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 TTNC 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 $\mathcal{C}_1$, and a horizon-localized conical contribution, governed by $\mathcal{W}_h$. The smooth coefficient controls the source-induced homogeneous projection of the TTNC Weyl Ward identity, whereas the logarithmic thermal entropy is controlled by $\mathcal{C}_{\rm therm}=\mathcal{C}_1+zL^2\mathcal{W}_h$. We give closed expressions for $\mathcal{C}_1$, $\mathcal{W}_h$, and $\mathcal{C}_{\rm therm}$ in 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.