Geometric and Spectral Desingularization

Abstract

There are two seemingly independent paradigms for resolving singularities: the geometric route — Arnold's σ-process, or blow-up, which replaces the singular point by the space of directions through it — and the spectral route — the Möbius transform, which turns sign-flipping (two-valued) objects into single-valued signals of finite spectrum on the double cover. In this article we show that the two are the two faces of one and the same structure: the total space of the projective blow-up of the plane is identical to the total space of the Möbius line bundle (the open Möbius strip), the boundary circle of a tubular neighborhood of the exceptional divisor is the double cover of the divisor, and the sector decomposition of the Möbius transform is precisely the diagonalization of the blow-up's monodromy — the action of the deck group. In short: the σ-process builds the stage; the Möbius transform is that stage's native sheet music. We extend the connection in two directions: (i) for branchings with Z_n monodromy we give the fractional-shift (a/n) GDFT family, with closed deck-group Fourier projection formulas and machine-precision numerical verification; (ii) we show that a blow-up performed along a circle with non-orientable normal bundle yields a Klein-bottle exceptional set, whose native harmonic analysis is exactly the Klein transform — with nematic disclination loops as a physical arena. A dedicated section treats the role of the inverse transform: synthesis as monodromy enforcement (topological index protection), the composition inverse∘forward as a constraint-restoring projection, and the Puiseux connection — the half-integer radial powers (r^(n+1/2), the sqrt(r) behavior) as a consequence of the inverse synthesis and regularity at the origin, verified numerically. Finally we introduce the blow-up transform L, which lifts the whole function onto the surface of the Möbius strip: an isometry with exact sector characterization (plane functions + orientation-field-type data) and an elementary half-integer Hankel radial calculus; with extensions to the RP^2 -> Klein bottle case, the solid Klein bottle, and the hyper-Klein 3-spaces — including the Casimir energy of the latter (E(K^3) = -0.1325576). The article delimits precisely the classical building blocks (Hironaka, Arnold, Milnor–Stasheff) from the present synthesis. A new eighth chapter connects the system of tensor invariants [13,14]: the deck transformation of the blown-up space is orientation-reversing, but only in even dimensions — which explains why the blow-up along a circle is needed — and among the jet/jad/dev slices only the dual reading rot switches sector. We derive the divisor theorem: on a punctured domain the master theorem acquires the term D_k = -(integral over S^(n-1)) A T-hat(omega) omega_k d(omega), finite at the threshold p = -(n-1), to which by the selection rule only the first (l = 1) spherical harmonic contributes — so the source term of the Gauss theorem and the Burgers vector of the Stokes theorem are two selector choices of the same integral. The divisor term of jad is half a Burgers vector (b/2 on the divisor, b on the nearby circle): the integral-theorem form of half-strength disclinations. The jet of the crack-tip sqrt(r) field is pure dev, with the trace and twist slices vanishing identically. Finally we show that the blow-up removes one degree from the measure, that the restoring weight |r| is exactly the measure of the transform L itself, that the MKT remains unitary there for any radial weight, and that the half-integer Hankel basis is orthogonal and complete in it.