Hyperarithmetic Complexity of Paraconsistent Kernels: A Technical Note

This technical note analyzes the descriptive complexity of the contradiction kernels arising from the paraconsistent sharp construction developed in the companion paper “Paraconsistent Construction of Sharp Models: A Conditional Framework”. Working under the Countable Root Axiom, we consider the constructible hierarchy Lᵢnftyᵣoot built from a paraconsistent root world via a consistency projection operator that separates classical cores from contradiction kernels. We show that the total kernel K — obtained as the union of all active contradiction kernels across the hierarchy — is a hyperarithmetic (Delta¹₁) subset of the natural numbers, relative to a real coding the root structure. Equivalently, the encoded kernel theory Sigma is Delta¹₁ in that real. The proof proceeds by coding the entire hierarchy as a real, showing that validity of such a code is arithmetical relative to the root, and then observing that membership in K admits both a Sigma¹₁ and a Pi¹₁ definition thanks to uniqueness of the hierarchy. This yields a Delta¹₁ classification by standard results in descriptive set theory. Conceptually, the result shows that the paraconsistent framework does not “create complexity from nothing”: the contradiction kernels are at most as complex as the branching information present at the paraconsistent root. The construction systematically transcodes modal branching behavior into set-theoretic structure while keeping the resulting kernel theory at the same descriptive complexity level as classical sharp-like theories. We briefly discuss consequences for constructibility, the role of the Church–Kleene ordinal relative to the root real, and the structural parallel between paraconsistent kernels and the theories associated with classical sharps.