Relative Consistency Without Collapse: The Canonical ZF-Coded Model of NCZFs

This note proves the relative consistency of the canonical support-coded formulation of NCZFₛ with Zermelo–Fraenkel set theory. The central result is that if ZF is consistent, then the canonical formulation of NCZFₛ is consistent as well. The proof constructs, from any model of ZF, a canonical reconstruction model NC (M). In this model, stabilized objects are codes for the sets of the original ZF model, structural equality corresponds to ordinary equality, and internal realization corresponds to ordinary membership. The construction explicitly represents the reconstruction apparatus through carrier codes, residual trace witnesses, admitted support, stabilization, raw realization, structural equality, and internal realization. This shows that the canonical version of NCZFₛ introduces no additional consistency strength beyond ZF. Together with the reconstruction direction from NCZFₛ back to the translated ZF layer, the result gives equiconsistency between ZF and the canonical support-coded formulation of NCZFₛ. The result should not be interpreted as reducing NCZFₛ to a mere notational variant of ZF. Rather, it proves formal safety relative to ZF while leaving open the possibility of non-canonical reconstruction models containing additional structural information about trace, support, stabilization, and realization. In this sense, NCZFₛ is consistent with ZF, but not necessarily exhausted by ZF. This note continues the Non-Closure Reconstruction program and clarifies the formal status of NCZFₛ as a reconstruction framework for the emergence of membership from residual support and stabilization.