Chapter 47: Tower Coherence and τ

Parts VIII–XI built the finite-stage data of τ: modular arithmetic on ℤ/M_kℤ, the split-complex scalar ring ℤ_τ[j] (the relevant definition, I.D20), the fundamental characters χ_+ and χ_- (the relevant definition, I.D37), and the four-valued logic Ω_τ. the relevant chapter showed that D-holomorphy — the Cauchy–Riemann analogue for split-complex scalars — gives sector independence but too much flexibility: any pair of independent sector functions qualifies. This chapter adds the crucial constraint: tower coherence. A τ-holomorphic function must be compatible with the primorial ladder at every stage. We define ω-germ transformers (the relevant definition, I.D45): functions from omega-tails to split-complex values. Tower coherence (the relevant definition, I.D46) requires that reducing the output first, or reducing the input first and then applying the transformer, gives the same result — a naturality condition on the primorial category. A τ-holomorphic function (HolFun) (the relevant definition, I.D47) is an ω-germ transformer that is both D-holomorphic in sector coordinates and tower-coherent. The CRT coherence constraint (the relevant theorem, I.T18) shows that tower coherence reduces the infinite-dimensional problem to a prime-by-prime determination: a τ-holomorphic function is determined by its action on each CRT factor ℤ/p_iℤ.