{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:2A5CUYSIXMVIU4WKQ7RWSB4V5G","short_pith_number":"pith:2A5CUYSI","canonical_record":{"source":{"id":"2409.13164","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-20T02:36:23Z","cross_cats_sorted":["math-ph","math.FA","math.MP"],"title_canon_sha256":"7167ae376a62e08812356dd9b75ff266b5f76a5046b4dd39fd06e84fbc738726","abstract_canon_sha256":"27d43b9bd6c9ef1355cb6e919c3155215a9ffdd9cfeda964c374d17e0565c7b3"},"schema_version":"1.0"},"canonical_sha256":"d03a2a6248bb2a8a72ca87e3690795e98286bb54220955119825daa7c72b97b0","source":{"kind":"arxiv","id":"2409.13164","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.13164","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"arxiv_version","alias_value":"2409.13164v3","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.13164","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"pith_short_12","alias_value":"2A5CUYSIXMVI","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"pith_short_16","alias_value":"2A5CUYSIXMVIU4WK","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"pith_short_8","alias_value":"2A5CUYSI","created_at":"2026-07-05T11:19:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:2A5CUYSIXMVIU4WKQ7RWSB4V5G","target":"record","payload":{"canonical_record":{"source":{"id":"2409.13164","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-20T02:36:23Z","cross_cats_sorted":["math-ph","math.FA","math.MP"],"title_canon_sha256":"7167ae376a62e08812356dd9b75ff266b5f76a5046b4dd39fd06e84fbc738726","abstract_canon_sha256":"27d43b9bd6c9ef1355cb6e919c3155215a9ffdd9cfeda964c374d17e0565c7b3"},"schema_version":"1.0"},"canonical_sha256":"d03a2a6248bb2a8a72ca87e3690795e98286bb54220955119825daa7c72b97b0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:19:21.072954Z","signature_b64":"WwsSGJeAz951XMaPWg7vwlu37AUlBtDSLOv/hYRZhtjTdWlmjcnuu+AlaSpO3fHbI4sN1Mea9EO/Kdl/w9W4Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d03a2a6248bb2a8a72ca87e3690795e98286bb54220955119825daa7c72b97b0","last_reissued_at":"2026-07-05T11:19:21.072509Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:19:21.072509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2409.13164","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:19:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KJkd/yHHfKrsLcINazDYlj+SAdSwJvRDV6T8kovxzqAW909mxTmZokOoZmnjNbyKonJwUZgh19s1+WDoLzUyBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T06:52:19.205754Z"},"content_sha256":"648b48f4e3f923a29cc6edc98dc243330ce6dae6361aac95e853d4432fd7f926","schema_version":"1.0","event_id":"sha256:648b48f4e3f923a29cc6edc98dc243330ce6dae6361aac95e853d4432fd7f926"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:2A5CUYSIXMVIU4WKQ7RWSB4V5G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Harmonic analysis of Mandelbrot cascades -- in the context of vector-valued martingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.PR","authors_text":"Xinxin Chen, Yanqi Qiu, Yong Han, Zipeng Wang","submitted_at":"2024-09-20T02:36:23Z","abstract_excerpt":"We solve a long-standing open problem of determining the Fourier dimension of the Mandelbrot canonical cascade measure (MCCM). This problem of significant interest was raised by Mandelbrot in 1976 and reiterated by Kahane in 1993. Specifically, we derive the exact formula for the Fourier dimension of the MCCM for random weights $W$ satisfying the condition $\\mathbb{E}[W^t]<\\infty$ for all $t>0$. As a corollary, we prove that the MCCM is Salem if and only if the random weight has a specific two-point distribution. In addition, we show that the MCCM is Rajchman with polynomial Fourier decay when"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.13164","kind":"arxiv","version":3},"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/2409.13164/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":"2026-07-05T11:19:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B8nJBeim19nWs0asxtSXxcsd6sTyTH7Ro8A7U6c8fiXjo0/+Miol8FQP532VHMeNbz7h+oK+0D+iRi7EeTv3Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T06:52:19.206337Z"},"content_sha256":"110f693a8bad46c5a9a53a068f6ccaef0f1e3de9e3d99c707f7e489f8293a3d2","schema_version":"1.0","event_id":"sha256:110f693a8bad46c5a9a53a068f6ccaef0f1e3de9e3d99c707f7e489f8293a3d2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2A5CUYSIXMVIU4WKQ7RWSB4V5G/bundle.json","state_url":"https://pith.science/pith/2A5CUYSIXMVIU4WKQ7RWSB