{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:E3TGE3LXMVBG4XD6GIL4W7FJZT","short_pith_number":"pith:E3TGE3LX","canonical_record":{"source":{"id":"2605.30616","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-28T22:08:29Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"eea1b9fd5ca475e83d19100955bee3fdf571e7a4a045da4414659392cb57b3b5","abstract_canon_sha256":"ca0bf510fadcf29018c0bfc452733cf2283aa664cd2b3037fd06f2a9fc2e584f"},"schema_version":"1.0"},"canonical_sha256":"26e6626d7765426e5c7e3217cb7ca9ccd199c778469baaeeaba3d0c6fa0a002a","source":{"kind":"arxiv","id":"2605.30616","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.30616","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"arxiv_version","alias_value":"2605.30616v1","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.30616","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"pith_short_12","alias_value":"E3TGE3LXMVBG","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"pith_short_16","alias_value":"E3TGE3LXMVBG4XD6","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"pith_short_8","alias_value":"E3TGE3LX","created_at":"2026-06-01T01:03:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:E3TGE3LXMVBG4XD6GIL4W7FJZT","target":"record","payload":{"canonical_record":{"source":{"id":"2605.30616","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-28T22:08:29Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"eea1b9fd5ca475e83d19100955bee3fdf571e7a4a045da4414659392cb57b3b5","abstract_canon_sha256":"ca0bf510fadcf29018c0bfc452733cf2283aa664cd2b3037fd06f2a9fc2e584f"},"schema_version":"1.0"},"canonical_sha256":"26e6626d7765426e5c7e3217cb7ca9ccd199c778469baaeeaba3d0c6fa0a002a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T01:03:04.138644Z","signature_b64":"H4KAL61S/fO1hkXHDjuIcwjP6xXHa6yfuHpifiHO+hA805W+rnDvDnFHCqJ4SH6jvl4hof++0pOa7fhAkFG5Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26e6626d7765426e5c7e3217cb7ca9ccd199c778469baaeeaba3d0c6fa0a002a","last_reissued_at":"2026-06-01T01:03:04.137883Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T01:03:04.137883Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.30616","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":"2026-06-01T01:03:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kAldSAq85gQ8nfA7HiJj1jGcAbPuiMKmhM2QASMvz8Sz5ASo+h2sFgw2dx8/ts6lsQQ8ZXD2bcUuSlGoIXR6Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T17:33:14.232725Z"},"content_sha256":"280f1493cb1bc1bed210a22a74ff07c8d8af9680148b81dc3bba8042d90624d0","schema_version":"1.0","event_id":"sha256:280f1493cb1bc1bed210a22a74ff07c8d8af9680148b81dc3bba8042d90624d0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:E3TGE3LXMVBG4XD6GIL4W7FJZT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Brownian motion: the hyperbolic number setting","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Daniel Alpay, Ilwoo Cho, Liora Mayats-Alpay","submitted_at":"2026-05-28T22:08:29Z","abstract_excerpt":"The purpose of this paper is to define normal Gaussian variables in the setting of hyperbolic probabilities, and introduce an associated Brownian motion, when both the index and the values of the process lie in the real algebra $\\mathbb{H}$ of hyperbolic numbers.\n  In Hida's white noise space, we construct two probability measures (say $P_1$ and $P_2$), and associate to them two families of $N(0,1)$ variables $(Z_n)_{n\\in\\mathbb N_0}$ (independent with respect to $P_1$) and $(W_n)_{n\\in\\mathbb N_0}$ (independent with respect to $P_2$). An important feature is that the $Z_n$ and $W_m$ need not "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30616","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/2605.30616/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":"2026-06-01T01:03:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q1Yf9DwHeP60yRkkqW9gXjsyPw10r/jfWxYjociIAAwYKhte/Lsn54BLy56GD+J7eJBITldlVNZTm7+yX25IAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T17:33:14.233540Z"},"content_sha256":"d544f2ffada84d3e8ab699da6c3f629a3cde2543ef5424269d869c0b69b6bcf9","schema_version":"1.0","event_id":"sha256:d544f2ffada84d3e8ab699da6c3f629a3cde2543ef5424269d869c0b69b6bcf9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E3TGE3LXMVBG4XD6GIL4W7FJZT/bundle.json","state_url":"https://pith.science/pith/E3TGE3LXMVBG4XD6GIL4W7FJZT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E3TGE3LXMVBG4XD6GIL4W7FJZT/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":"2026-06-07T17:33:14Z","links":{"resolver":"https://pith.science/pith/E3TGE3LXMVBG4XD6GIL4W7FJZT","bundle":"https://pith.science/pith/E3TGE3LXMVBG4XD6GIL4W7FJZT/bundle.json","state":"https://pith.science/pith/E3TGE3LXMVBG4XD6GIL4W7FJZT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E3TGE3LXMVBG4XD6GIL4W7FJZT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:E3TGE3LXMVBG4XD6GIL4W7FJZT","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":"ca0bf510fadcf29018c0bfc452733cf2283aa664cd2b3037fd06f2a9fc2e584f","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-28T22:08:29Z","title_canon_sha256":"eea1b9fd5ca475e83d19100955bee3fdf571e7a4a045da4414659392cb57b3b5"},"schema_version":"1.0","source":{"id":"2605.30616","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.30616","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"arxiv_version","alias_value":"2605.30616v1","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.30616","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"pith_short_12","alias_value":"E3TGE3LXMVBG","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"pith_short_16","alias_value":"E3TGE3LXMVBG4XD6","created_at":"2026-06-01T01:03:04Z"},{"alias_kind":"pith_short_8","alias_value":"E3TGE3LX","created_at":"2026-06-01T01:03:04Z"}],"graph_snapshots":[{"event_id":"sha256:d544f2ffada84d3e8ab699da6c3f629a3cde2543ef5424269d869c0b69b6bcf9","target":"graph","created_at":"2026-06-01T01:03:04Z","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/2605.30616/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The purpose of this paper is to define normal Gaussian variables in the setting of hyperbolic probabilities, and introduce an associated Brownian motion, when both the index and the values of the process lie in the real algebra $\\mathbb{H}$ of hyperbolic numbers.\n  In Hida's white noise space, we construct two probability measures (say $P_1$ and $P_2$), and associate to them two families of $N(0,1)$ variables $(Z_n)_{n\\in\\mathbb N_0}$ (independent with respect to $P_1$) and $(W_n)_{n\\in\\mathbb N_0}$ (independent with respect to $P_2$). An important feature is that the $Z_n$ and $W_m$ need not ","authors_text":"Daniel Alpay, Ilwoo Cho, Liora Mayats-Alpay","cross_cats":["math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-28T22:08:29Z","title":"Brownian motion: the hyperbolic number setting"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30616","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":"sha256:280f1493cb1bc1bed210a22a74ff07c8d8af9680148b81dc3bba8042d90624d0","target":"record","created_at":"2026-06-01T01:03:04Z","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":"ca0bf510fadcf29018c0bfc452733cf2283aa664cd2b3037fd06f2a9fc2e584f","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-28T22:08:29Z","title_canon_sha256":"eea1b9fd5ca475e83d19100955bee3fdf571e7a4a045da4414659392cb57b3b5"},"schema_version":"1.0","source":{"id":"2605.30616","kind":"arxiv","version":1}},"canonical_sha256":"26e6626d7765426e5c7e3217cb7ca9ccd199c778469baaeeaba3d0c6fa0a002a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"26e6626d7765426e5c7e3217cb7ca9ccd199c778469baaeeaba3d0c6fa0a002a","first_computed_at":"2026-06-01T01:03:04.137883Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-01T01:03:04.137883Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H4KAL61S/fO1hkXHDjuIcwjP6xXHa6yfuHpifiHO+hA805W+rnDvDnFHCqJ4SH6jvl4hof++0pOa7fhAkFG5Bg==","signature_status":"signed_v1","signed_at":"2026-06-01T01:03:04.138644Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.30616","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:280f1493cb1bc1bed210a22a74ff07c8d8af9680148b81dc3bba8042d90624d0","sha256:d544f2ffada84d3e8ab699da6c3f629a3cde2543ef5424269d869c0b69b6bcf9"],"state_sha256":"f380c5f21b0515b184d5937f563ab29de959990eda893e96af42ee5c1f43638b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Adokyn7gZ5hjgTMcZlQxsRU9TJtooTcvTs4UvsqopXvT38M4INnrMtMCD9KggJZHaK7ZiK9exPNOGBa/4liADw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T17:33:14.237577Z","bundle_sha256":"7ecae8b637cf75002e3dfdd171319dca487dc82ca6effd729e5eb285f7231122"}}