We present a PHP-based notification engine that couples a token-bucket rate limiter, exponential back-off retry policy, and a height-balanced binary search tree for queue ordering. The design preserves a strict height-balanced invariant and key-ordered invariant, guaranteeing O(log N) complexity for enqueue, dequeue, and retry scheduling. We formalize state transitions for notification objects, including attempt counters and next-attempt timestamps, and prove that the queue remains height-balanced after each operation. Static analysis of the source yields 253 SLOC and 46 function blocks, with an AST depth of 12. No empirical timing data is reported; all claims are supported by formal proofs and static bounds.
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_designing_a_notification_system_que,
title={Height-Balanced Token-Bucket Invariants for PHP-Based Notification Queues with Exp...},
author={Yonathan, Mateus},
journal={Polyglot Systems Lab Research Preprints},
year={2026},
publisher={Polyglot Systems Lab},
url={https://myonathanlinkedin.github.io/papers/2026-10-16_designing_a_notification_system_que/},
howpublished={\url{https://raw.githubusercontent.com/myonathanlinkedin/mini-papers/main/papers/2026-10-16_designing_a_notification_system_que/2026-10-16_designing_a_notification_system_que.pdf}}
}