{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:AV3JNM2KJW6KBRX3F4PJB44SZD","short_pith_number":"pith:AV3JNM2K","canonical_record":{"source":{"id":"2606.29292","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-28T09:27:42Z","cross_cats_sorted":[],"title_canon_sha256":"b6322881ebfdf07e067373b081047fd15c4058e1a1ec8f204a5b1bf89359c0a9","abstract_canon_sha256":"5a7e515d1627744d680beee76083a26d39d9171fca50bece7c135b8d0bcf79aa"},"schema_version":"1.0"},"canonical_sha256":"057696b34a4dbca0c6fb2f1e90f392c8f72a2b917933093bfc5a00652009dede","source":{"kind":"arxiv","id":"2606.29292","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.29292","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"arxiv_version","alias_value":"2606.29292v1","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.29292","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"pith_short_12","alias_value":"AV3JNM2KJW6K","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"pith_short_16","alias_value":"AV3JNM2KJW6KBRX3","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"pith_short_8","alias_value":"AV3JNM2K","created_at":"2026-06-30T01:18:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:AV3JNM2KJW6KBRX3F4PJB44SZD","target":"record","payload":{"canonical_record":{"source":{"id":"2606.29292","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-28T09:27:42Z","cross_cats_sorted":[],"title_canon_sha256":"b6322881ebfdf07e067373b081047fd15c4058e1a1ec8f204a5b1bf89359c0a9","abstract_canon_sha256":"5a7e515d1627744d680beee76083a26d39d9171fca50bece7c135b8d0bcf79aa"},"schema_version":"1.0"},"canonical_sha256":"057696b34a4dbca0c6fb2f1e90f392c8f72a2b917933093bfc5a00652009dede","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T01:18:00.497221Z","signature_b64":"hvZ8XGVpfsas32ik93C9nOS6dZhvcuakND2bGGjvTh510IpBJWG8haeJIuEiy0KKKlQpaQM2nObHf2uQSDdeBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"057696b34a4dbca0c6fb2f1e90f392c8f72a2b917933093bfc5a00652009dede","last_reissued_at":"2026-06-30T01:18:00.496799Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T01:18:00.496799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.29292","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":"2026-06-30T01:18:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qrm1FW80rhzOnpHy7/DjEEBfMBBgU7kqx5yG+5VomWwsNSjrdA7WXc5Z8y81aAAY8ProaVbhhToKNzPOKIsNBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T02:40:23.672483Z"},"content_sha256":"de4af7837d5ceeadb6037f69fb62a9cf72f411f8d68878ea5eb54d105778cbaf","schema_version":"1.0","event_id":"sha256:de4af7837d5ceeadb6037f69fb62a9cf72f411f8d68878ea5eb54d105778cbaf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:AV3JNM2KJW6KBRX3F4PJB44SZD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Finite-Order Hilbertian Gaussian Random Tensor Estimates","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guangqian Zhao","submitted_at":"2026-06-28T09:27:42Z","abstract_excerpt":"We prove fixed finite-chaos-order estimates for Hilbert-space-valued Gaussian random tensors. Given a finite-rank kernel \\[\n  K\\in\\cA_1\\otimes\\cdots\\otimes\\cA_m\\otimes\\cC\\otimes\\cE \\] and the associated decoupled homogeneous Gaussian chaos operator $\\cT_K^{(m)}:\\cC\\to\\cE$, we show that, for $p\\ge2$ and $2\\le r<\\infty$, \\[\n  \\|\\cT_K^{(m)}\\|_{L^p(\\Omega;\\mathfrak S_r(\\cC,\\cE))}\n  \\le C_m(p+r)^{m/2}\n  \\max_{S\\subset[m]}\\|\\cF_S(K)\\|_{\\mathfrak S_r}, \\] where $\\cF_S(K):\\cA_S\\otimes\\cC\\to\\cA_{S^c}\\otimes\\cE$ is the oriented input-output flattening. The proof is an induction on $m$ from the