Realized and Mediated Response Across Dynamical Domains: Minimal Models and Context-Dependent Thresholds

This paper continues the philosophical-physical framework developed in Preface to the Trilogy on the Principle of Dynamical Sensitivity, The Principle of Dynamical Sensitivity of Structures within a Shared Physical Environment (Paper I), Operationalizing the Principle of Dynamical Sensitivity: Response, Coupling, and Cross-Scale Interaction (Paper II), and Mediated Interaction in a Shared Dynamical Environment: A Mechanistic Extension of the Principle of Dynamical Sensitivity (Paper III). Its focus is a consequence already implicit in the earlier response-based formulation: the threshold of realized response need not always be treated as a single fixed criterion.

The paper argues that thresholding should be treated in graded and context-dependent form, so that the same realized response may be evaluated differently depending on whether the relevant analytical aim concerns detectability, effective interaction, persistence, or structural consequence. On that basis, the paper develops a common abstract response skeleton and illustrates it in three connected minimal settings: a two-oscillator Kuramoto-type model, a candidate dynamical-network formulation, and a minimal mediated three-node chain.

Across these cases, the paper distinguishes inherited sensitivity as a guiding horizon, realized response as an operational category, and threshold-dependent effective regimes as analytically separable levels. In the mediated case, it further distinguishes latent channels from dynamically effective pathways and shows how graded thresholds may be extended to mediated realized response.

The paper does not propose new synchronization equations, a new universal response theorem, or a new physical force. Its contribution is instead a clarified cross-domain synthesis: inherited sensitivity, realized response, differentiated thresholding, and mediated response are brought into one common analytical language equipped with minimal formal illustrations.

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Post-publication author’s note: In Section 6.2, the output-response measure contains a typographical error. The correct expression uses delta y_t rather than delta h_t: I_y(T) := (1/T) sum from t = 0 to T - 1 of ||delta y_t||^2. This correction is purely notational and does not affect the paper’s argument or conclusions.

Revised section:
A second natural level is the output-response level. If the relevant question concerns what the network actually expresses outwardly, the same logic may be applied to the output variable.
I_y(T) := (1/T) sum from t = 0 to T - 1 of ||delta y_t||^2, and the corresponding output-level criterion is I_y(T) > Theta_y.
Here, Theta_y denotes the threshold above which the realized response becomes relevant at the level of observable output.