Geometric and Spectral Desingularization

Two paradigms resolve singularities: the geometric route – Arnold's σ-process, or blow-up, replacing the singular point by the space of directions through it – and the spectral route – the Möbius transform, turning sign-flipping objects into single-valued signals on the double cover. We show how the two meet: the total space of the projective blow-up of the plane is identical to the total space of the Möbius line bundle, the exceptional divisor is its base circle, and the sections of this bundle over the divisor are exactly the antiperiodic signals the Möbius transform analyses — the σ-process builds the stage, and the Möbius transform is that stage's native sheet music. Sign-monodromy data around the singular point live on a different double cover, where the same transform acts ring by ring. The connection is extended in two directions: for branchings with Z_n monodromy we give the fractional-shift (a/n) GDFT family with closed projection formulas, and we set up the blow-up transform — the pullback along the blow-down map — an isometry with exact sector characterization. A chapter connects the system of tensor invariants: blowing up a point gives a non-orientable space only in even dimensions, and on a punctured domain the master theorem acquires a divisor term — the classical δ-term of homogeneous distributions, read on the ladder — finite at the threshold p=-(n-1) and subject to a selection rule that keeps only the first harmonic. The divisor term of jad is the full Burgers vector; its halving on the projective divisor is a convention. Further chapters carry the theorem to blow-up along a curve, where the divisor is a Klein bottle and a double selection rule appears, and along a surface, where the Euler class of the normal bundle decides how many components of the vector term exist, while the scalar term always survives. For codimension-two centres, the unit-normal divisor is non-orientable exactly when the ambient manifold is non-orientable along the submanifold. The classical building blocks are delimited from the present synthesis, and every numerical value quoted is produced by a deterministic script, with the blocks that record an identity by construction labelled as such both in the code and in the text.