A Condensed Variational Framework on CPTP Mapping Spaces: Measure-Theoretic Foundations, Discrete-Continuum Correspondence, and Parameter Self-Consistency

We study the extremal variational problem on the space of completely positive trace-preserving (CPTP) maps over finite directed 2-complexes---minimizing the face action over all CPTP assignments. Using Clausen-Scholze condensed mathematics, we establish a rigorous mathematical foundation for this variational framework.Finite-dimensional rigorization. We construct a condensed Choquet probability measure on CPTP mapping spaces (Theorem 2.1), introduce condensed equivalence that replaces pointwise limits with condensed fibrations, prove a condensed Connes trace theorem establishing the discrete-continuum correspondence for Seeley-DeWitt coefficients (Theorem 3.1), and verify the seven axioms of condensed spectral triples (Appendix E). The algebraic foundations are fully rigorous; the geometric limit (axiom 3, condensed Rellich-Kondrachov) is an explicit open problem.Condensed Schwinger-Dyson fixed point and parameter self-consistency. We prove a condensed fixed-point theorem (Theorem C.1): in the parameter region where the low-order moments of the cutoff function are much smaller than the high-order moments, the condensed Schwinger-Dyson equations possess a unique fixed point that determines all coupling parameters. The KO-theoretic fiber classification (Section 5.2) identifies the topological origins of fermion generation structure.Infinite-dimensional extension. We extend the finite-dimensional results to approximately finite (AF) algebras and hyperfinite II_1 factors via the liquid limit of the condensed variational framework (Appendix B), establishing a rigorous mathematical bridge between finite-dimensional noncommutative geometry on 2-complexes and infinite-dimensional operator algebras.