Consistency Projections and Contradiction Kernels in Pair-Valued Semantics

This work develops a set-theoretic algebraic framework for paraconsistent valuations modeled as pairs of subsets (T, F) of a universe U. A natural consistency projection C(T, F) = (T \ F, F \ T) decomposes every state uniquely into a classical core and a pure contradiction kernel (K, K) where K = T ∩ F. In the finite case, the theory recovers known bilattice and four-valued semantics in algebraic form. In the infinite-cardinal setting, it reveals additional structure: stratification of contradiction kernels by cardinality, transfinite families of partial “lambda-collapse” operators generating non-idempotent hierarchies, a semigroup structure for collapse operators, and symmetry centralizers under equivariance assumptions. The interaction between classical information and contradiction is shown to factor independently, and the resulting “cardinal geometry of contradiction” reflects—but does not decide—the ambient cardinal arithmetic of the surrounding set-theoretic universe.