An independent check of the local Fourier compiler in OpenAI's preprint "An explicit power saving for the exact discrete Fourier transform": exact and numerical tests, depth, cost, and limits

This study presents an independent computational investigation of the local Fourier compiler described in OpenAI's September 25, 2026 preprint, "An explicit power saving for the exact discrete Fourier transform." The investigation focuses on the Newton reduction of the discrete Fourier transform to triangular Toeplitz factors, the exact-width block Toeplitz construction using convolution-DAG replay, and the C-slot shear decomposition. Exact computations in cyclotomic fields establish equality between the implemented operator and the DFT matrix for eight selected transform sizes (r = 4, 7, 8, 9, 12, 15, 16, 32). Additional finite-field experiments provide necessary-condition checks at larger sizes, including a streaming verification at r = 1024. The numerical analysis reveals severe finite-precision instability. Although the DFT matrix itself is well-conditioned, the investigated construction loses approximately 142 decimal digits at r = 256. In float64 arithmetic, the observed relative error reaches approximately 8.1 × 10²³ at r = 64. The study further examines fixed circuit topology, arithmetic operation counts, parallel depth under ASAP scheduling, and the relationship between the preprint's computational model and its associated Lean formalization. Scope and limitations: The study verifies a local component of OpenAI's construction, not the full tensor-network algorithm responsible for the claimed asymptotic improvement. The findings neither prove nor refute the main theorem, which concerns exact arithmetic rather than finite-precision numerical stability. The research is accompanied by publicly available implementations, unit tests, experimental results, reproducibility scripts, and verification manifests. Research code: https://github.com/cyberdid/fft-power-saving-verification Reproducibility package DOI: https://doi.org/10.5281/zenodo.23375601