Certified Spectral Computation for Semilinear Evolution: Finite Records and Polynomial Bit Cost

We study finite records whose accuracy for an infinite-dimensional nonlinear evolution can be checked without rerunning the producing algorithm. The model is a fixed Neumann reaction–diffusion equation with quadratic reaction on a fixed time interval. Inputs are rational cosine polynomials satisfying a common exponentially weighted coefficient bound. Exact rational tests of continuity and coefficient bounds together with the initial discrepancy and Galerkin residual certify the trajectory error for the original equation and the error of its terminal mean and point value. Acceptance depends on the submitted record independently of the algorithm that produced its coefficients. A rational Taylor construction yields an accepted record for every admissible input and requested accuracy. Construction cost and record length are polynomial in the complete input length and the number of requested accuracy bits. Checking is polynomial in the input and submitted record lengths. The proof controls quantized restarts and the bit lengths of intermediate numerators and denominators. A separate rational check certifies quadratic accounts with positive transport defects and signed injection and approximation increments. Finite response approximations also include operator-tail and rounding bounds in the acceptance test. These results give a computational realization of the generated spectral framework for the encoded Neumann family. A rational finite-dimensional system provides a second realization through quantized Picard iteration. Construction and checking programs accompany both models.Reading and research materialsSSRN preprintZenodo archiveArticle and supplementary materialsJinku Guo · Northwestern Polytechnical University · ORCID 0009-0000-6600-6171.This metadata record provides a research summary and links to the work. Full texts, software and supplementary files are available at the linked repositories under their stated licences.