Chapter 49: Hartogs Extension in H_τ

the relevant chapter restated the boundary ring characters as idempotent-supported objects (the relevant definition, II.D59): every character decomposes canonically as χ = e_+ · χ_+ + e_- · χ_- (Proposition [prop:character-decomposition], II.P13). This chapter proves that each idempotent-supported character extends uniquely to the interior of τ³, and that the extension lives in the split-complex codomain H_τ (not classical ℂ). The construction has two stages: first, the Extension Lemma (Lemma [lem:extension-h-tau], II.L12) builds the extension channel-by-channel using the BndLift operator (the relevant definition, II.D36) from Part VI; second, the Hartogs Extension Uniqueness Theorem (the relevant theorem, II.T37) proves that the extension is the only one — any τ-holomorphic function on τ³ whose boundary restriction is χ must coincide with the constructed extension. Uniqueness follows from two independent arguments: the Code/Decode bijection (the relevant theorem, II.T35) and, more illuminatingly, the Mutual Determination Theorem (the relevant theorem, II.T27). Both arguments exploit the same structural feature: bipolar channel independence. The B-channel and C-channel extensions are each determined independently by their respective boundary data, and there is no room for an alternative. This chapter establishes the boundary → interior direction of the Central Theorem. The reverse direction (interior → boundary) and the full isomorphism O(τ³) ≅ A_spec(𝕃) are assembled in the relevant chapter.