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When $\\eta^i$ is smooth, it is well-known that there is a unique connection 1-form $\\omega$ verifying Cartan's first structural equations $d\\eta^i = (*\\eta^i) \\wedge \\omega$, and Cartan's second structural equation $d\\omega = K_g dvol_g$. We prove that this statement remains valid when the frame is $C^0 \\cap H^{\\frac12}$, where the structural equations are understood in the sense of distributions. From this, we deduce that the Gauss equation $\\mathrm{Det}\\, D^2 f = K_g (1+|Df|^2)^2$ holds for"},"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":"2607.02412","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T16:44:12Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"003d45893a3c1d2ceb5a839c94955332d97f179e7771e5d833e59a09f2d9ce14","abstract_canon_sha256":"19bf96357763202ae36805028003f33fa537d9d6be0b3c68a810e8c51094d3a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:58.446684Z","signature_b64":"AjfpSkqNCFocMeein6O65L1P3D11wQv6PKkBP7aIGWxq3u1HFVjm7LniphQX8lb6S6v82wC7SOJpohPm51xpDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd16be99489b7393c28590b63ff92712d6df76016d41e83e9bb1ef64b08573cc","last_reissued_at":"2026-07-03T01:17:58.446243Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:58.446243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cartan's and Gauss's equations and rigidity theorems for isometric embeddings in low Sobolev regularity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Isaac Newell, Luc Nguyen","submitted_at":"2026-07-02T16:44:12Z","abstract_excerpt":"Let $\\{\\eta^i\\}_{i=1}^2$ be a an orthonormal coframe on a domain $U$ on a smooth surface $(\\Sigma,g)$. 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