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

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. 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)}. Only the second can alter the local dynamical spectrum \operatorname{spec}(M_q). We prove a structural–dynamical non-implication theorem within the specified model class, and supply two explicit, numerically verified counterexamples on K_3. We give two elementary realizations of the map w^\ast:\mathcal F\to\mathbb R, 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. 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. 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)}. Only the second can alter the local dynamical spectrum \operatorname{spec}(M_q). We prove a structural–dynamical non-implication theorem within the specified model class, and supply two explicit, numerically verified counterexamples on K_3. We give two elementary realizations of the map w^\ast:\mathcal F\to\mathbb R, 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.