OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
An explicit power saving for the exact discrete Fourier transform
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.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
We construct exact nonuniform Fourier circuits of size o(nlog n) along an unbounded sequence of lengths, counting every addition, subtraction, and scalar multiplication. This refutes the \Omega(nlog n) lower bound in the unrestricted complex linear-circuit model. The construction uses a finite tensor saving: a tensor power of…
Full source abstract
We construct exact nonuniform Fourier circuits of size o(nlog n) along an unbounded sequence of lengths, counting every addition, subtraction, and scalar multiplication. This refutes the \Omega(nlog 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.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.