Tensor-Sector Non-Inheritance and Functional Renormalization in History-Dependent Gravity: TT Vertex Structure, Regulated Loop Kernel, and Spectral Reconstruction (Paper LXXIV)

We formulate and analyze the tensor-sector functional renormalization-group (FRG) flow generated by a history-dependent curvature form factor in quantum gravity. The central discovery is scalar non-inheritance: the scalar nonlocal operator R·F(□)·R does not trivially transmit its momentum dependence into the transverse-traceless (TT) tensor sector. We prove that the cubic tensor vertex generated by this operator vanishes upon TT projection, while the quartic vertex remains nonzero, forcing the tensor flow to proceed through mixed channels and tadpole contributions rather than direct scalar inheritance. We derive the dimensionless flow equation ∂ₜg̃ₖ(z) = 2g̃ₖ(z) + 2zg̃ₖ'(z) + K(√z)/(32π²Z_T) with frozen tensor normalization, where K(q) is the regulated loop kernel. The kernel construction includes: (i) piecewise representation across three momentum regions determined by regulator support; (ii) polynomial division isolating logarithmic structures; (iii) cancellation-safe radial representation eliminating apparent singularities; (iv) asymptotic expansion with both even and odd inverse powers (q⁻³, q⁻⁵, ...); and (v) rigorous C¹ regularity proof at the kinematic matching point q=2 through bulk/support decomposition with explicit boundary-term cancellation. The framework provides a controlled route from FRG flow to tensor spectral density via Stieltjes inversion. We emphasize that this work establishes the analytical and computational architecture within a specified truncation; complete gauge-independent predictions require additional ingredients (ghosts, running Z_T, regulator independence) addressed in subsequent work. All numerical implementations, regression tests, and benchmark values are provided for reproducibility.