Tensor-Sector Reconstruction and Spectral Non-Inheritance in History-Dependent Gravity (Paper LXXIII)

We prove that the scalar spectral density ρ_R(μ) obtained from the FRG flow of R F_R(□) R does not determine the tensor graviton propagator, because the scalar curvature perturbation vanishes identically for transverse-traceless fluctuations on Minkowski space. We therefore introduce a minimal TT-sensitive truncation containing R_μν g_k(□) R^μν and derive its TT Hessian and the functional structure of the tensor-Laplacian vertices. The resulting vertices involve divided differences of the tensor form factor, including a second divided difference at quartic order. This establishes a tensor FRG flow structurally distinct from the scalar projection, although the non-equality of the fully integrated beta functions is left to the explicit computation. We formulate the conditions under which the infrared tensor form factor admits a dispersive/Stieltjes spectral representation and define the corresponding tensor spectral density. The proportionality relation ρ_T = C_T ρ_R is retained as an open, quantitatively testable hypothesis for the subsequent numerical FRG analysis. This is Paper LXXIII in the History-Dependent Gravity series.