_record":{"source":{"id":"2607.26446","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-07-29T03:51:18Z","cross_cats_sorted":[],"title_canon_sha256":"b8bfe64262ffdd0cd512b460895e8d62d7c1a7a6b5f8145dab134402fca1bece","abstract_canon_sha256":"168aae18813e01cee4e5ad4b83b582eb8f5fa675558382fcae6098714a035244"},"schema_version":"1.0"},"canonical_sha256":"64a1a38752aa22bc9af8313304617411ce74159e313d8d2e5e029cedf9016d37","receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64a1a38752aa22bc9af8313304617411ce74159e313d8d2e5e029cedf9016d37","last_reissued_at":"2026-07-30T01:20:31.644139Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:20:31.644139Z"},"source_kind":"arxiv","source_id":"2607.26446","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":"2026-07-30T01:20:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OSk+epA9wjuaBC9gLD5ez9A3Y+SH5gM1uDJS12+AiNycaEyT70enavrdrIw1eM1OZX3p1awYAouc0AMbxaMTBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T22:50:19.862333Z"},"content_sha256":"dbdfb5fafc047b50803f43fa56273edd9afd56c8f9dfb37913650d359c7f08de","schema_version":"1.0","event_id":"sha256:dbdfb5fafc047b50803f43fa56273edd9afd56c8f9dfb37913650d359c7f08de"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:MSQ2HB2SVIRLZGXYGEZQIYLUCH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the denseness of distal points","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Hanfeng Li, Ruiwen Li, Su Gao","submitted_at":"2026-07-29T03:51:18Z","abstract_excerpt":"We give an answer to a question of Xu and Ye (Disjointness with all minimal systems under group actions, to appear in Israel J. Math., arxiv:2212.07830) on the denseness of distal points in the Bernoulli shift $2^G$ for a countable discrete group $G$. For a related but stronger notion of almost automorphic points, we answer the similar question by showing that the corresponding collection of the groups coincides with maximal almost periodic ones. These characterizations allow us to construct 2-step nilpotent groups for which the answers to the Xu-Ye question differ. In search for an intrinsic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26446","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/2607.26446/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":"2026-07-30T01:20:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2EEWuPDv6A18v6xXg56I2aNSYiyGrOjW5NRqz5GGhkbhUHJp6hcJnkwD3ZZnky6L1b0kMRTtUmLkIJPlFDUjBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T22:50:19.862848Z"},"content_sha256":"a1f7b826ee023fc381c2a55eab22b3509f4ae25f1355f97b5d3ea0cb93454fb9","schema_version":"1.0","event_id":"sha256:a1f7b826ee023fc381c2a55eab22b3509f4ae25f1355f97b5d3ea0cb93454fb9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MSQ2HB2SVIRLZGXYGEZQIYLUCH/bundle.json","state_url":"https://pith.science/pith/MSQ2HB2SVIRLZGXYGEZQIYLUCH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MSQ2HB2SVIRLZGXYGEZQIYLUCH/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":"2026-08-23T22:50:19Z","links":{"resolver":"https://pith.science/pith/MSQ2HB2SVIRLZGXYGEZQIYLUCH","bundle":"https://pith.science/pith/MSQ2HB2SVIRLZGXYGEZQIYLUCH/bundle.json","state":"https://pith.science/pith/MSQ2HB2SVIRLZGXYGEZQIYLUCH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MSQ2HB2SVIRLZGXYGEZQIYLUCH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MSQ2HB2SVIRLZGXYGEZQIYLUCH","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":"168aae18813e01cee4e5ad4b83b582eb8f5fa675558382fcae6098714a035244","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-07-29T03:51:18Z","title_canon_sha256":"b8bfe64262ffdd0cd512b460895e8d62d7c1a7a6b5f8145dab134402fca1bece"},"schema_version":"1.0","source":{"id":"2607.26446","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26446","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26446v1","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26446","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"pith_short_12","alias_value":"MSQ2HB2SVIRL","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"pith_short_16","alias_value":"MSQ2HB2SVIRLZGXY","created_at":"2026-07-30T01:20:31Z"},{"alias_kind":"pith_short_8","alias_value":"MSQ2HB2S","created_at":"2026-07-30T01:20:31Z"}],"graph_snapshots":[{"event_id":"sha256:a1f7b826ee023fc381c2a55eab22b3509f4ae25f1355f97b5d3ea0cb93454fb9","target":"graph","created_at":"2026-07-30T01:20:31Z","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/2607.26446/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give an answer to a question of Xu and Ye (Disjointness with all minimal systems under group actions, to appear in Israel J. Math., arxiv:2212.07830) on the denseness of distal points in the Bernoulli shift $2^G$ for a countable discrete group $G$. For a related but stronger notion of almost automorphic points, we answer the similar question by showing that the corresponding collection of the groups coincides with maximal almost periodic ones. These characterizations allow us to construct 2-step nilpotent groups for which the answers to the Xu-Ye question differ. In search for an intrinsic ","authors_text":"Hanfeng Li, Ruiwen Li, Su Gao","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-07-29T03:51:18Z","title":"On the denseness of distal points"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26446","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:dbdfb5fafc047b50803f43fa56273edd9afd56c8f9dfb37913650d359c7f08de","target":"record","created_at":"2026-07-30T01:20:31Z","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":"168aae18813e01cee4e5ad4b83b582eb8f5fa675558382fcae6098714a035244","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-07-29T03:51:18Z","title_canon_sha256":"b8bfe64262ffdd0cd512b460895e8d62d7c1a7a6b5f8145dab134402fca1bece"},"schema_version":"1.0","source":{"id":"2607.26446","kind":"arxiv","version":1}},"canonical_sha256":"64a1a38752aa22bc9af8313304617411ce74159e313d8d2e5e029cedf9016d37","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64a1a38752aa22bc9af8313304617411ce74159e313d8d2e5e029cedf9016d37","first_computed_at":"2026-07-30T01:20:31.644139Z","kind":"pith_receipt","last_reissued_at":"2026-07-30T01:20:31.644139Z","receipt_version":"0.3","signature_status":"unsigned_v0"},"source_id":"2607.26446","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dbdfb5fafc047b50803f43fa56273edd9afd56c8f9dfb37913650d359c7f08de","sha256:a1f7b826ee023fc381c2a55eab22b3509f4ae25f1355f97b5d3ea0cb93454fb9"],"state_sha256":"1f6fe749cea43684dda2b59841ee3a383cd90d18aa30ed3a5137f62ad9fe44aa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ve8K3Mhsjcfeslx0cW4SIlICbpBgepqbAyidHij46qUMXg3UAdbMQLeWS+109ZBwCoMIKjoUzMfdWd9YMPWSCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T22:50:19.868170Z","bundle_sha256":"6ea773c1ecc43aed3af03f5a479f08bae2a54d194dd9e8ef2cd6eb5eb2d9c327"}}