Classical complex analysis expands holomorphic functions in the monomial basis {z^n : n ≥ 0}. The choice of basis is canonical once the coordinate z is fixed, but the coordinate itself is a convention: any biholomorphism z ↦ w(z) produces a different monomial basis. In Category τ, there is no ambient complex plane and no coordinate freedom. The algebraic structure of τ³—the bipolar idempotent decomposition (I.D21, Book I), the CRT tower (I.T18, Book I), and the normal-form addressing of cylinders (the relevant definition, II.D10)—forces a unique basis for the space of τ-holomorphic functions. This chapter constructs the canonical holomorphic basis B_τ (II.D45): classical monomials {z^n} are replaced by cylinder generators E_n,v^(B/C) (II.D46), each indexed by a (stage, peel token, channel) triple. the relevant theorem (II.T31): every τ-holomorphic function has **finite spectral support—forced by Book I’s unique infinity (I.T36). The Projection Formula (II.P08) reduces holomorphic computations to finite linear algebra at each stage via a discrete Fourier transform on cyclic fibers. The basis is canonical: no choices are made.