Dynamical Threshold Theory of Sleep v0.3.2 is a claim-bounded public repository refresh of a reduced toy-network diagnostic framework for sleep-like recovery transitions, instability recurrence, hysteresis, and rescue-regime behavior. The archived framework studies a limited theoretical question: in a simplified dynamical toy network, under what conditions does a sleep-like recovery transition succeed, fail, recur, or require a stronger rescue-like intervention? The model treats sleep-like recovery as a threshold-regulated transition in a reduced non-equilibrium surrogate system, rather than as a clinical or physiological sleep model. The package integrates the earlier DTTS manuscript and audit materials into a cleaner public-facing repository structure. The v0.3.2 release mainly improves public readability, metadata, claim-boundary statements, GitHub Pages compatibility, manifest consistency, and discoverability. It does not introduce new medical, clinical, diagnostic, or treatment claims. The core interpretive result is conservative: within the tested toy-network setting, recovery-like transitions can exhibit threshold failure, hysteresis, recurrence, and rescue-regime dependence. This should be understood as a hypothesis-generating reduced-surrogate observation about dynamical recovery transitions, not as a model of human sleep, a sleep-disorder explanation, a treatment proposal, or a recommendation for behavioral or medical intervention. This release does not claim to diagnose insomnia, sleep disorders, fatigue, circadian disruption, psychiatric conditions, neurological conditions, or any medical state. It does not claim to predict human sleep from EEG, wearable data, subjective reports, or clinical measurements. It does not provide sleep advice, sleep-improvement methods, treatment protocols, medical recommendations, or clinical decision support. Any future empirical extension would require externally defined datasets, clinical oversight where applicable, predeclared metrics, physiological validation, appropriate baselines, safety review, and expert domain evaluation. Until then, DTTS should be read strictly as a claim-bounded toy-network diagnostic archive for thinking about threshold-regulated recovery transitions.