No Vacuum Screening Without an Infrared Price: A Zero-Mode Dichotomy for the Cosmological Constant Problem

Vacuum-energy screening must distinguish an arbitrary constant shift of the matter effective action from finite-wavelength excitations that gravitate normally. We formulate this as a support-discrimination problem and prove a zero-mode dichotomy for universal, differentiable response: a transfer function continuous at k = 0 that annihilates the vacuum shift necessarily suppresses a band of finite-wavelength sources, while any autonomous rule for the zero-mode sector is a functional of the spacetime average of the source — global information, with the exact projector 1 − P0 as idealized endpoint. No finite-order local operator, nor any response with symbol analytic at the origin, realizes the latter. Every admissible smooth screening takes the normalized mean-subtraction form K = δ − G with ∫G = 1: the dichotomy is finite averaging range versus global data, and a kernel-shape-independent recovery floor k⋆L1 ≥ 1 − ε follows from the first moment alone. Local traceless projections do not escape: Bianchi integrability restores an undetermined global constant. For the (L, α) degravitation family we derive the finite-band penalty and, under a horizon-squeeze necessity heuristic with declared anchors, a conditional benchmark αmax(k⋆, ε, H0) ≈ 0.26 — a coincidence problem reappears as the filter scale is pinned to the present horizon. Closing the non-universality exit through equivalence-principle charge decompositions, the result is a structural trilemma: radiatively stable vacuum screening cannot isolate the zero mode without exposing an infrared, global, or universality cost. We apply it to unimodular gravity, degravitation, vacuum-energy sequestering and the recent lapse-based constructions.