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We show that the curvature of $\\mathcal{W}$ is holomorphic along $C$ if and only if the curvature of $\\mathcal{W}_n$ is holomorphic along $C$. When $\\mathcal{W}_n$ is non-degenerate along $C$, we prove that $K(\\mathcal{W}_n)$, and hence $K(\\mathcal{W})$, is holomorphic along $C.$ We deduce that, if $\\mathcal{W}_n$ is irreducible and $\\mathrm{mult}\\le"},"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":"2606.13373","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-06-11T13:57:11Z","cross_cats_sorted":["math.CV","math.DG"],"title_canon_sha256":"bf4be0e538f3b663a051fb36989ee664a2ca703cf2b34d37bafa22e1f7ffa446","abstract_canon_sha256":"a7801fa4c7363923c11e93e5b19ae86d90d27f822ff1f993cf258a6714fea4b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:09:58.279312Z","signature_b64":"aZ11lLnCRw4o6tjiLmdw1ud9Flz2QstoAAOvvqyk1n/Aou4QxXmsD3lgISm5I7MMmxmjOpUCoweaINiparBHBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e427b959d27bda02a595371651a03fcb714f7464373f744edff885fe4369f72","last_reissued_at":"2026-06-12T01:09:58.278481Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:09:58.278481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the holomorphy of the curvature of planar webs along an invariant curve","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CV","math.DG"],"primary_cat":"math.DS","authors_text":"Samir Bedrouni","submitted_at":"2026-06-11T13:57:11Z","abstract_excerpt":"Let $\\mathcal{W}=\\mathcal{W}_{n}\\boxtimes\\mathcal{W}_{d-n}$ be a $d$-web on $(\\mathbb{C}^2,0)$, where $\\mathcal{W}_n$ is an $n$-web with a totally invariant irreducible curve~$C$, and $\\mathcal{W}_{d-n}$ is a regular $(d-n)$-web transverse to $C$. We show that the curvature of $\\mathcal{W}$ is holomorphic along $C$ if and only if the curvature of $\\mathcal{W}_n$ is holomorphic along $C$. When $\\mathcal{W}_n$ is non-degenerate along $C$, we prove that $K(\\mathcal{W}_n)$, and hence $K(\\mathcal{W})$, is holomorphic along $C.$ We deduce that, if $\\mathcal{W}_n$ is irreducible and $\\mathrm{mult}\\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13373","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/2606.13373/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":"2606.13373","created_at":"2026-06-12T01:09:58.278619+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.13373v1","created_at":"2026-06-12T01:09:58.278619+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.13373","created_at":"2026-06-12T01:09:58.278619+00:00"},{"alias_kind":"pith_short_12","alias_value":"DZBHXFM5E662","created_at":"2026-06-12T01:09:58.278619+00:00"},{"alias_kind":"pith_short_16","alias_value":"DZBHXFM5E662AKSZ","created_at":"2026-06-12T01:09:58.278619+00:00"},{"alias_kind":"pith_short_8","alias_value":"DZBHXFM5","created_at":"2026-06-12T01:09:58.278619+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/DZBHXFM5E662AKSZKNYWKGQD7S","json":"https://pith.science/pith/DZBHXFM5E662AKSZKNYWKGQD7S.json","graph_json":"https://pith.science/api/pith-number/DZBHXFM5E662AKSZKNYWKGQD7S/graph.json","events_json":"https://pith.science/api/pith-number/DZBHXFM5E662AKSZKNYWKGQD7S/events.json","paper":"https://pith.science/paper/DZBHXFM5"},"agent_actions":{"view_html":"https://pith.science/pith/DZBHXFM5E662AKSZKNYWKGQD7S","download_json":"https://pith.science/pith/DZBHXFM5E662AKSZKNYWKGQD7S.json","view_paper":"https://pith.science/paper/DZBHXFM5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.13373&json=true","fetch_graph":"https://pith.science/api/pith-number/DZBHXFM5E662AKSZKNYWKGQD7S/graph.json","fetch_events":"https://pith.science/api/pith-number/DZBHXFM5E662AKSZKNYWKGQD7S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DZBHXFM5E662AKSZKNYWKGQD7S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DZBHXFM5E662AKSZKNYWKGQD7S/action/storage_attestation","attest_author":"https://pith.science/pith/DZBHXFM5E662AKSZKNYWKGQD7S/action/author_attestation","sign_citation":"https://pith.science/pith/DZBHXFM5E662AKSZKNYWKGQD7S/action/citation_signature","submit_replication":"https://pith.science/pith/DZBHXFM5E662AKSZKNYWKGQD7S/action/replication_record"}},"created_at":"2026-06-12T01:09:58.278619+00:00","updated_at":"2026-06-12T01:09:58.278619+00:00"}