Smooth isometric immersion of every closed Riemannian surface into R^4

Theorem 1.1: every closed smooth Riemannian surface (compact, without boundary, not necessarily orientable or connected, any Gaussian curvature) admits a $C^\infty$ isometric immersion into $\mathbb R^4$ ; self-intersections are allowed. This lowers Gromov's $\mathbb R^5$ immersion dimension for closed surfaces to four. Not shown: embeddings, noncompact or bounded surfaces (the paper says it treats the compact boundaryless case of Gromov's question), or analytic regularity.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Smooth isometric immersion of every closed Riemannian surface into R^4 · Closed surfaces isometrically into R4

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Smooth isometric immersion of every closed Riemannian surface into R^4

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No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against the question as the manuscript cites it from Gromov. The construction (positive rank-one metric decomposition, oscillatory corrections with controlled normal geometry, smoothing) was not refereed. The challenge lean/ComparatorChallenges/SurfaceImmersion.json (theorem OAI.ClosedSurfaceR4.FiniteOrderSmoothing.smooth_isometric_immersion, solution module OAI.Geometry.SurfaceImmersion.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml; it is found through lean/docs/333.md. Its statement was read here: for every compact Hausdorff second-countable smooth manifold modelled on $\mathbb R^2$ without boundary and every smooth Riemannian metric, there is a smooth map to $\mathbb R^4$ whose differential preserves the inner product on every tangent space. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.