Single Obstruction Theory: Projection Nonlinearity, ℓ¹ Rigidity, and Density Scaling

Projection-based approximations such as mean-field theory and product-state ansätze simplify many-body systems by discarding correlations. This paper studies the structural obstruction created by such projections. We prove three results: Projection non-dilability: a nonlinear projection composed with a linear semigroup cannot be conjugate to any linear semigroup. Categorical rigidity: the ℓ¹ coboundary norm is the unique additive faithful diagnostic of the resulting defect. Finite-time density scaling: for finite-range non-commuting lattice interactions the obstruction density remains nonzero over an O(1) time interval independent of system size. The framework uses Banach-valued presheaves over finite posets and connects geometric projection curvature, sheaf-theoretic obstruction, and correlation growth in many-body dynamics.