Geometric and Spectral Desingularization II: Equations, Distributions, Cones and the Limits of the Sector Method

This article continues Geometric and Spectral Desingularization. It asks which singularities of functions and of linear and semilinear equations the sector decomposition of the Möbius–Klein transform resolves exactly, and names the tool that takes over elsewhere. A singularity is resolved when the object becomes regular on a modified space, or when the equation holds on the original space as an identity of distributions with explicit point terms. Under the conformal branched cover the Laplace, Poisson, Helmholtz and Schrödinger equations keep smooth coefficients, and bounded solutions of every sector become smooth. An anisotropic principal part with an anti-holomorphic traceless term leaves a finite, explicit order of smoothness. On the base, homogeneous singularities extend uniquely within their harmonic except at critical degrees, sorted by the antipodal sector of the link. There a scale or a point-source datum must be supplied, and the article states which constant it fixes. The divisor theorem holds at every depth beyond the threshold, with a selection rule that admits one sector of the profile, and the point terms separate by slice of the derivative tensor. Unbounded half-integer modes are the fields of twisted point sources on the cover, and the cover adds invisible constants unless the pole condition holds. Logarithms arise only from multiple roots and from collisions inside a sector, resonant sources included. Infinite monodromy splits into characters and Jordan chains, resolved on the strip of the logarithm. Cone points of rational angle become smooth problems on a cyclic cover. Puiseux characteristic exponents are the jumps of the character group, and inverse-square potentials and point interactions are classified by sector. The weighted norm and the Mellin finite part coincide on the weighted space and stay additive over sectors beyond it. Essential singularities, products of point terms and singularities formed by evolution lie outside the method.