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 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)=-0.132\,557\,6). 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=-_S^n-1 A T_k\, d, finite at the threshold p=-(n-1), to which by the selection rule only the first (=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 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. A further chapter carries the divisor theorem over to blow-up along a curve, where the divisor is a Klein bottle: the threshold is unchanged, but a double selection rule appears (only the first harmonic angularly, only the average over s longitudinally), and — unlike the point case — only the component of the divisor term lying in the trivial summand of the normal bundle is well defined, because the bundle splits as trivial ⊕ Möbius. The longitudinal circulation of jad therefore vanishes identically: this reproduces, in integral-theorem form, the known statement of defect topology that a Burgers vector cannot be carried along a non-orientable direction. The closing section of Chapter 9 gives a general ambient criterion: the divisor is non-orientable if and only if the ambient manifold is non-orientable along the submanifold (w_1(divisor)=^*(w_1(M)|_P)). It follows that there is no twisted divisor either along a curve in R^3 or along a surface in R^4 — the construction requires a non-orientable ambient — and that the dimensional parity rule of Section 8.1 is a special case of it.