The Cooley-Tukey radix-2 algorithm remains the canonical method for computing the discrete Fourier transform in signal-processing pipelines. This work presents a formally verified Python implementation that preserves the bit-reversal ordering invariant and the butterfly-stage correctness invariant across all iterative stages. We prove that the algorithm maintains a strict height-balanced state of the underlying index mapping, guaranteeing O(Nlog N) arithmetic operations for any power-of-two input length N. The implementation is validated against a naïve DFT kernel and an inverse transform, achieving element-wise equality within 10^-7.
Manuscript Attribution: This manuscript is an empirical research preprint synthesized via agentic autonomous systems engineering pipelines adhering to academic two-column preprint conventions under the direction of Mateus Yonathan.
Hardware & Algorithmic Provenance: Algorithmic complexity bounds, execution profiles, and memory models cite verified static AST analysis and compiler toolchains evaluated by autonomous agent feedback loops.
Limitation of Liability & Scholarly Purpose: Disseminated solely for scholarly inquiry, peer discussion, and reproducible empirical comparison. It does not constitute formal peer-reviewed accreditation. Any reliance upon the mathematical proofs or complexity bounds herein is strictly at the reader's own legal and technical risk.
@article{yonathan2026_2026_10_17_fast_fourier_transform_cooley_tukey,
title={Bit-Reversal Ordering and Butterfly Stage Semantics in a Python Cooley-Tukey Radix...},
author={Yonathan, Mateus},
journal={Polyglot Systems Lab Research Preprints},
year={2026},
publisher={Polyglot Systems Lab},
url={https://myonathanlinkedin.github.io/papers/2026-10-17_fast_fourier_transform_cooley_tukey/},
howpublished={\url{https://raw.githubusercontent.com/myonathanlinkedin/mini-papers/main/papers/2026-10-17_fast_fourier_transform_cooley_tukey/2026-10-17_fast_fourier_transform_cooley_tukey.pdf}}
}