Chapter 74: Paraconsistent Logic at Boundaries

At the lemniscate boundary 𝕃 = S¹ ∨ S¹, the two spectral sectors of the bipolar algebra can disagree: one sector confirms a predicate while the other denies it. The result is the overdetermined truth value B—a structurally real contradiction. This chapter develops the philosophical import of Book I’s four-valued logic (Definition I.D21) and the explosion barrier (Theorem I.T13): the natural logic of τ is not Boolean but four-valued, contradictions at boundaries are genuine phenomena rather than defects, and yet the system does not collapse because the spectral decomposition blocks explosion. Classical logic governs the interior; paraconsistent logic governs the boundary. These are not rival logics but different regions of the same structural landscape.