Part XI: τ
Parts VI–VII proved that the primes carry a canonical bipolar polarity (B-dominant vs. C-dominant), and Parts IX–X developed the split-complex scalars, the spectral decomposition, and the master constant ι_τ that governs the B/C asymmetry.
This Part harvests a {logic} from polarity. The key observation is that the two spectral sectors (B and C) can agree, disagree, both confirm, or neither confirm a given predicate. This yields four truth values — T (true), F (false), B (both), N (neither) — arranged in a diamond lattice T > B, N > F.
The resulting logic, Truth4, is earned from the bipolar structure, not axiomatized. It has a built-in explosion barrier: the overdetermined value B (“true and false simultaneously”) does not collapse to triviality, because the B-sector witnesses and C-sector witnesses live in orthogonal spectral components. Classical Boolean logic is recovered as a quotient that collapses B and N.
Truth4 will serve as the subobject classifier Ω_τ in the earned topos structure of Part XIII.