{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:QSSUL6XLIB3H72YJ3JJBLTWQ4F","short_pith_number":"pith:QSSUL6XL","canonical_record":{"source":{"id":"2506.12164","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-13T18:30:18Z","cross_cats_sorted":[],"title_canon_sha256":"7ba0f2e8bd859aca33f2dcc8c36a9ed9b0c134f61df9e07bd2d05c0e9153bf4d","abstract_canon_sha256":"226260f2ae5ab6ad5e34051fae3c2009a590f064d2bd814ebf35aa79118060e2"},"schema_version":"1.0"},"canonical_sha256":"84a545faeb40767feb09da5215ced0e14ae392676442303f5e7db1dd8e2fdb0c","source":{"kind":"arxiv","id":"2506.12164","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.12164","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"arxiv_version","alias_value":"2506.12164v1","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.12164","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"pith_short_12","alias_value":"QSSUL6XLIB3H","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"pith_short_16","alias_value":"QSSUL6XLIB3H72YJ","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"pith_short_8","alias_value":"QSSUL6XL","created_at":"2026-07-05T11:21:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:QSSUL6XLIB3H72YJ3JJBLTWQ4F","target":"record","payload":{"canonical_record":{"source":{"id":"2506.12164","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-13T18:30:18Z","cross_cats_sorted":[],"title_canon_sha256":"7ba0f2e8bd859aca33f2dcc8c36a9ed9b0c134f61df9e07bd2d05c0e9153bf4d","abstract_canon_sha256":"226260f2ae5ab6ad5e34051fae3c2009a590f064d2bd814ebf35aa79118060e2"},"schema_version":"1.0"},"canonical_sha256":"84a545faeb40767feb09da5215ced0e14ae392676442303f5e7db1dd8e2fdb0c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:21:22.337287Z","signature_b64":"gX1b/sLVqoM1MSyRZ8+ferfaN4NaSSdBTserd8TsfpVNCXsZdw3yYNRPF1qhTZQHEOoYtGNAj16ciK/mCEZAAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84a545faeb40767feb09da5215ced0e14ae392676442303f5e7db1dd8e2fdb0c","last_reissued_at":"2026-07-05T11:21:22.336819Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:21:22.336819Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.12164","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":"2026-07-05T11:21:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZkVM/baBvENIoT8ZVKwFHacFyqHwRtDcyiopvYjtOtCw3PpIu29UCxFDOBmUYQvGttJH6U25CKap8qS6inD5BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T01:45:10.821216Z"},"content_sha256":"0c55bc33d18cfb53690d20e9459d0235027b13f91f6c3152465f404afd9f7c5c","schema_version":"1.0","event_id":"sha256:0c55bc33d18cfb53690d20e9459d0235027b13f91f6c3152465f404afd9f7c5c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:QSSUL6XLIB3H72YJ3JJBLTWQ4F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The boundedness of rough generalized commutators with Lipschitz functions on homogeneous variable exponent Herz type spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ferit Gurbuz","submitted_at":"2025-06-13T18:30:18Z","abstract_excerpt":"With the development of science, many nonlinear problems have emerged. At this time, the classical function space has certain restrictions. For example, it has lost its effectiveness for nonlinear problems under nonstandard growth conditions. In the process of studying such nonlinear problems, scholars are paying more and more attention to the transition from classical function space to variable exponent function space. Also, there is a big difference between variable exponent space and classical function space, mainly because variable exponent function space has lost translation invariance. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.12164","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/2506.12164/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":"2026-07-05T11:21:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6qMgRjMp16+R3vNp2YX4RLgQyuYwQN1+zujmRpm5fWBRQqcMgm1yiVkXn+QIdQnA6IJ1FK7ITe2H6Ui2nqD5Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T01:45:10.821974Z"},"content_sha256":"3bc474950cf930c08dcfd9c80c37acc0c64be49fa4577d90a7f46170180584b9","schema_version":"1.0","event_id":"sha256:3bc474950cf930c08dcfd9c80c37acc0c64be49fa4577d90a7f46170180584b9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QSSUL6XLIB3H72YJ3JJBLTWQ4F/bundle.json","state_url":"https://pith.science