We present a formally verified implementation of a balanced k-D tree in Python that supports efficient nearest-neighbor queries and dynamic insertion. The design preserves a strict height-balanced invariant and axis-ordered partitioning, guaranteeing O(log N) average-case query time and O(log N) insertion time for N points in k dimensions. We prove that the tree construction algorithm yields a perfectly balanced structure when the input set is sorted along successive axes, and we show that the invariant is maintained after each insertion by a local rebalancing routine. Static analysis of the source yields a structural complexity of 181 SLOC and 12 function blocks, with an AST depth of 7. The implementation is fully reproducible and available at
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_18_k_d_tree_spatial_partitioning_and_n,
title={Height-Balanced Invariant Analysis of a Python K-D Tree for Nearest Neighbor Search},
author={Yonathan, Mateus},
journal={Polyglot Systems Lab Research Preprints},
year={2026},
publisher={Polyglot Systems Lab},
url={https://myonathanlinkedin.github.io/papers/2026-10-18_k_d_tree_spatial_partitioning_and_n/},
howpublished={\url{https://raw.githubusercontent.com/myonathanlinkedin/mini-papers/main/papers/2026-10-18_k_d_tree_spatial_partitioning_and_n/2026-10-18_k_d_tree_spatial_partitioning_and_n.pdf}}
}