Exact Fourier transforms below n log n

What OpenAI claims

Gives a deterministic length-n discrete Fourier transform algorithm using O(n(log n)^1-δ) operations for every n, with explicit δ=10^-13. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing. (Lean)

Papers (2)

1. An explicit power saving for the exact discrete Fourier transform

   25 September 2026 · Preprint, not peer reviewed · PDF

   We give a deterministic algorithm that computes the discrete Fourier transform at every length n in O(n(log n)^1-10-13) operations. The model uses exact complex arithmetic, unrestricted coefficients, specified Fourier roots, and unit-cost logarithmic-size indexing; scalar preparation and array organization are included.

2. Finite tensor savings and exact Fourier circuits

   25 September 2026 · Preprint, not peer reviewed · PDF

   We construct exact nonuniform Fourier circuits of size o(n log n) along an unbounded sequence of lengths, counting every addition, subtraction, and scalar multiplication. This refutes the Ω(n log n) lower bound in the unrestricted complex linear-circuit model. The construction uses a finite tensor saving: a tensor power of some invertible nonmonomial complex matrix can be computed with fewer matrix calls than the standard tensor-axis algorithm on the same coordinates, when invertible monomial maps are allowed freely between calls.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.