Geometric and Spectral Desingularization III: Systems, Singular Sets and Cores

This article carries the sector decomposition of the Möbius–Klein transform to systems, singular sets with structure and the cores of defects. On the double cover of a plane crack tip a single Kolosov–Muskhelishvili term stays non-smooth, a Lipschitz function proportional to the in-plane stress intensity factors. The lifted Lamé operator acquires a coefficient of degree zero through its divergence part, and no unimodular phase removes the term. The term explains the 4 crack-tip enrichment functions and reads the stress intensity factors free of the Kolosov constant. The Airy function and the plate deflection at a free crack are smooth on the cover; a tip point force adds one logarithmic term. At a conducting edge, finite energy is membership in the first Sobolev space on the cover. On a cone the spinor lies in the twisted sector of the frame rotation. For the spin structure that extends over the apex, a free self-adjoint datum exists exactly when the cone angle exceeds a full turn. Subtracting the twisted multipole removes the slow convergence of the half-integer Hankel series at a tip. The Duffy map is the blow-up chart, and the cover completes it. At a degenerate equilibrium, odd weighted degree with odd weights is the twisted sector and forces a characteristic direction; for homogeneous fields of even degree, an invariant line. Along a curved front curvature enters with explicit coefficients; a variable opening angle brings squared logarithms. The loop energy of a disclination has an explicit finite part, and anisotropy selects the harmonics of the profile. Half-integer lines cannot escape: the twisted sector has no constant mode. At a vertex an even number of twisted edges meet, and on a singular surface the leading coefficient has at least half as many zeros as the normal Euler number in modulus.