From Coherence to Finite Propagation: The α = d Locality Threshold

We derive a sharp locality constraint from coherence stability under composition. For composed systems on a d-dimensional lattice, we define uniform restoration as cycle-stable contraction of deviation with a contraction factor that does not degrade with system size. We prove that uniform restoration implies a thermodynamic-limit obstruction to extensive long-range coupling: for translation-invariant power-law interaction tails K(r) ~ r^(−α), the total long-range influence diverges for α ≤ d (logarithmically at α = d), forcing loss of restoration contractivity in the large-system limit unless coupling strength is scaled to zero with size. Coherent regimes therefore require quasi-local interactions satisfying α > d, which immediately yields finite effective propagation speed. This "α = d barrier" constitutes a falsifiable prediction: platforms with tunable interaction range should exhibit sharp coherence degradation as α → d⁺ and restoration failure for fixed-strength α ≤ d. Unlike frameworks that assume locality or derive it from already-local Hamiltonians, locality here emerges as a theorem from a stability condition that makes no prior reference to spatial structure.