Structural Closure of the Isolation Program Pseudo-Deterministic XOR Isolation, Rank Amplification, and Inherent Barriers

This project investigates the structural limits of isolation-based approaches to Unique-SAT and related reductions, with a particular focus on XOR-based and pseudo-deterministic constructions. Across four interrelated papers, we develop a unified algebraic framework for isolation, formalizing rank amplification as the central mechanism by which XOR constraints enforce uniqueness. We show that when implemented uniformly and deterministically, rank amplification constitutes a natural property in the sense of Razborov and Rudich, leading to inherent structural barriers analogous to those in the Natural Proofs framework. The project further explains why pseudo-deterministic approaches can partially circumvent these barriers, while fully deterministic XOR-based isolation cannot close the randomness–determinism gap unconditionally. Rather than proposing new algorithms, this work provides a structural and conceptual closure of the classical isolation program, clarifying which directions are fundamentally limited and which remain open for future exploration.