Siberian Snakes in the Sector Calculus: Half-Integer Spin Tune, Error Selection and Field Parity

The Möbius Klein Transformation (MKT) splits fields on a space with a twist into the untwisted and the twisted sectors of a double cover. It is applied here to polarized beams in storage rings and to their magnets. A full Siberian snake fixes the spin tune at one half for every energy. In the spinor group SU(2) the rotation by π about the vertical axis grades the one-turn map: vertical-axis rotations are even, full snakes odd. An odd number of full snakes therefore gives the spin tune one half for any number and size of even errors. With 1 snake the spin component across the stable axis lies in the twisted sector of the turn circle. The spinor map squares to −1: a Z_2 twist for vectors and a Z_4 twist for spinors. At spin tune one half a single error shifts the tune by an exact closed formula. Every single error whose axis is perpendicular to the stable spin axis leaves the tune at one half to all orders. For 2 orthogonal snakes this protects 1 rotation-angle error, 1 axis tilt or 1 horizontal-axis kick; axis-angle errors and horizontal orbit angles shift the tune at 1st order. On the field side, the harmonics a coil error can excite follow from the characters of the signed rotation group of the coil. At 2nd order the characters multiply, which locates the 2nd-order harmonic families. A helical dipole is symmetric under a transverse half-turn with current reversal, which selects one parity of its harmonics. A canted-cosine-theta layer pair is the even projection of 1 layer under a transverse half-turn. To 1st order its forbidden integrated harmonics arise only from the differential part of a winding error, and never from a common-mode one. Deterministic scripts check every statement.