Numerical Verification Reports for the Möbius–Klein Framework

Three self-contained reports: a comparative measurement of orientation-field coding, the Casimir energy of the flat Klein bottle, and the vacuum energies of the eight spin sectors of the 3-torus This record brings together three self-contained numerical verification reports, each supporting a specific quantitative claim of the Möbius–Klein framework. (1) Orientation-field coding: a like-for-like comparison between the Möbius spectrum and the established angle-doubling procedure around a +1/2 index singularity; the measurement also maps the crossover point where the two methods swap places. (2) The Casimir energy of the flat Klein bottle: the vacuum energy of a scalar and of a twisted (pseudoscalar) field by zeta regularisation, in closed form, verified by three independent methods. (3) The eight spin sectors of the 3-torus: the vacuum energy of all eight sectors of the three-way 4pi Möbius construction, by theta-decomposed analytic continuation, with a fourfold numerical cross-check and a closed sum rule. Each report ships a runnable Python script and states precisely which claim of which article it verifies. The reports are deliberately short and self-contained: they are not theoretical papers but verification protocols.