# Improving OpenAI's bound on the exact discrete Fourier transform below n log n

> Source: <https://twitter.com/ryaneshea/status/2108239154572394772>
> Published: 2026-10-09 00:43:06+00:00

We’re publishing a research draft on OpenAI Problem #130: computing the exact discrete Fourier transform below n log n.
Our draft proposes an all-length bound of
T(n) = O(n(log n)^(1−δ)), with δ = 7.3×10⁻⁵.
That’s a 730-million-fold increase in the exponent saving over OpenAI’s published δ = 10⁻¹³.
Building on Swapnil Jain’s round-six complex network for Problem #109 and its cited predecessors, we propose transferring those advances from integer multiplication to the Fourier setting.

Sixth update to OpenAI problem #109 (integer multiplication): we are now past 2^-15.
κ > 2⁻¹⁵  (tightened from κ = 2⁻¹⁸²)
The exact witness is 3.667 × 10⁻⁵, about 2.4 fold over our previous 1.548 × 10⁻⁵, and a 2¹⁶⁷ fold improvement over the original OAI result. 
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