Certified Fourier rounding for the real Grothendieck inequality

We give a computer-assisted proof that the real Grothendieck constant is strictly smaller than 1.77987164. The proof provides a complete interval-arithmetic realization of Ren's five-variable construction. A finite Fourier calculation encloses every scalar coefficient through degree 403; a fourth-derivative estimate on the entire unit circle bounds the remaining coefficient sum by 10^-4. Exact rational arithmetic then certifies the final inequality. We include the transfer to all finite real matrices and arbitrary vector dimensions, the analytic quadrature errors, the rational parameters, and source code for checking and regenerating the certificate. Files: paper PDF, arXiv source archive with ancillary material, and standalone code archive. Numerical regeneration uses Python 3.12 and python-flint 0.8.0. The reconstruction command retrieves twelve arithmetic dependencies from the pinned upstream commit and verifies their SHA-256 hashes; a verified cache supports offline execution.