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.