Spatial indexing of high-dimensional bounding boxes is central to computer vision, geographic information systems, and robotics. We present a compact, height-balanced R-Tree implementation in Python that preserves strict MBR invariants during insertion, deletion, and node splitting. The design guarantees that every node's minimum bounding rectangle (MBR) encloses all descendant rectangles, and that the tree height remains logarithmic in the number of stored objects. Formal proofs establish that point and range queries, as well as insertions, execute in O(log N) time, while the memory footprint is linear.
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_r_tree_spatial_index_for_multidimen,
title={Height-Balanced Invariants and Search Complexity in a Python R-Tree for Multidimen...},
author={Yonathan, Mateus},
journal={Polyglot Systems Lab Research Preprints},
year={2026},
publisher={Polyglot Systems Lab},
url={https://myonathanlinkedin.github.io/papers/2026-10-18_r_tree_spatial_index_for_multidimen/},
howpublished={\url{https://raw.githubusercontent.com/myonathanlinkedin/mini-papers/main/papers/2026-10-18_r_tree_spatial_index_for_multidimen/2026-10-18_r_tree_spatial_index_for_multidimen.pdf}}
}