A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

We give a computer-assisted proof that the very-compressed locus with Hilbert function (1,4,10,10) is the reduced support of a generically nonreduced irreducible component of the Hilbert scheme of 25 points on affine four-space over an algebraically closed field of characteristic zero. The support is A^4 times Gr(10,20) and has dimension 104. This supplies the case a=10 omitted from Jelisiejew's published theorem for (1,4,10,a), a=6,7,8,9, and identifies the generic component throughout the interval in AIM Problem 31. The new computation gives 244 primary-obstruction quadrics in 46 normal variables over F_3. Exact F4 rounds produce a positive pure leading power of every variable, proving that the normal obstruction algebra is zero-dimensional. A properness argument transfers this certificate to characteristic zero, where the published Bialynicki-Birula criterion applies. The general obstruction framework and the four earlier cases are established prior work. We do not compute the generic nilpotent local algebra or claim absolute priority. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. The two decisive computational runs use the same patched solver. No independent ideal-membership kernel or computation of the generic nilpotent local algebra is claimed. The manuscript and original materials are CC BY 4.0; the bundled msolve patch is subject to the separately identified upstream GPL terms. Corpus identifier: AIM-ARITHMETIC_GEOMETRY-0078. Paper page: https://eulersolve.org/papers/aim-arithmetic-geometry-0078/