4V5G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2A5CUYSIXMVIU4WKQ7RWSB4V5G/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":"2026-08-09T06:52:19Z","links":{"resolver":"https://pith.science/pith/2A5CUYSIXMVIU4WKQ7RWSB4V5G","bundle":"https://pith.science/pith/2A5CUYSIXMVIU4WKQ7RWSB4V5G/bundle.json","state":"https://pith.science/pith/2A5CUYSIXMVIU4WKQ7RWSB4V5G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2A5CUYSIXMVIU4WKQ7RWSB4V5G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:2A5CUYSIXMVIU4WKQ7RWSB4V5G","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":"27d43b9bd6c9ef1355cb6e919c3155215a9ffdd9cfeda964c374d17e0565c7b3","cross_cats_sorted":["math-ph","math.FA","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-20T02:36:23Z","title_canon_sha256":"7167ae376a62e08812356dd9b75ff266b5f76a5046b4dd39fd06e84fbc738726"},"schema_version":"1.0","source":{"id":"2409.13164","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.13164","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"arxiv_version","alias_value":"2409.13164v3","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.13164","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"pith_short_12","alias_value":"2A5CUYSIXMVI","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"pith_short_16","alias_value":"2A5CUYSIXMVIU4WK","created_at":"2026-07-05T11:19:21Z"},{"alias_kind":"pith_short_8","alias_value":"2A5CUYSI","created_at":"2026-07-05T11:19:21Z"}],"graph_snapshots":[{"event_id":"sha256:110f693a8bad46c5a9a53a068f6ccaef0f1e3de9e3d99c707f7e489f8293a3d2","target":"graph","created_at":"2026-07-05T11:19:21Z","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/2409.13164/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We solve a long-standing open problem of determining the Fourier dimension of the Mandelbrot canonical cascade measure (MCCM). This problem of significant interest was raised by Mandelbrot in 1976 and reiterated by Kahane in 1993. Specifically, we derive the exact formula for the Fourier dimension of the MCCM for random weights $W$ satisfying the condition $\\mathbb{E}[W^t]<\\infty$ for all $t>0$. As a corollary, we prove that the MCCM is Salem if and only if the random weight has a specific two-point distribution. In addition, we show that the MCCM is Rajchman with polynomial Fourier decay when","authors_text":"Xinxin Chen, Yanqi Qiu, Yong Han, Zipeng Wang","cross_cats":["math-ph","math.FA","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-20T02:36:23Z","title":"Harmonic analysis of Mandelbrot cascades -- in the context of vector-valued martingales"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.13164","kind":"arxiv","version":3},"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":"sha256:648b48f4e3f923a29cc6edc98dc243330ce6dae6361aac95e853d4432fd7f926","target":"record","created_at":"2026-07-05T11:19:21Z","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":"27d43b9bd6c9ef1355cb6e919c3155215a9ffdd9cfeda964c374d17e0565c7b3","cross_cats_sorted":["math-ph","math.FA","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-09-20T02:36:23Z","title_canon_sha256":"7167ae376a62e08812356dd9b75ff266b5f76a5046b4dd39fd06e84fbc738726"},"schema_version":"1.0","source":{"id":"2409.13164","kind":"arxiv","version":3}},"canonical_sha256":"d03a2a6248bb2a8a72ca87e3690795e98286bb54220955119825daa7c72b97b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d03a2a6248bb2a8a72ca87e3690795e98286bb54220955119825daa7c72b97b0","first_computed_at":"2026-07-05T11:19:21.072509Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:19:21.072509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WwsSGJeAz951XMaPWg7vwlu37AUlBtDSLOv/hYRZhtjTdWlmjcnuu+AlaSpO3fHbI4sN1Mea9EO/Kdl/w9W4Bg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:19:21.072954Z","signed_message":"canonical_sha256_bytes"},"source_id":"2409.13164","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:648b48f4e3f923a29cc6edc98dc243330ce6dae6361aac95e853d4432fd7f926","sha256:110f693a8bad46c5a9a53a068f6ccaef0f1e3de9e3d99c707f7e489f8293a3d2"],"state_sha256":"a4dc53be9ec4e0faae4a6b342a141dae76e529373b1a5128f198365aab9b6ecf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hHzO90n8Rxg89kzXtF52nx34k0WzIttMCPw3yfuq6LMmkJ1xjwrBeChJpgPTjjMBxW+ODtZYVfHTaLEVrSkbBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T06:52:19.210636Z","bundle_sha256":"fe2965d2a200184543fe74fdd283a1f81f9a9efad03f880bad09268c22c68ac2"}}