Tau-Identity Theorem: Agreement at One Depth Implies Agreement Everywhere
If two τ-holomorphic functions agree at any single depth level, they agree at all depths — the τ-analogue of the identity theorem for holomorphic functions.
Overview
I.T21 (Tau-Identity Theorem) proves that if two τ-holomorphic functions f and g agree on any single depth stratum of τ³, they agree on all of τ³. This is the τ-analogue of the classical identity theorem for holomorphic functions (agreement on an open set implies agreement everywhere). The result is foundational for proving uniqueness statements throughout the series.
Detail
The classical identity theorem in complex analysis states that a holomorphic function on a connected domain is uniquely determined by its values on any open subset, or equivalently by agreement with another holomorphic function on any convergent sequence. I.T21 extends this to τ: τ-holomorphic functions are uniquely determined by their values on any single depth level of the tower structure τ³. The depth levels correspond to the tower-grading by enrichment level: depth 0 = E₀ data, depth 1 = E₁ data, etc. Agreement at depth k implies agreement at all depths because the τ-propagator (the mechanism by which data at depth k generates data at depth k+1) is injective. The proof uses the prime polarity structure: γ-even and η-odd components at depth k independently determine the same components at depth k+1, so any difference between f and g at depth k+1 would force a difference at depth k.
Result Statement
I.T21: If two τ-holomorphic functions agree on any single depth stratum of τ³, they agree on all of τ³. τ-analogue of the classical identity theorem.