Result 130, Theoretical computer science

Exact Fourier transforms below nlog⁡nn\log n

Gives a deterministic length-n discrete Fourier transform algorithm using O(n(log⁡n)1−δ)O(n(\log n)^{1-\delta}) operations for every n, with explicit δ=10−13\delta=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 formalization Algorithm or complexity result

The bigger picture

Why it matters

A Fourier transform rewrites a list of numbers in terms of frequencies. An unreviewed manuscript claims an algorithm that beats an n log n operation count, under an idealized model of exact arithmetic.

What changes?

For every input length n, the manuscript reports a deterministic discrete Fourier transform algorithm using O(n (log n)^(1 - 10^-13)) operations. The saving is a fixed but extremely small reduction in the logarithm's exponent. The model allows exact complex arithmetic and unrestricted coefficients, with the required Fourier root of unity supplied. It includes scalar preparation and array organization in the count, and treats indexing with logarithmic-size words as unit-cost work.

What does that help mathematicians do?

A companion manuscript reports length-specific Fourier circuits, meaning sequences of arithmetic operations, whose sizes grow strictly more slowly than n log n along an unbounded sequence of lengths. It counts additions, subtractions and scalar multiplications. This would rule out a universal n log n lower bound for unrestricted complex linear circuits. Researchers seeking such a lower bound would therefore need restrictions that exclude these constructions.

Are there practical applications?

The immediate value is foundational: the claimed results change the operation-count limits for exact Fourier computation in the stated models. They do not demonstrate faster numerical software. Exact complex arithmetic and unrestricted coefficients differ from finite-precision computation, and the tiny exponent saving alone gives no practical runtime guarantee at usable input sizes.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

An explicit power saving for the exact discrete Fourier transform

September 25, 2026 31 pages

We give a deterministic algorithm that computes the discrete Fourier transform at every length n in O(n(log⁡n)1−10−13)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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026,
  author = {{OpenAI}},
  title = {{An explicit power saving for the exact discrete Fourier transform}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf}{OAI:An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026}},
  year = {2026}
}

Finite tensor savings and exact Fourier circuits

September 25, 2026 49 pages Main result formalized in Lean

We construct exact nonuniform Fourier circuits of size o(nlog⁡n)o(n\log n) along an unbounded sequence of lengths, counting every addition, subtraction, and scalar multiplication. This refutes the Ω(nlog⁡n)\Omega(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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026,
  author = {{OpenAI}},
  title = {{Finite tensor savings and exact Fourier circuits}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf}{OAI:Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/130.md.

Exact Fourier transforms below ‘nlog⁡n‘`n\log n`

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result gives exact discrete Fourier transforms with arbitrarily small normalized circuit cost along an unbounded sequence of lengths. For every c>0c>0 and every cutoff N0≥2N_0\ge2, some n≥N0n\ge N_0 has a circuit computing the unnormalized DFT with fewer than cnlog⁡2ncn\log_2 n gates. Addition, subtraction, and multiplication by a predetermined complex scalar each cost one gate; diagonal scalings are charged. The result is subsequential, with no all-length, bounded-coefficient, conditioning, or bit-complexity claim.

Comparator links

Result Comparator statement
Subsequential savings for exact Fourier circuits ExactFourier.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.