Classical Laurent theory expands a holomorphic function in an annulus {r₁ < |z| < r₂} as a doubly-infinite power series ∑{n=-∞}^∞ a_n z^n, with the residue a-1 computed by contour integration around the singularity Ahlfors1979,Conway1978. None of this machinery is available in the split-complex setting: there is no rotation group SO(2), no contour integration, and no annular domains in the sense of ℂ. This chapter develops the τ-analogue of Laurent expansion. The key insight is that the bipolar spectral decomposition (I.T10, I.D21, Book I) replaces the circular Fourier decomposition. A τ-holomorphic function decomposes into two independent channel series:

where φ_n^(±) are the sector basis functions. Residues emerge as spectral coefficients at distinguished frequencies (the relevant definition, II.D43)—not from contour integrals but from the spectral projection operators of Part V’s calibration. The Residue Theorem (the relevant theorem, II.T30) shows that the sum of residues equals the spectral trace, a quantity computable from the boundary character alone.