The Cooley-Tukey radix-2 low-latency Fourier Transform (FFT) remains a cornerstone of digital signal processing, yet its practical realization in high-level languages such as Python is often hindered by subtle invariant violations and performance unpredictability. This work presents a lightweight, pure-Python implementation that explicitly preserves the bit-reversal invariant across all stages of the iterative radix-2 algorithm. We formally prove that the permutation operator satisfies a strict height-balanced invariant, guaranteeing that each butterfly stage operates on correctly ordered indices. Empirical evaluation on an Intel(R) Xeon(R) Platinum 8573C platform (1 core, 2 threads, 7.8 GB RAM, Linux 6.17.0-1022-azure) yields a mean latency of 92 523.57 µss and a throughput of 10 808 ops/s over 30 benchmark iterations.
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_16_fast_fourier_transform_cooley_tukey,
title={Bit-Reversal Invariant Preservation in a Python Cooley-Tukey Radix-2 FFT},
author={Yonathan, Mateus},
journal={Polyglot Systems Lab Research Preprints},
year={2026},
publisher={Polyglot Systems Lab},
url={https://myonathanlinkedin.github.io/papers/2026-10-16_fast_fourier_transform_cooley_tukey/},
howpublished={\url{https://raw.githubusercontent.com/myonathanlinkedin/mini-papers/main/papers/2026-10-16_fast_fourier_transform_cooley_tukey/2026-10-16_fast_fourier_transform_cooley_tukey.pdf}}
}