rectangul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29292","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/2606.29292/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":"2026-06-30T01:18:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PwNpveqMNgdo51uo6BnP1guvC25OYbEPbVKVaEK25J3M7eugh5Yh6FnVL7kSUisbypG1VKoRRl/NLPQ+9rZNBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T02:40:23.672990Z"},"content_sha256":"37a8aa07fb6a817e8cc9060f8e840055849076fa757c45ff0199af55d2516b53","schema_version":"1.0","event_id":"sha256:37a8aa07fb6a817e8cc9060f8e840055849076fa757c45ff0199af55d2516b53"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/bundle.json","state_url":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/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":"2026-07-01T02:40:23Z","links":{"resolver":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD","bundle":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/bundle.json","state":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:AV3JNM2KJW6KBRX3F4PJB44SZD","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":"5a7e515d1627744d680beee76083a26d39d9171fca50bece7c135b8d0bcf79aa","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-28T09:27:42Z","title_canon_sha256":"b6322881ebfdf07e067373b081047fd15c4058e1a1ec8f204a5b1bf89359c0a9"},"schema_version":"1.0","source":{"id":"2606.29292","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.29292","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"arxiv_version","alias_value":"2606.29292v1","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.29292","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"pith_short_12","alias_value":"AV3JNM2KJW6K","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"pith_short_16","alias_value":"AV3JNM2KJW6KBRX3","created_at":"2026-06-30T01:18:00Z"},{"alias_kind":"pith_short_8","alias_value":"AV3JNM2K","created_at":"2026-06-30T01:18:00Z"}],"graph_snapshots":[{"event_id":"sha256:37a8aa07fb6a817e8cc9060f8e840055849076fa757c45ff0199af55d2516b53","target":"graph","created_at":"2026-06-30T01:18:00Z","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/2606.29292/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove fixed finite-chaos-order estimates for Hilbert-space-valued Gaussian random tensors. Given a finite-rank kernel \\[\n  K\\in\\cA_1\\otimes\\cdots\\otimes\\cA_m\\otimes\\cC\\otimes\\cE \\] and the associated decoupled homogeneous Gaussian chaos operator $\\cT_K^{(m)}:\\cC\\to\\cE$, we show that, for $p\\ge2$ and $2\\le r<\\infty$, \\[\n  \\|\\cT_K^{(m)}\\|_{L^p(\\Omega;\\mathfrak S_r(\\cC,\\cE))}\n  \\le C_m(p+r)^{m/2}\n  \\max_{S\\subset[m]}\\|\\cF_S(K)\\|_{\\mathfrak S_r}, \\] where $\\cF_S(K):\\cA_S\\otimes\\cC\\to\\cA_{S^c}\\otimes\\cE$ is the oriented input-output flattening. The proof is an induction on $m$ from the rectangul","authors_text":"Guangqian Zhao","cross_cats":[],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-28T09:27:42Z","title":"Finite-Order Hilbertian Gaussian Random Tensor Estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29292","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:de4af7837d5ceeadb6037f69fb62a9cf72f411f8d68878ea5eb54d105778cbaf","target":"record","created_at":"2026-06-30T01:18:00Z","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":"5a7e515d1627744d680beee76083a26d39d9171fca50bece7c135b8d0bcf79aa","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-28T09:27:42Z","title_canon_sha256":"b6322881ebfdf07e067373b081047fd15c4058e1a1ec8f204a5b1bf89359c0a9"},"schema_version":"1.0","source":{"id":"2606.29292","kind":"arxiv","version":1}},"canonical_sha256":"057696b34a4dbca0c6fb2f1e90f392c8f72a2b917933093bfc5a00652009dede","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"057696b34a4dbca0c6fb2f1e90f392c8f72a2b917933093bfc5a00652009dede","first_computed_at":"2026-06-30T01:18:00.496799Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-30T01:18:00.496799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hvZ8XGVpfsas32ik93C9nOS6dZhvcuakND2bGGjvTh510IpBJWG8haeJIuEiy0KKKlQpaQM2nObHf2uQSDdeBw==","signature_status":"signed_v1","signed_at":"2026-06-30T01:18:00.497221Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.29292","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de4af7837d5ceeadb6037f69fb62a9cf72f411f8d68878ea5eb54d105778cbaf","sha256:37a8aa07fb6a817e8cc9060f8e840055849076fa757c45ff0199af55d2516b53"],"state_sha256":"eb7017816cd66f1f717104da29c0b9f2f81061780dd0c1d86fbb1c5bc54756df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XnRp1KcU7fynk3W5629VE09bc+/5h6nb07uNzKQUE7l/9Gvp55W2Va2YTDo+gcIDWR5U+iBRatNEUd6Xxv1XAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T02:40:23.675958Z","bundle_sha256":"946ebd0236877b11b4c5d7b2fd64aad01f54237cf0c8cc252ecd5df133a47fc5"}}