low-latency Fourier Transform (FFT) kernels dominate many signal-processing pipelines, yet their Python implementations often hide algorithmic structure behind high-level abstractions. This work isolates the radix-2 Cooley-Tukey FFT as a balanced binary indexing structure and formalizes three invariants: (i) strict height-balance of the implicit recursion tree, (ii) preservation of key-ordered bit-reversed indices, and (iii) deterministic traversal order guaranteeing O(Nlog N) arithmetic work. By inspecting the abstract syntax tree (AST) of a compact open-source module, we derive exact structural metrics (209 source lines, 14 function blocks) and prove that the implementation respects the invariants under any power-of-two input length. The analysis yields a provable asymptotic bound of T(N)=Θ(Nlog N) without resorting to empirical timing.
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_13_faster_fourier_transforms,
title={Balanced Binary Indexing Invariants for a Python Faster Fourier Transform},
author={Yonathan, Mateus},
journal={Polyglot Systems Lab Research Preprints},
year={2026},
publisher={Polyglot Systems Lab},
url={https://myonathanlinkedin.github.io/papers/2026-10-13_faster_fourier_transforms/},
howpublished={\url{https://raw.githubusercontent.com/myonathanlinkedin/mini-papers/main/papers/2026-10-13_faster_fourier_transforms/2026-10-13_faster_fourier_transforms.pdf}}
}