Several recent papers have studied the decay rate of the expectation of Wilson loop and Wilson line observables in lattice gauge theory and the lattice Higgs model. These results have all been perturbative in the sense that the parameters need to scale with the length of the loop or line for the error term to be smaller than the estimate. In this paper, we further develop ideas from~\cite{fv2023} and~\cite{f2024} to give more detailed asymptotics for the expectation of Wilson loop and Wilson line observables, which do not require the parameters to be very large or very small for the error term to be small, thus improving the results of~\cite{flv2022, flv2023, flv2020, f2022b}. In particular, we show that Wilson line and loop observables have a pure parameter law in the Higgs and confinement phases of the lattice Higgs model.