{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:WUQVX6MDD6SEYQXNR3YK5MIADU","short_pith_number":"pith:WUQVX6MD","canonical_record":{"source":{"id":"2608.13392","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-13T15:52:11Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"2464e526e797912d640cb91a080ad72245dd50ac03fa8dc9908e00d70e3058d3","abstract_canon_sha256":"7f8c129590c5ce40c3339440e3603cbf064edf3f51eedfa5252224adaf449bba"},"schema_version":"1.0"},"canonical_sha256":"b5215bf9831fa44c42ed8ef0aeb1001d24cf18c7a6bd7753b3b57167156373f2","source":{"kind":"arxiv","id":"2608.13392","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.13392","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"arxiv_version","alias_value":"2608.13392v1","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.13392","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"pith_short_12","alias_value":"WUQVX6MDD6SE","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"pith_short_16","alias_value":"WUQVX6MDD6SEYQXN","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"pith_short_8","alias_value":"WUQVX6MD","created_at":"2026-08-14T01:04:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:WUQVX6MDD6SEYQXNR3YK5MIADU","target":"record","payload":{"canonical_record":{"source":{"id":"2608.13392","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-13T15:52:11Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"2464e526e797912d640cb91a080ad72245dd50ac03fa8dc9908e00d70e3058d3","abstract_canon_sha256":"7f8c129590c5ce40c3339440e3603cbf064edf3f51eedfa5252224adaf449bba"},"schema_version":"1.0"},"canonical_sha256":"b5215bf9831fa44c42ed8ef0aeb1001d24cf18c7a6bd7753b3b57167156373f2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-14T01:04:21.130401Z","signature_b64":"0SmiMoloj0NxXlEhkr8l/gUdMIQLcMLvYKKy0Y+K+fp9RmYTkzc4TSNbshmthF3QWcn22TkoyKx6hknzlk9CBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5215bf9831fa44c42ed8ef0aeb1001d24cf18c7a6bd7753b3b57167156373f2","last_reissued_at":"2026-08-14T01:04:21.128745Z","signature_status":"signed_v1","first_computed_at":"2026-08-14T01:04:21.128745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2608.13392","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":"2026-08-14T01:04:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WE+Pu+KdnCrk/BuRy6EAK2hY648UMdt+6eEqPCLN+TIKdTuh+rOWJIBsMF4aoBlonG6/wtlgyFgn7ZcUOenQDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T15:54:43.136742Z"},"content_sha256":"6e7c6ee58e537445be06d53d7bce7cfe48d9b03e23defcfad6cc93cf4060e0ee","schema_version":"1.0","event_id":"sha256:6e7c6ee58e537445be06d53d7bce7cfe48d9b03e23defcfad6cc93cf4060e0ee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:WUQVX6MDD6SEYQXNR3YK5MIADU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted Besov Spaces on Homogeneous Lie Groups and Applications to Parabolic Anderson Models","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.AP","authors_text":"Harprit Singh","submitted_at":"2026-08-13T15:52:11Z","abstract_excerpt":"We develop an intrinsic theory of weighted, inhomogeneous Besov spaces on general homogeneous Lie groups without recourse to group-specific arguments. Starting from a definition in terms of localised test functions, we establish an equivalent, wavelet-like, multiscale characterisation. This provides a unified mechanism for deriving Besov embeddings, a Taylor-remainder characterisation, Young-type product estimates, Schauder estimates for convolution semigroups, and a weighted Kolmogorov criterion for random distributions.\n  We apply this framework to parabolic Anderson-type equations associate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.13392","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/2608.13392/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":"2026-08-14T01:04:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HcSpCS7NfZoVnksHmPPU/bWTeEbaVU242lUgQptN1pjoAILvZU+rM/Bh6lWoqga9hP+u/3Jm1WH7gfvyis8uBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T15:54:43.137673Z"},"content_sha256":"864740aa745c0d6214e774bce0593880960cdbebe7ac446571e76f3486c99783","schema_version":"1.0","event_id":"sha256:864740aa745c0d6214e774bce0593880960cdbebe7ac446571e76f3486c99783"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WUQVX6MDD6SEYQXNR3YK5MIADU/bundle.json","state_url":"https://pith.science/pith/WUQVX6MDD6SEYQXNR