Physical Interpretation of the Tensor Sector in History-Dependent Gravity: From Nonlocal Memory to Loop Topologies (Paper LXXVI)

We develop a theoretical synthesis of the tensor sector of History-Dependent Gravity (HDG), combining the tensor-continuation analysis, the non-inheritance structure of the tensor form factor, and the numerical characterization of the transverse-traceless (TT) loop flow established in our preceding works. The central question is how temporal nonlocality and its spectral representation are related to the momentum-dependent tensor-sector dynamics without assuming that the nonlocal structure of the scalar sector is inherited unchanged by the TT sector. We establish that the tensor-sector form factor g_k(Q^2) cannot be obtained by direct inheritance from another gravitational sector; it must be determined by its own functional flow and tensor projection. We then interpret the previously validated sunset-to-tadpole ratio R_TS ≡ T_tad / T_sun as a topology-level diagnostic of the implemented B.27-C TT kernel. For the Litim regulator, the previously established values are numerically stable under variations of the RG scale at fixed p/k, but the comparison with an exponential regulator produces a shift of approximately 9.7%–13.8%, demonstrating that numerical stability within a fixed scheme does not imply regulator independence. The combined picture separates three logically distinct levels of the analysis: the HDG-specific memory structure, the tensor-sector dynamics encoded in the corresponding scale-dependent effective action and tensor two-point function, and generic regulator- and truncation-dependent features of the FRG tensor flow. The resulting interpretation identifies R_TS as a useful diagnostic of the relative weight of surviving TT loop topologies rather than as a direct observable or a universal physical constant. We discuss the implications of this structure for nonlocal tensor response and gravitational-wave propagation, while keeping explicit the distinction between derived theoretical quantities and experimentally testable predictions. The analysis provides a conceptual bridge from temporal memory and spectral nonlocality to tensor-sector dynamics and identifies the closure of the dressed TT truncation as the next step toward physical predictions.