Chapter 43: Four Truth Values from Polarity Stabilization

The spectral decomposition of the relevant chapter (the relevant theorem, I.T12) splits every element of the algebraic lemniscate 𝕃 into B-sector and C-sector components via the characters χ_+ and χ_- (the relevant definition, I.D37). This chapter harvests a logic from that decomposition. Given a predicate P on the τ-index, the two spectral sectors can independently confirm or deny P. Four combinations arise: both confirm (T, true), both deny (F, false), one confirms and the other denies incompatibly (B, both — overdetermined), neither is decisive (N, neither — underdetermined). These four truth values form a diamond lattice T > B, N > F, with conjunction as meet, disjunction as join, and the polarity swap σ as negation (T ↔ F, B ↔ N). The resulting logic, Truth4, is earned from the bipolar structure, not axiomatized. Truth4 will serve as the subobject classifier Ω_τ in the earned topos of Part XIII.