Mutual Determination: Five Characterizations of Holomorphic Functions
Five characterizations of τ-holomorphic functions are mutually equivalent, making the theory structurally over-determined.
Overview
The Mutual Determination Theorem (II.T27) proves that five a priori distinct characterizations of τ-holomorphic functions are equivalent: (R) refinement sequence convergence, (S) spectral decomposition on τ³, (G) omega-germ transformation, (C) boundary character on L = S¹ ∨ S¹, and (H) Hartogs extension from the boundary. Equivalence of all five is the structural core of Book II, demonstrating that holomorphic functions in τ are not defined by arbitrary choice but by a multiply-convergent characterization.
Detail
In classical complex analysis, holomorphic functions are defined by Cauchy-Riemann equations, equivalent to power-series representation, equivalent to conformal mappings, etc. Book II establishes an analogous but deeper equivalence for τ-holomorphic functions. The five characterizations are: (R) a refinement sequence {f_n} of τ-objects converges spectrally; (S) the spectral decomposition on τ³ separates into γ-even and η-odd components; (G) the function is an omega-germ transformer, mapping germs to germs via ω-sector transport; (C) the restriction to the lemniscate boundary L satisfies the boundary character equations; and (H) any function on the boundary extends uniquely to the interior by Hartogs extension (II.T31). The chain (R)⟺(S)⟺(G)⟺(C)⟺(H) is II.T27. The theorem is used in every subsequent application of holomorphy in Books III–V.
Result Statement
Five characterizations of τ-holomorphic functions are mutually equivalent (II.T27): (R) refinement sequence, (S) spectral decomposition, (G) omega-germ transformer, (C) boundary character on L, (H) Hartogs extension.