3YK5MIADU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WUQVX6MDD6SEYQXNR3YK5MIADU/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":"2026-08-19T15:54:43Z","links":{"resolver":"https://pith.science/pith/WUQVX6MDD6SEYQXNR3YK5MIADU","bundle":"https://pith.science/pith/WUQVX6MDD6SEYQXNR3YK5MIADU/bundle.json","state":"https://pith.science/pith/WUQVX6MDD6SEYQXNR3YK5MIADU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WUQVX6MDD6SEYQXNR3YK5MIADU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:WUQVX6MDD6SEYQXNR3YK5MIADU","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":"7f8c129590c5ce40c3339440e3603cbf064edf3f51eedfa5252224adaf449bba","cross_cats_sorted":["math.FA","math.PR"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-13T15:52:11Z","title_canon_sha256":"2464e526e797912d640cb91a080ad72245dd50ac03fa8dc9908e00d70e3058d3"},"schema_version":"1.0","source":{"id":"2608.13392","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.13392","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"arxiv_version","alias_value":"2608.13392v1","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.13392","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"pith_short_12","alias_value":"WUQVX6MDD6SE","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"pith_short_16","alias_value":"WUQVX6MDD6SEYQXN","created_at":"2026-08-14T01:04:21Z"},{"alias_kind":"pith_short_8","alias_value":"WUQVX6MD","created_at":"2026-08-14T01:04:21Z"}],"graph_snapshots":[{"event_id":"sha256:864740aa745c0d6214e774bce0593880960cdbebe7ac446571e76f3486c99783","target":"graph","created_at":"2026-08-14T01:04:21Z","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/2608.13392/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop an intrinsic theory of weighted, inhomogeneous Besov spaces on general homogeneous Lie groups without recourse to group-specific arguments. Starting from a definition in terms of localised test functions, we establish an equivalent, wavelet-like, multiscale characterisation. This provides a unified mechanism for deriving Besov embeddings, a Taylor-remainder characterisation, Young-type product estimates, Schauder estimates for convolution semigroups, and a weighted Kolmogorov criterion for random distributions.\n  We apply this framework to parabolic Anderson-type equations associate","authors_text":"Harprit Singh","cross_cats":["math.FA","math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-13T15:52:11Z","title":"Weighted Besov Spaces on Homogeneous Lie Groups and Applications to Parabolic Anderson Models"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.13392","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":"sha256:6e7c6ee58e537445be06d53d7bce7cfe48d9b03e23defcfad6cc93cf4060e0ee","target":"record","created_at":"2026-08-14T01:04:21Z","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":"7f8c129590c5ce40c3339440e3603cbf064edf3f51eedfa5252224adaf449bba","cross_cats_sorted":["math.FA","math.PR"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-13T15:52:11Z","title_canon_sha256":"2464e526e797912d640cb91a080ad72245dd50ac03fa8dc9908e00d70e3058d3"},"schema_version":"1.0","source":{"id":"2608.13392","kind":"arxiv","version":1}},"canonical_sha256":"b5215bf9831fa44c42ed8ef0aeb1001d24cf18c7a6bd7753b3b57167156373f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b5215bf9831fa44c42ed8ef0aeb1001d24cf18c7a6bd7753b3b57167156373f2","first_computed_at":"2026-08-14T01:04:21.128745Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-14T01:04:21.128745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0SmiMoloj0NxXlEhkr8l/gUdMIQLcMLvYKKy0Y+K+fp9RmYTkzc4TSNbshmthF3QWcn22TkoyKx6hknzlk9CBw==","signature_status":"signed_v1","signed_at":"2026-08-14T01:04:21.130401Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.13392","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e7c6ee58e537445be06d53d7bce7cfe48d9b03e23defcfad6cc93cf4060e0ee","sha256:864740aa745c0d6214e774bce0593880960cdbebe7ac446571e76f3486c99783"],"state_sha256":"b59b0b6b47b0d0920a1f0f38714995256f5147db9562612ab52498d2e3f30611"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jvch915b35qVdT82NyBt9BPi9zWyQ563gKGdjQ0m+ukN8yLXYScIGXgIycSirtf3Aa9YgZHGN9IqoJ1lJfnkAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T15:54:43.145204Z","bundle_sha256":"5c9e87185068ae7c0b8e191224d3743da49f8c38f7a6b67335068282a043b7f5"}}