Inter-Universal Teichmüller Theory Comes Home

This paper gives a paraconsistent formulation of Inter-Universal Teichmüller Theory in which the nonclassical carrier is part of the construction itself rather than a semantic reinterpretation imposed after a Boolean calculation. The native state space is $\\mathcal X=\\mathcal P(\\{T,F,t,f\\})$, equipped with a non-explosive consequence relation; Belnap--Dunn FOUR and the Boolean pair $\\{F,T\\}$ occur only as successive restrictions of this larger carrier. Within this setting, the $\\Theta$-link is represented as transport across locally incompatible arithmetic structures without identifying them, while decomposition, branchwise transformation, recombination, filtration, and invariant retention are expressed by ordinary maps on $\\mathcal X$ and enriched spaces over it. The formulation yields three central results. First, contradiction and indeterminacy remain local states of the calculation rather than triggers for global collapse. Second, graph-theoretic reconnection is distinct from classical agreement: the bounded-discrepancy execution can reconnect while terminating in the non-Boolean state $Ftf=\\{F,t,f\\}$. Third, filtration position is mathematical data, not dispensable notation. Suppressing the index that carries the $j^2$ weights produces a tensor-blind scalar comparison and can render the resulting inequality vacuous. Mochizuki's rejection of the Scholze-Stix commuting-loop surrogate is therefore sharpened here as a locality and information-retention statement: incompatible local data need not be identified or eliminated in order for the global construction to remain well formed. Classicalization appears only at the terminal readout, as a lossy projection of a richer paraconsistent computation.