Structural Freedom and Dynamical Stability: A Branch of the Negative-Entropy Construction

Abstract

We develop a branch of the Negative-Entropy Construction that formalizes the notion of structural freedom in constrained network flow systems. Given a network with conservation law Bf=0, central constraints Af=0, and capacity bounds 0\le f\le C, we define the bidirectional local freedom D_{\rm bi}(f) as the dimension of the subspace of perturbations feasible in both directions. We prove an effective-constraint theorem: adding a constraint reduces D_{\rm bi} if and only if the constraint is linearly independent from the existing constraint rows. This separates nominal centralization from structural centralization. We further show that D_{\rm bi}(t), as a function of the flow trajectory, is a piecewise-constant state variable with locally finite active-set events. A single-event transition 1\to0\to1 is realized explicitly on K_3. The construction is extended to time-dependent capacity envelopes C(t), where active-set events can be generated either by flow increase or by capacity contraction, without assuming any evolution law for C(t). From L_f, the subspace realizing D_{\rm bi}, two distinct and parallel coupling channels emanate: \mathcal B_R|_{L_f}:L_f\to\mathbb C^3\quad\text{(additive forcing)}, \mathcal J_{\rm mult}|_{L_f}:L_f\to T_{M_q}\mathcal M\quad\text{(multiplicative feedback)}. Within the model class considered here, only the multiplicative channel changes M_q directly and therefore provides the explicit structural-to-spectral interface studied in this paper. We prove a structural–dynamical non-implication theorem within the specified model class, supported by two numerically verified K_3 counterexamples. We give the characteristic polynomial of M_q, the Hopf condition a_1a_2=a_3, the explicit Hopf threshold \lambda_H=1.1440860215\ldots, and the transversality condition h'(\lambda)\neq0. Two elementary realizations of w^\ast:\mathcal F\to\mathbb R are given, one producing r_M=1 and one producing r_M=0, confirming that D_{\rm bi}>0 does not imply r_M>0. No numerical predictions, empirical calibration, or historical judgments are offered.