CNVS, Formal Mathematical Synthesis.

This paper introduces Closed Native Verification Systems (CNVS), a novel formal architecture for privacy-preserving verification where raw structural membership is primitive, while state validity relies strictly on local admissibility, relational consistency, and global invariant constraints. CNVS shifts the security paradigm from traditional algebraic reconstruction (e.g., Shamir’s Secret Sharing) to restricted local observation, hidden invariant binding, and bounded topological leakage. The system employs a context-preserving recursive decomposition that generates atomic evaluation primitives—distributed across rigorously defined hierarchical depths, accurately mapping the progression between terminal fragments and nodal reconstruction from level $S(0)$ to n—and injects them into a randomized routing topology. External verifiers do not receive the internal semantic payload; instead, they operate on measurement grids that require convergent observation. A core theoretical contribution of this work is the formalization of the Theorem of Emergent Security, which proves that deterministic unauthorized reconstruction is mathematically blocked provided the residual conditional min-entropy of critical fragments remains strictly positive under a worst-case adversarial view. The foundational axioms and integration theorems of the CNVS framework have been rigorously validated using the Lean 4 theorem prover. Furthermore, empirical validation through Monte Carlo simulations demonstrates the architecture's robustness, achieving resistance thresholds of 80% to 90% against dependent collusion attacks under specific fragmentation conditions. Ultimately, this synthesis opens a mathematically grounded design space for distributed, confidentiality-preserving evaluation distinct from traditional threshold cryptography.