{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:AV3JNM2KJW6KBRX3F4PJB44SZD","short_pith_number":"pith:AV3JNM2K","schema_version":"1.0","canonical_sha256":"057696b34a4dbca0c6fb2f1e90f392c8f72a2b917933093bfc5a00652009dede","source":{"kind":"arxiv","id":"2606.29292","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.29292","created_at":"2026-06-30T01:18:00.496862+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.29292v1","created_at":"2026-06-30T01:18:00.496862+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.29292","created_at":"2026-06-30T01:18:00.496862+00:00"},{"alias_kind":"pith_short_12","alias_value":"AV3JNM2KJW6K","created_at":"2026-06-30T01:18:00.496862+00:00"},{"alias_kind":"pith_short_16","alias_value":"AV3JNM2KJW6KBRX3","created_at":"2026-06-30T01:18:00.496862+00:00"},{"alias_kind":"pith_short_8","alias_value":"AV3JNM2K","created_at":"2026-06-30T01:18:00.496862+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD","json":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD.json","graph_json":"https://pith.science/api/pith-number/AV3JNM2KJW6KBRX3F4PJB44SZD/graph.json","events_json":"https://pith.science/api/pith-number/AV3JNM2KJW6KBRX3F4PJB44SZD/events.json","paper":"https://pith.science/paper/AV3JNM2K"},"agent_actions":{"view_html":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD","download_json":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD.json","view_paper":"https://pith.science/paper/AV3JNM2K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.29292&json=true","fetch_graph":"https://pith.science/api/pith-number/AV3JNM2KJW6KBRX3F4PJB44SZD/graph.json","fetch_events":"https://pith.science/api/pith-number/AV3JNM2KJW6KBRX3F4PJB44SZD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/action/storage_attestation","attest_author":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/action/author_attestation","sign_citation":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/action/citation_signature","submit_replication":"https://pith.science/pith/AV3JNM2KJW6KBRX3F4PJB44SZD/action/replication_record"}},"created_at":"2026-06-30T01:18:00.496862+00:00","updated_at":"2026-06-30T01:18:00.496862+00:00"}