Chapter 48: The Diagonal-Free Protection Theorem

the relevant chapter showed that classical D-holomorphy — split-complex holomorphy with j² = +1 — is crippled by zero divisors: the idempotents e_+ = (1+j)/2 and e_- = (1-j)/2 satisfy e_+ · e_- = 0, allowing pathological functions that vanish on one sector and behave arbitrarily on the other. the relevant chapter defined the τ-holomorphic functions (HolFun, I.D47) as D-holomorphic functions that additionally satisfy tower coherence. This chapter proves that HolFun is protected from the zero-divisor pathology by two independent structural guards: the diagonal discipline (axiom K5, I.D03) and the Prime Polarity Theorem (I.T05). The No Simultaneous Projection proposition (I.P23) shows that no compatible omega-tail can project nontrivially onto both sectors through a τ-holomorphic function. The Diagonal-Free Protection Theorem (I.T19) elevates this to a full structural guarantee: the zero-divisor product e_+ · e_- = 0 cannot arise as T(t₁) · T(t₂) for any T ∈ HolFun and any compatible omega-tails t₁, t₂. The chapter concludes by proving that HolFun is closed under composition (I.T20) and that composition is associative (I.P24), giving HolFun a monoid structure that will serve as the substrate for the earned category Cat_τ in Part XIII.