Projection Obstruction for Factorized States: Local Correlation Lower Bounds, Additive Diagnostics, and Observer Recovery

Title. Projection Obstruction for Factorized States: Local Correlation Lower Bounds, Additive Diagnostics, and Observer Recovery Abstract. For a finite quantum system with a time-independent Hamiltonian containing one- and two-site interactions and a product initial state, we bound the departure of each exact two-site reduced state from the product of its own marginals. Define Fe(t)=ρe(t)−ρa(t)⊗ρb(t),De(t)=∥Fe(t)∥1,F_e(t)=\rho_e(t)-\rho_a(t)\otimes\rho_b(t), \qquad D_e(t)=\|F_e(t)\|_1, where e={a,b}e=\{a,b\} and ∥⋅∥1\|\cdot\|_1 is the full trace norm. With local interaction norm g=max⁡v∑X∋v∥hX∥∞g=\max_v\sum_{X\ni v}\|h_X\|_\infty and first-order coefficient νe=∥Fe′(0)∥1\nu_e=\|F_e'(0)\|_1, the bound is De(t)≥νet−24g2t2.D_e(t)\ge\nu_et-24g^2t^2. More generally, if Fe(j)(0)=0F_e^{(j)}(0)=0 for 0≤j<k0\le j