Chapter 42: Code/Decode and Diagonal Protection

A holomorphic function on τ³ is determined by its boundary behavior—this is the lesson of the Mutual Determination Theorem (II.T27) and the Global Hartogs Theorem (I.T31, Book I). But how is the boundary data organized? This chapter answers: as a bipolar boundary coefficient stream, decomposed by the idempotent pair e_+, e_- into two independent channels. A code (II.D51) is such a stream—a pair (c^+, c^-) of spectral coefficient sequences, one per channel, satisfying tower coherence. A decode (II.D52) takes a code and reconstructs the unique holomorphic function whose boundary spectral coefficients match. the relevant theorem (II.T35): Code and Decode are mutually inverse—a holomorphic function **is its bipolar boundary stream, and every coherent bipolar stream is a holomorphic function. This is the Holomorphic Content Theorem.

The chapter closes with the most delicate conceptual point of Part VII: why does split-complex algebra not collapse? The codomain H_τ has zero divisors (e_+ · e_- = 0), so arbitrary projections onto the zero-divisor ideals could destroy information. The answer is the diagonal discipline (I.X05, Book I): K5 forbids the diagonal map δ : x ↦ (x, x) in the solenoidal generators, which means that e_+ and e_- are the only available projections. The B-channel and C-channel carry independent information precisely because no diagonal conflates them. Diagonal protection (Remark [rem:diagonal-protection], II.R13) is the foundational safeguard that makes the Code/Decode bijection possible.