Matter one-loop logarithms and homogeneous TTNC scale response of Lifshitz black branes

We compute the logarithmic one-loop matter contribution to the thermodynamics and homogeneous twistless torsional Newton–Cartan scale response of a four-dimensional Lifshitz black brane. The background is the neutral planar member of the analytic Einstein–Maxwell–dilaton Lifshitz black-brane family, while the quantum fields are treated as probes: a real scalar with arbitrary mass and nonminimal curvature coupling, a four-component Dirac spinor, and an Abelian Maxwell field with its Faddeev–Popov ghosts. For each spin sector, the logarithmic coefficient separates into a smooth radial heat-kernel contribution, governed by ${\text{calC}}_1$, and a horizon-localized conical contribution, governed by ${\text{calW}}_h$. This separation identifies which part of the matter one-loop logarithm is visible in the boundary Lifshitz/TTNC Ward identity and which part instead belongs to horizon replica entropy. The smooth coefficient ${\text{calC}}_1$ controls the source-induced homogeneous projection of the TTNC Weyl Ward identity, namely the smooth boundary-source contribution to the Ward combination $z\epsilon -2p$, whereas the thermodynamic logarithmic entropy is controlled by ${\text{calC}}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}={\text{calC}}_1+z{L}^2{\text{calW}}_h$. We give closed expressions for ${\text{calC}}_1$, ${\text{calW}}_h$, and ${\text{calC}}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}$ for the scalar, Dirac, and Maxwell probe sectors, identify the gauge-field contact contribution in the conical entropy, and verify that the smooth source response vanishes in the relativistic planar $z=1$ limit while the standard horizon heat-kernel entropy coefficient remains.