/pith/QSSUL6XLIB3H72YJ3JJBLTWQ4F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QSSUL6XLIB3H72YJ3JJBLTWQ4F/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":"2026-08-13T01:45:10Z","links":{"resolver":"https://pith.science/pith/QSSUL6XLIB3H72YJ3JJBLTWQ4F","bundle":"https://pith.science/pith/QSSUL6XLIB3H72YJ3JJBLTWQ4F/bundle.json","state":"https://pith.science/pith/QSSUL6XLIB3H72YJ3JJBLTWQ4F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QSSUL6XLIB3H72YJ3JJBLTWQ4F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:QSSUL6XLIB3H72YJ3JJBLTWQ4F","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":"226260f2ae5ab6ad5e34051fae3c2009a590f064d2bd814ebf35aa79118060e2","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-13T18:30:18Z","title_canon_sha256":"7ba0f2e8bd859aca33f2dcc8c36a9ed9b0c134f61df9e07bd2d05c0e9153bf4d"},"schema_version":"1.0","source":{"id":"2506.12164","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.12164","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"arxiv_version","alias_value":"2506.12164v1","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.12164","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"pith_short_12","alias_value":"QSSUL6XLIB3H","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"pith_short_16","alias_value":"QSSUL6XLIB3H72YJ","created_at":"2026-07-05T11:21:22Z"},{"alias_kind":"pith_short_8","alias_value":"QSSUL6XL","created_at":"2026-07-05T11:21:22Z"}],"graph_snapshots":[{"event_id":"sha256:3bc474950cf930c08dcfd9c80c37acc0c64be49fa4577d90a7f46170180584b9","target":"graph","created_at":"2026-07-05T11:21:22Z","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/2506.12164/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"With the development of science, many nonlinear problems have emerged. At this time, the classical function space has certain restrictions. For example, it has lost its effectiveness for nonlinear problems under nonstandard growth conditions. In the process of studying such nonlinear problems, scholars are paying more and more attention to the transition from classical function space to variable exponent function space. Also, there is a big difference between variable exponent space and classical function space, mainly because variable exponent function space has lost translation invariance. T","authors_text":"Ferit Gurbuz","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-13T18:30:18Z","title":"The boundedness of rough generalized commutators with Lipschitz functions on homogeneous variable exponent Herz type spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.12164","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":"sha256:0c55bc33d18cfb53690d20e9459d0235027b13f91f6c3152465f404afd9f7c5c","target":"record","created_at":"2026-07-05T11:21:22Z","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":"226260f2ae5ab6ad5e34051fae3c2009a590f064d2bd814ebf35aa79118060e2","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-13T18:30:18Z","title_canon_sha256":"7ba0f2e8bd859aca33f2dcc8c36a9ed9b0c134f61df9e07bd2d05c0e9153bf4d"},"schema_version":"1.0","source":{"id":"2506.12164","kind":"arxiv","version":1}},"canonical_sha256":"84a545faeb40767feb09da5215ced0e14ae392676442303f5e7db1dd8e2fdb0c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"84a545faeb40767feb09da5215ced0e14ae392676442303f5e7db1dd8e2fdb0c","first_computed_at":"2026-07-05T11:21:22.336819Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:21:22.336819Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gX1b/sLVqoM1MSyRZ8+ferfaN4NaSSdBTserd8TsfpVNCXsZdw3yYNRPF1qhTZQHEOoYtGNAj16ciK/mCEZAAw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:21:22.337287Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.12164","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0c55bc33d18cfb53690d20e9459d0235027b13f91f6c3152465f404afd9f7c5c","sha256:3bc474950cf930c08dcfd9c80c37acc0c64be49fa4577d90a7f46170180584b9"],"state_sha256":"a148c7f7d9fef70b392f4a3c32f01f6bed5ff0107f98ae2f4a403204a9ec8e77"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X0FxLi0vX6bQ7AtyqFGatXuTVyzXeBci2Aek2q2G69Ahwcj0JyQXHPaNMcLF0ZUCIuCDA6xgeLGNgSbhSpZ3Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T01:45:10.828731Z","bundle_sha256":"3999842ab730a4cd2c264e8fc76086b24ce06648622c3d7b0792bc4b3a0a429b"}}