Structural Drift and Projective Dynamics

We refine the notion of structural freedom by replacing the dimension count Dbi(f) with a projective object: the orthogonal projector Π_L(x) onto the local allowed-perturbation subspace L_x= 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 Dbi and Π_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 G_x=Dw^M_q∘ Dw^_x∘Π_L(x) and its Hilbert–Schmidt norm Γ_M, and we bound the first-order spectral displacement of a simple eigenvalue by κ_μΓ_M. On K_3, two realizations with identical Dbi=1 and identical Π_L produce Γ_Mᵐᵉᵃⁿ=2.9845 and Γ_Mᵈⁱᶠᶠ=0, confirming that dynamical coupling depends on the observable-to-dynamics map Dw^|L_x, not on the freedom count or the freedom subspace alone. No numerical predictions, empirical calibration, or historical judgments are offered.