Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains

We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation $$\mathcal{K}u={\mathrm{\Delta }}_{x}u+x\cdot {\mathrm{\nabla }}_{y}u-{\mathrm{\partial }}_{t}u=0$$ in non-characteristic intrinsic Lipschitz graph domains whose defining functions may depend on the higher-order variable $y_m$. Earlier theory required independence of $y_m$. We replace this symmetry by quantitative asymptotic cylindricality, measured by a scale-invariant $C_{y_m}^{0,1\mathrm{/}3}$ defect. The perturbative comparison theorem applies whenever this defect is sufficiently small, and hence at all sufficiently small scales for $C_{y_m}^{0,\alpha }$ graphs with $\alpha >1\mathrm{/}3$, as well as under a vanishing critical cylindrical modulus. We obtain local comparability and intrinsic Hölder continuity of quotients of positive solutions, together with a Hölder estimate for their logarithmic quotient. The proof combines local boundary estimates, valid without $y_m$-independence, with a global rigidity argument for cylindrical blow-up limits. The central issue is uniform balance of forward and backward reference values. If this balance fails at collapsing scales, maximal-scale selection and a second rescaling produce a non-negative polynomial-growth Dirichlet solution in an unbounded cylindrical domain that vanishes at the backward reference point but not at the forward one. We exclude this configuration using boundary-energy and mean-value estimates, finite-dimensionality of spaces of polynomial-growth solutions, and translation invariance in $y_m$. The resulting finite-dimensional translation representation yields real-analytic dependence on $y_m$. Propagation of zeros, analytic continuation along invariant fibres, and a Tikhonov uniqueness argument then force the limiting solution to vanish identically. The exponent $1\mathrm{/}3$ is critical because $y_m$ has homogeneous degree three.