Chapter 32: Mutual Determination (5-Way Equivalence)

This chapter proves the central unification theorem of Book II. Five apparently distinct descriptions of a holomorphic object on τ³ turn out to be the same thing: (1) a refinement tail in the primorial tower, (2) a spectral tail in the character decomposition, (3) an ω-germ at the profinite limit, (4) a boundary character on the boundary ring, (5) a Hartogs continuation from boundary to interior. The proof proceeds by a chain of four lemmas (II.L02–II.L05), each establishing a pairwise equivalence. The unifying mechanism is the split-complex polarity: the bipolar idempotents e_± = (1 ± j)/2 decompose each description into two independent channels, and it is this decomposition that makes all five perspectives mutually determining. The Mutual Determination Theorem (II.T27) is the algebraic engine behind the Central Theorem of Part IX.