We introduce a continuous analogue of the kernel method for boundary-interrupted affine tridiagonal production rules arising in weighted lattice paths and related models. In the classical kernel method, an unknown catalytic boundary series is eliminated by substituting an algebraic root of the kernel. Here affine height dependence turns the kernel into a differential kernel: the catalytic boundary trace is selected instead by following a characteristic to the boundary and imposing regularity there. This produces a first-kind Abel--Volterra equation and a reconstruction formula for height-refined generating functions. For affine Motzkin triangles, the framework identifies the balanced case as the exact locality condition, explains integer imbalance through height shifts and finite operator compression, and yields closed shifted-Pearson collapses for arbitrary real boundary index, including the secant-power hierarchy interpolating Euler and higher-order Euler numbers. The same boundary index also appears naturally in Jacobi recurrence data and affine birth--death models, showing that the theory is intrinsically continuous rather than purely integral.