This article develops a strengthened version of Non-Closure Reconstruction as a structural reconstruction of Zermelo–Fraenkel set theory. Its central claim is that membership is not taken as a primitive relation, but emerges from residual reconstruction trace, admitted support, stabilization, raw realization, structural equality, and quotienting. The paper introduces internal realization as a derived relation: raw realization is first generated by residual support components, structural equality is then obtained from realization profiles, and final internal realization is defined as raw realization modulo structural equality. In this way, the article aims to avoid merely renaming set-theoretic membership and instead gives a structural mechanism for its emergence. On this basis, the article reconstructs the basic ZF principles: Extensionality, Empty Set, Foundation, Pairing, Union, Power Object, Separation, Replacement, and Infinity. The theory is organized into a hierarchy of fragments: basic objecthood, finite support reconstruction, hierarchical reconstruction, power-object stabilization, definable structural stability, successor-chain stabilization, and full ZF reconstruction. The article also provides two model-theoretic directions: a finite-tree model for the basic fragments and a closure-system model schema for stronger fragments. It identifies the main open problems: formalizing structural equality, constructing full closure-system models, proving the hard relative consistency direction, and analyzing conservativity over ZF. This work continues the earlier Non-Closure Reconstruction program, especially the structural reconstruction of ZF presented in the previous Zenodo article DOI: 10.5281/zenodo.21176803.
Luka Gluvić (2026) studied this question.