Nonlinear TLMM Framework v3.8: Bounded-Optimum Structural Suppression under Nonlinear, Stochastic, and Detectability-Constrained Dynamics

This repository presents version 3.8 of the nonlinear Threshold-Limited Mode Modulation (TLMM) framework for bounded-optimum structural suppression. The TLMM describes a class of dynamical systems in which an internal structural mode couples to an externally driven envelope, with effective stiffness determined by a nonlinear saturation function of the time-averaged envelope ⟨E⟩τ. Unlike linear-response models, the framework predicts that structural suppression is not maximized by simply increasing stimulation amplitude, envelope strength, or coupling strength. Instead, effective suppression emerges only within a constrained operating regime defined by nonlinear saturation, optimal entrainment windows, proxy-constrained coupling intervals, detectability thresholds, and practical safety limits. The framework derives: • a bounded-optimum stiffness response with saturation asymptote k0 + β/γ,• a non-monotonic suppression functional,• an optimal entrainment window,• a proxy-constrained viable interval for the cross-scale coupling coefficient α,• a feasibility proposition defining existence conditions for viable operating points,• a protocol-design workflow integrating stimulation parameters, envelope strength, coupling feasibility, and suppression outcome,• sensitivity and robustness analyses across ν, λ, μ, and θR,• and practical application guidelines for realistic stimulation scenarios. Figures 1–6 summarize the theoretical structure, operational workflow, validation logic, parameter sensitivity, and practical design interpretation of the framework. This repository includes:• the full manuscript PDF,• LaTeX source,• Python figure-generation code,• all generated figures,• and supporting README documentation. The framework is intentionally minimal and exploratory. Current limitations include one-dimensional dynamics, phenomenological suppression modeling, absence of full stochastic phase dynamics, lack of multi-mode coupling, and absence of patient-specific calibration. Future versions may extend the framework toward stochastic TLMM dynamics, Kuramoto-derived phase coupling, structural hysteresis, multi-mode suppression, and closed-loop protocol optimization. All numerical examples in this work are illustrative and are not derived from clinical datasets.