Kinematic distributions in Higgs decays to four charged leptons, the so called `golden channel, are a powerful probe of the tensor structure of its couplings to neutral electroweak gauge bosons. In this study we construct the first part of a comprehensive analysis framework designed to maximize the information contained in this channel in order to perform direct extraction of the various possible Higgs couplings. To that end we first complete an earlier analytic calculation of the leading order fully differential cross sections for the golden channel signal and background to include the 4e and 4μ final states with interference between identical final states. We also examine the relative fractions of the different possible combinations of scalar-tensor couplings by integrating the fully differential cross section over all kinematic variables as well as show various doubly differential spectra for both the signal and background. From these analytic expressions we then construct a `generator level' analysis framework based on the maximum likelihood method. We demonstrate the ability of our framework to perform multi-parameter extractions of all the possible effective couplings of a spin-0 scalar to pairs of neutral electroweak gauge bosons including any correlations. This framework provides a powerful method for study of these couplings and can be readily adapted to include the relevant detector and systematic effects which we demonstrate in an accompanying study to follow.