The Extraction Paradigm: A Formal Mathematical Framework for Property-Based Quantum State Extraction

This paper develops a formal mathematical foundation for quantum computation based on the Extraction Paradigm, the principle that quantum algorithms do not construct solutions but extract them from a fully instantiated Hilbert space. The framework begins with three axioms describing answer-space existence, simultaneous instantiation of computational possibilities, and measurement as projection. From these axioms, the work defines the concept of a property predicate and introduces the Extraction Map, a mathematical operator that isolates solution-bearing subspaces through geometric filtering. The paper proves core results including Completeness (correct answers exist from initialization), Filter Isolation (property filters produce separable subspaces), and the Extraction Guarantee (the correct answer can be recovered with probability approaching unity). A Coverage Condition is derived as a necessary corollary, culminating in a unifying main theorem. All results are formulated independently of physical hardware, positioning the paradigm as a foundational theoretical framework for understanding quantum algorithms as processes of structured state selection rather than sequential construction.