Chapter 50: Extensions Are ω-Germ Transformers

the relevant chapter proved the Hartogs Extension Uniqueness Theorem (II.T37): every idempotent-supported boundary character extends uniquely to a holomorphic function on the interior of τ³, with the extension living in the split-complex codomain H_τ. This chapter takes the next step. We prove that these Hartogs extensions are ω-germ transformers—the algebraic objects defined in Book I Part XIII (I.D45–I.D47). The identification is not a metaphor: the extension f_χ of a boundary character χ determines, and is determined by, a unique element of End(d(τ³)), the endomorphism monoid of the ω-germ space. The key ingredient is Stagewise Naturality (II.L13): the Hartogs extension respects the CRT reduction maps at every stage of the primorial tower, because the boundary character that generates it is tower-coherent by definition. Stagewise naturality ensures that the stage-k actions of the extension form a coherent system, which in the inverse limit becomes an ω-germ transformer. The converse—every regular ω-germ transformer comes from a Hartogs extension—closes the second link in the Central Theorem chain:

The next chapter will close the loop by showing that ω-germ transformers are holomorphic functions.