Book II · Part VI

Part VI: Local Hartogs and the Holomorphic Interior

Book II

Part VI is pivotal: split-complex holomorphy becomes load-bearing. Eight chapters build the Local Hartogs extension theory, prove the central Mutual Determination Theorem unifying five descriptions of holomorphic functions, and establish the canonical function representation.

the relevant chapter restates the split-complex codomain H_τ = A_τ^(B) × A_τ^(C) (I.D20) with concrete numerical calibration from Part V: π, e, j, ι_τ are now known. Bipolar idempotents e_± have geometric meaning.

the relevant chapter defines the BndLift_n operator: boundary datum at stage n → interior datum at stage n+1. The Chinese Remainder Theorem (I.T18) on primorial completions enables the lift; bipolar idempotents e_± split into two independent channels. ι_τ governs coupling strength.

the relevant chapter proves Theorem II.T27: the Mutual Determination Theorem (the five-way equivalence). Refinement tail = Spectral tail = ω-germ = Boundary character = Hartogs continuation. All five descriptions are unified by split-complex polarity; each determines all others uniquely. This is the central unification theorem of Book II.

the relevant chapter defines the evolution operator via ω-germ transport and shows that B/C asymmetry provides a causal arrow (which elliptic ℂ lacks). Forward propagation follows characteristic curves of the wave equation.

the relevant chapter establishes the category structure: composition = stagewise composition of ω-germs; identity = constant family; Theorem II.T29 proves associativity from the program monoid (I.P02). The category HolEnd_τ is formed.

the relevant chapter develops Laurent series with bipolar spectral decomposition. Residues emerge via e_±-sectors (not classical circular contours). The residue theorem is verified in split-complex setting.

the relevant chapter constructs the canonical holomorphic basis B_τ: classical monomials {z^n} are replaced by cylinder generators E_n,v^(B/C), each indexed by a (stage, peel token, channel) triple. Every τ-holomorphic function has finite spectral support (Theorem II.T31), and the DFT projection formula reduces holomorphic computations to finite linear algebra at each stage.

the relevant chapter establishes sheaf axioms from ω-germ stagewise compatibility. Gluing lemmas extend local holomorphic data to global sections.

Part VI closes with split-complex structure proven load-bearing. Interior points, continuous maps, holomorphic functions, and the sheaf structure all rest on the e_± decomposition.

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