Geometric and Spectral Desingularization IV: Moving Singularities, Caustics and Parameter Space

This paper places the cover over time, phase space and parameter space. A moving branch point carries its cover, and while it stays isolated each sector component keeps its sector; while the branches stay distinct, branch points appear and disappear only in sets with monodromy product 1. An in-plane crack running below the Rayleigh speed is built from 2 contracted variables, each smooth on its own cover, and no cover makes both smooth. At zero speed the remainder becomes the non-smooth term of the resting crack. At a movable singular point of an ordinary differential equation, rational exponents with the compatibility conditions give a local cover in complex time. Each free constant carries a sector, and the characteristic exponents form a Puiseux tower; a failed condition brings an infinite monodromy. At a fold of a Lagrangian manifold the complexified projection is a branched double cover, and the quarter turn of the sheet amplitude on the fourfold cover is the Maslov phase. The canonical operator leads back to the field. The Maslov parity is the orientation character, so orientation-reversing loops carry a quarter shift in the quantization. In a weighted blow-up the cusp caustic becomes 2 folds and the field a single entire Pearcey function. The birth of a shock in Burgers' equation is the same cusp in space–time: the cover gives 3 branches, viscosity chooses one, and the solution scales with the blow-up weights. At a conical intersection the eigenpairs are polynomials on the cover, and the E⊗e pseudorotation number is a half-integer. At an exceptional point the eigenpairs live on the double cover and the bilinearly normalized eigenvectors on the fourfold cover. On a Klein-bottle parameter space a two-level real family reaches only 2 of the 4 eigenline classes; 3 levels, or a gluing symmetry, reach the others.