Chapter 44: The Explosion Barrier

In classical logic, a single contradiction destroys everything: from P ∧ ¬ P one can derive any proposition Q whatsoever. This is the principle of ex falso quodlibet — from falsehood, anything follows. Classical logic has no structural defense against inconsistency. the relevant chapter introduced the four truth values T, F, B, N (the relevant definition, I.D21), where B represents the overdetermined state “both true and false.” The critical question is: does B trigger explosion? This chapter proves that it does not. The explosion barrier (the relevant theorem, I.T13) states that from val(P) = B, one cannot derive val(Q) = T for arbitrary Q. The structural reason is algebraic: B-witnesses and ¬B-witnesses live in orthogonal idempotent sectors (e_+ · e_- = 0), so contradictions cannot propagate across the spectral decomposition. The explosion barrier is a theorem, not an axiom choice — it is earned from the spectral structure of the algebraic lemniscate 𝕃.