Chapter 49: The Identity Theorem

In classical complex analysis, the identity theorem states: if two holomorphic functions agree on a set with an accumulation point, they agree everywhere. This rigidity — the determination of a global function by local data — is the hallmark of holomorphy. Does τ-holomorphy enjoy a similar rigidity? This chapter proves that it does, and in a stronger form. The τ-Identity Theorem (the relevant theorem, I.T21) shows that agreement at a single primorial depth d₀ forces global equality on all omega-tails. The proof uses tower coherence (the relevant definition, I.D46) and the CRT coherence structure (the relevant theorem, I.T18) rather than Taylor series or analytic continuation. The key intermediate result is Tail Agreement Propagation (Lemma [lem:tail-agreement-propagation], I.L07): agreement at depth d₀ propagates upward through the primorial tower to all depths. This is the opposite direction from classical analysis, where agreement propagates via analytic continuation. In τ, the Chinese Remainder Theorem forces upward propagation. We also define the space Hol(𝕃) (the relevant definition, I.D49) of τ-holomorphic functions on the algebraic lemniscate, whose elements are uniquely determined by finite-depth data — a consequence that will be exploited in the Global Hartogs Theorem .