Fourier Transverse Modes Obstruct Volume-Uniform Critical Coercivity in a Flat Lattice Gauge Block Form

Let a fine periodic lattice have side LN' and let Q_L be the L^{-d}-normalized block average of length-L line integrals. In four dimensions, the rescaling Q_L -> LQ_L repairs the elementary constant-field scaling obstruction to a Poincare estimate. We prove that it cannot yield coercivity on the full one-cochain space with a constant uniform in the block side.For every L >= 2, every fixed N' >= 1, and N_c >= 2, we embed the first within-block Fourier phase zeta_L = exp(2 pi i/L) in a real two-plane of the internal coordinate space and construct a transverse one-cochain A_L. It satisfies Q_L A_L = 0 and div A_L = 0. For every dimension d >= 2,||A_L||^2 = (LN')^d, = (LN')^d lambda_L,where lambda_L = 4 sin^2(pi/L) <= 4 pi^2/L^2.Thus the Rayleigh quotient of K_0 + (s_L Q_L)^*(s_L Q_L) is exactly lambda_L for every scalar rescaling s_L. Every admissible full-space Poincare constant obeys C_P(L) >= 1/lambda_L >= L^2/(4 pi^2), so no constant chosen before L can work. Here "volume-uniform" means uniform while the block/fine side L varies at an arbitrary fixed positive coarse side N'; it is not a claim about N' -> infinity at fixed L.The construction, kernel identities, exact Hodge energy, Rayleigh identity, and quantified no-go theorem are formalized in Lean 4. The focused axiom audit reports only propext, Classical.choice, and Quot.sound. The theorem concerns the stated flat full-domain form; it does not contradict gauge-restricted propagator constructions and makes no infinite-volume, continuum, or Yang-Mills mass-gap claim.