Structural Drift and Projective Dynamics

We refine the notion of structural freedom by replacing the dimension count D_{\rm bi}(f) with a projective object: the orthogonal projector \Pi_L(x) onto the local allowed-perturbation subspace L_x=\ker J_x. This upgrade exposes a three-level structure of structural freedom: dimension, orientation, and dynamical accessibility. We distinguish two regimes. In the active-set network model of the companion paper [6], J_x is piecewise constant, hence D_{\rm bi} and \Pi_L are both piecewise constant, and structural change occurs through active-set jumps. In a generalized state-dependent constraint geometry J_x=Dg(x), the projector may vary continuously at constant rank, giving rise to continuous structural drift. We define the structural gradient field \mathcal G_x=D_{w^\ast}M_q\circ Dw^\ast_x\circ\Pi_L(x) and its Hilbert–Schmidt norm \Gamma_M, and we bound the first-order spectral displacement of a simple eigenvalue by \kappa_\mu\Gamma_M. On K_3, two realizations with identical D_{\rm bi}=1 and identical \Pi_L produce \Gamma_M^{\rm mean}=2.9845 and \Gamma_M^{\rm diff}=0, confirming that dynamical coupling depends on the observable-to-dynamics map Dw^\ast|_{L_x}, not on the freedom count or the freedom subspace alone. No numerical predictions, empirical calibration, or historical judgments are offered.