Local Smoothability Without Global Smoothability: An Explicit Two-Chart Surface and Its Fourteen-Dimensional Formal Deformation Space

We construct an explicit separated nonnormal surface over a field of characteristic zero by gluing two affine semigroup surfaces. Although both affine charts admit flat smoothings, the resulting surface admits no flat finite-type smoothing over an equicharacteristic-zero discrete valuation ring. We compute the complete unrestricted first-order deformation space, establishing that it has dimension fourteen, and construct an explicit formally miniversal family over a smooth fourteen-dimensional formal base. The construction combines multigraded conormal linear algebra, finite free algebra constructions, determinantal presentations, exact overlap identities, and a complete Kodaira–Spencer calculation. Every deformation in the formal family carries a distinguished section along which the relative cotangent module requires at least three generators. Formal versality transports this section to any putative DVR smoothing, yielding a contradiction to smoothness of the generic fiber. The manuscript includes the complete mathematical proof, independent verification certificates, exact computational scripts and outputs, detailed experiment ledgers, and historical calculations documenting both successful and unsuccessful approaches. All supporting materials are consolidated into a single self-contained PDF to facilitate independent verification and reproducibility. The theorem concerns the explicit abstract surface constructed in the manuscript. Its identification with the motivating algebraic-cycle geometry remains an open problem.