Projection Obstruction for Factorized States: Nonlinear Dynamics, Conditional ℓ1 Diagnostics, and Finite-Time Defect Bounds

This preprint studies the structural defect introduced by projection-based approximations such as mean-field theory, product-state ansatze, and factorized variational methods. These methods replace correlated states with product states through a nonlinear retraction. The paper models the resulting discrepancy as an observer-relative coboundary obstruction over finite posets and finite-dimensional lattice systems. Three results are established. First, for a C² retraction onto a product-state manifold, the induced projected generator is generally nonlinear in ambient affine coordinates under a non-degeneracy condition on the Jacobian. This is an affine-coordinate obstruction result, not a claim excluding all possible nonlinear smooth conjugacies. Second, under explicit replica-extensive, edge-decomposable, symmetry-compatible observer assumptions, the ℓ¹ edge-sum coboundary diagnostic is selected up to edge weights; the unweighted form requires additional edge-uniformity or normalization. Third, for finite-range, finite-dimensional lattice systems with non-commuting interactions, the projection-induced defect has non-vanishing density over a finite local time window independent of system size, with constants controlled by local dimension, interaction strength, active bond fraction, graph density, non-degeneracy, and Lieb–Robinson bounds. The paper is finite-dimensional, finite-time, and explicitly conditional on declared observer assumptions. It does not claim a thermodynamic-limit result, a universal ℓ¹ metric, a global nonlinearization obstruction, or a new physical theory. Its purpose is to give a precise structural account of projection-induced error and to embed that account into the broader finite obstruction-calculus framework.