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.

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.