Chapter 65: Spectral Coefficients and the Restriction Map

The spectral decomposition (the relevant theorem, I.T12) splits every element of 𝕃 into B-sector and C-sector components via the characters χ_+ and χ_- (the relevant definition, I.D37). This chapter lifts the spectral machinery from elements to functions: every f ∈ Hol(𝕃) (the relevant definition, I.D49) has unique spectral coefficients (a_k, b_k) at each primorial stage (the relevant definition, I.D65). The restriction map (the relevant definition, I.D66) restricts a function on 𝕃 to the complement 𝕃 ∖ K. The Spectral Determination Theorem (the relevant theorem, I.T29) proves that a τ-holomorphic function is uniquely determined by its spectral coefficients: the characters form a basis at each stage (I.T12) and tower coherence (the relevant definition, I.D46) forces cross-stage agreement. This is the uniqueness engine behind the thinness criterion and the Global Hartogs Theorem .