Complex Analysis from Three Primitives

Complex analysis is derived entirely from the Tree of Continua C and the threeprimitives — same, different, opposite. No postulate of complex analysis is assumed.The complex numbers C are the IPG reading at ∞ of the cyclotomic fields Q(ωN ) ⊂Per(C), generated by the chiral involution φ(s) = −s on the centered alphabet. A functionf : C → C is a labeling of cylinder sets of C by values in Q(ωN ).A function is holomorphic — satisfying the Cauchy–Riemann equations — if andonly if it is a labeling that is chiral-compatible: its values are consistent with the chiralstructure of C under the cylinder set filtration. The Cauchy–Riemann equations arethe compatibility conditions for this chiral-consistent labeling, forced by the oppositeprimitive.Cauchy’s theorem — γ f dz = 0 for a holomorphic f on a simply connected domain— is Stokes’ theorem applied to a chiral-compatible labeling, already established in theRiemann integration paper. No separate proof is needed: holomorphicity is the conditionthat makes the integrand exact, and exactness gives zero integral around any closed curve.The residue theorem is the First IsomorphismTheorem for the integration TolFilt morphism around a pole: the integral γ f dz measures the kernel depth of the labelingat the singularity. The residue is the kernel depth, and the theorem equates the integralto the sum of kernel depths (residues) inside the contour.Taylor series, Laurent series, the identity theorem, and the maximum modulus principleall follow as cylinder set statements: a holomorphic function is determined by its valueson any sequence of cylinder sets accumulating to a point (identity theorem); its maximumis attained on the boundary of any cylinder set region (maximum modulus).The fundamental theorem of algebra is a corollary: every polynomial of degree n hasexactly n roots in C, counting multiplicity — a statement about the kernel structure ofthe polynomial as a TolFilt morphism.Three primitives. One complex analysis.