{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:A7J2SA5Z2KP4P2VGMEDSIIP5QL","short_pith_number":"pith:A7J2SA5Z","schema_version":"1.0","canonical_sha256":"07d3a903b9d29fc7eaa661072421fd82e97ca4cfa6c558a7aa4f190e01124738","source":{"kind":"arxiv","id":"2606.26133","version":1},"attestation_state":"computed","paper":{"title":"Fa\\`a di Bruno is Taylor Composition","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Heinrich Hartmann","submitted_at":"2026-06-18T14:06:50Z","abstract_excerpt":"We prove that reduced Taylor polynomials compose: for $C^k$ maps $\\phi: E \\to F$ and $\\psi: F \\to G$ between Banach spaces, $T_{\\ast}^k(\\psi\\circ\\phi;\\, x) = \\pi_{\\leq k}\\bigl(T_{\\ast}^k(\\psi;\\, y) \\circ T_{\\ast}^k(\\phi;\\, x)\\bigr)$ where $y = \\phi(x)$.\n  The proof is a direct estimate of the Peano remainder and requires no combinatorics or partition arguments. From this we derive the multivariate Fa\\`a di Bruno formula in partition form (Levy 2006), by polarization, and in multi-index form (Constantine-Savits 1996) by coefficient extraction.\n  As an application we give a higher-order product "},"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":"2606.26133","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-06-18T14:06:50Z","cross_cats_sorted":[],"title_canon_sha256":"7b830de2ad56d2ea07b3323ecc71ca792500343f7ac636a05095b3893d045b01","abstract_canon_sha256":"a131ff7a69d38fdef757b12cd040ecbfa5be1a517ff77af44d1b114a92f20f52"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-26T00:15:27.015483Z","signature_b64":"nQ47/nnLs8WM+ISDONoRVT/x2EC/ixD1JEENBW891F2AkWS+CEuEj4dH/AuZoUvs1TGt7/LAlwh0G2aPSlW4AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07d3a903b9d29fc7eaa661072421fd82e97ca4cfa6c558a7aa4f190e01124738","last_reissued_at":"2026-06-26T00:15:27.014974Z","signature_status":"signed_v1","first_computed_at":"2026-06-26T00:15:27.014974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fa\\`a di Bruno is Taylor Composition","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Heinrich Hartmann","submitted_at":"2026-06-18T14:06:50Z","abstract_excerpt":"We prove that reduced Taylor polynomials compose: for $C^k$ maps $\\phi: E \\to F$ and $\\psi: F \\to G$ between Banach spaces, $T_{\\ast}^k(\\psi\\circ\\phi;\\, x) = \\pi_{\\leq k}\\bigl(T_{\\ast}^k(\\psi;\\, y) \\circ T_{\\ast}^k(\\phi;\\, x)\\bigr)$ where $y = \\phi(x)$.\n  The proof is a direct estimate of the Peano remainder and requires no combinatorics or partition arguments. From this we derive the multivariate Fa\\`a di Bruno formula in partition form (Levy 2006), by polarization, and in multi-index form (Constantine-Savits 1996) by coefficient extraction.\n  As an application we give a higher-order product "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.26133","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/2606.26133/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":"2606.26133","created_at":"2026-06-26T00:15:27.015050+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.26133v1","created_at":"2026-06-26T00:15:27.015050+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.26133","created_at":"2026-06-26T00:15:27.015050+00:00"},{"alias_kind":"pith_short_12","alias_value":"A7J2SA5Z2KP4","created_at":"2026-06-26T00:15:27.015050+00:00"},{"alias_kind":"pith_short_16","alias_value":"A7J2SA5Z2KP4P2VG","created_at":"2026-06-26T00:15:27.015050+00:00"},{"alias_kind":"pith_short_8","alias_value":"A7J2SA5Z","created_at":"2026-06-26T00:15:27.015050+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/A7J2SA5Z2KP4P2VGMEDSIIP5QL","json":"https://pith.science/pith/A7J2SA5Z2KP4P2VGMEDSIIP5QL.json","graph_json":"https://pith.science/api/pith-number/A7J2SA5Z2KP4P2VGMEDSIIP5QL/graph.json","events_json":"https://pith.science/api/pith-number/A7J2SA5Z2KP4P2VGMEDSIIP5QL/events.json","paper":"https://pith.science/paper/A7J2SA5Z"},"agent_actions":{"view_html":"https://pith.science/pith/A7J2SA5Z2KP4P2VGMEDSIIP5QL","download_json":"https://pith.science/pith/A7J2SA5Z2KP4P2VGMEDSIIP5QL.json","view_paper":"https://pith.science/paper/A7J2SA5Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.26133&json=true","fetch_graph":"https://pith.science/api/pith-number/A7J2SA5Z2KP4P2VGMEDSIIP5QL/graph.json","fetch_events":"https://pith.science/api/pith-number/A7J2SA5Z2KP4P2VGMEDSIIP5QL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A7J2SA5Z2KP4P2VGMEDSIIP5QL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A7J2SA5Z2KP4P2VGMEDSIIP5QL/action/storage_attestation","attest_author":"https://pith.science/pith/A7J2SA5Z2KP4P2VGMEDSIIP5QL/action/author_attestation","sign_citation":"https://pith.science/pith/A7J2SA5Z2KP4P2VGMEDSIIP5QL/action/citation_signature","submit_replication":"https://pith.science/pith/A7J2SA5Z2KP4P2VGMEDSIIP5QL/action/replication_record"}},"created_at":"2026-06-26T00:15:27.015050+00:00","updated_at":"2026-06-26T00:15:27.015050+00:00"}