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Does the exact discrete Fourier transform require order n log n operations in unrestricted linear circuits?

Principal (Theorem 1.1): in exact complex linear circuits with arbitrary prechosen coefficients, $\liminf_n L(n)/(n\log_2 n)=0$ , along lengths that are products of distinct primes, so the unrestricted $\Omega(n\log n)$ lower bound is false. Companion (Theorem 1.1): one deterministic algorithm computes $F_nx$ at every length in $O(n(\log n)^{\theta}(\log\log n)^{4-\theta})$ operations with $\theta<1-10^{-13}$ , counting scalar preparation and indexing, given one specified root of unity. Neither result concerns bounded coefficients (where Morgenstern's bound stands), numerical stability, bit complexity or finite fields, and the constants are astronomically large.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Does the exact discrete Fourier transform require order n log n operations in unrestricted linear circuits? · The n log n lower bound for the DFT

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Does the exact discrete Fourier transform require order n log n operations in unrestricted linear circuits?

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the question; it states $\liminf L(n)/(n\log_2 n)=0$ in the unrestricted model, refuting the lower bound. lean/formalization.yaml lists comparator ExactFourier, declaration OAI.ExactFourier.main_theorem. The statement was read here: for every $c>0$ and $N_0\ge2$ there are $n\ge N_0$ and a circuit (a DAG of add, subtract and scale-by-any-complex gates) computing the unnormalised $F_n$ exactly with fewer than $cn\log_2 n$ gates. That is the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The model allows unbounded coefficients and conditioning; nothing is claimed for bit complexity or floating-point FFTs. The 7 October 2026 update added the companion's Comparator challenge UniformFourier (OAI.PowerSaving.transform_main and convolution_main, solution module OAI.Computability.FourierTransform.Main); read here, it states one fixed exact-complex RAM program computing the DFT (and one for convolution) at every positive length with work $O(n(\log n)^{1-10^{-13}})=o(n\log n)$ , given a supplied root of unity.