---
title: "Exact Fourier transforms below $n\\log n$"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 130; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "theoretical-computer-science"
  - "lean4"
  - "formalized"
seed_rank: 1986
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 9
lineage: ai-mathematical-reasoning
cites:
  - title: "An explicit power saving for the exact discrete Fourier transform"
    url: "https://github.com/openai/math/blob/main/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Finite tensor savings and exact Fourier circuits"
    url: "https://github.com/openai/math/blob/main/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Exact Fourier transforms below $n\log n$

## One-sentence takeaway

OpenAI's result family 130 (Theoretical computer science) claims: Gives a deterministic length-n discrete Fourier transform algorithm using $O(n(\log n)^{1-\delta})$ operations for every n, with explicit $\delta=10^{-13}$.

## Why it matters here

Closest to the systems side of the library: complexity, algorithms and computability statements here can change what is worth trying in ano, the engines or GRID COMMAND tooling. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems. The Lean development makes the claim machine-checkable, which is the pattern Broadside cares about for agentic verification loops.

## Key ideas

- Family summary (repo): The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.
- *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.
- *Finite tensor savings and exact Fourier circuits*: 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 $\Omega(n\log n)$ 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.
- Lean scope (lean/docs/130.md): The formalized result gives exact discrete Fourier transforms with arbitrarily small normalized circuit cost along an unbounded sequence of lengths. For every $c>0$ and every cutoff $N_0\ge2$, some $n\ge N_0$ has a circuit computing the unnormalized DFT with fewer than $cn\log_2 n$ gates.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: formalized in part. The repo has a scope note listing 1 Comparator challenge statement(s), and that note defines which statements are covered. Do not read the Lean coverage as the full content of every manuscript in the family.
- A grep of the repo's `lean/OAI` tree found no `sorry` and no top-level `axiom` declarations. The library was not compiled for this card, and Comparator was not run.

## Links

- Manuscript: [An explicit power saving for the exact discrete Fourier transform](https://github.com/openai/math/blob/main/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf)
- Manuscript: [Finite tensor savings and exact Fourier circuits](https://github.com/openai/math/blob/main/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf)
- Lean scope: https://github.com/openai/math/blob/main/lean/docs/130.md
- Comparator statement (Subsequential savings for exact Fourier circuits): https://github.com/openai/math/blob/main/lean/ComparatorChallenges/ExactFourier.lean
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math