OpenAI · 130-02 · 2026-09-25 · Theoretical computer science
Finite tensor savings and exact Fourier circuits
preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026
Family 130. Fourier transforms below .
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Gives a deterministic length- discrete Fourier transform algorithm using operations for every , with explicit . The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.
We construct exact nonuniform Fourier circuits of size along an unbounded sequence of lengths, counting every addition, subtraction, and scalar multiplication. This refutes the 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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 130: Exact Fourier transforms below . The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.