Paraconsistent Construction of Sharp Models: A Conditional Framework

We develop a paraconsistent framework for constructing sharp-like inner models via a hybrid modal–paraconsistent multiverse. Starting from a distinguished apophatic root world w0 governed by the Logic of Paradox (LP), we define a constructible hierarchy Lᵣoot^∞ using a consistency projection operator C that factors pair-valued states into classical cores and contradiction kernels. Under a countable-root assumption, we prove that the idempotency height of C is bounded by the first uncountable ordinal omega₁, so the hierarchy stabilizes at some stage hC below omega₁. The “waste products” of this stabilization—the contradiction kernels—form a Boolean-closed theory Sigma of glutty formulas. We define a canonical map from kernel entries to ordinal stages, obtaining a kernel entry spectrum I inside the stabilized model L^∞, and we show that the classical constructible universe L embeds as a classical core of L^∞. Assuming an Infinite Branching Axiom for the multiverse, a homogeneity hypothesis for the projection C, and a quantifier-closure conjecture for Sigma, the pair (L^∞, I) carries a sharp-like Ehrenfeucht–Mostowski blueprint with a club-like set of order indiscernibles. This yields a structural analogue of “zero sharp” (0#) internal to the paraconsistent multiverse. We close with a programmatic analysis of the expected consistency strength and an equiconsistency conjecture between suitable multiverse axioms and ZFC + 0#.