Chapter 38: σ

This chapter opens the Hodge block (Chapters 41–43) by posing the Hodge question in the language of Category τ. The classical Hodge Conjecture asks which cohomology classes are algebraic; in τ, the question becomes: which boundary characters on 𝕃 = S¹ ∨ S¹ are NF-addressable within each sector of the 4+1 decomposition? The σ-involution on the split-complex algebra H_τ = ℤ[j]/(j² - 1) acts on characters by swapping the idempotent components e_+ and e_-. The σ-fixed characters—those invariant under this swap—form the diagonal sublattice of ℤ^2 and are precisely the balanced characters of the spectral trichotomy (the relevant theorem, Ch. 19). We prove that the Hodge question is trivial at enrichment level E₀, where the Central Theorem of Book II already guarantees global NF-addressability, and becomes substantive only at E₁, where sector admissibility predicates introduce a non-trivial “which” question. We then define the sector addressability predicate and state the τ-Hodge conjecture: every σ-fixed character is sector-addressable in every primitive sector.