Axiomatic Topological Inverse Query and Complexity Maximization Theory: A Paradigm Shift toward Non-Commutative Algebraic Manifold and Chaotic Stream Synthesis

Abstract

For over half a century, the core paradigm of query optimization has been defined by a monotonic, scalar minimization convergence model aimed at suppressing computational resource consumption. This paper presents a radical paradigm shift that fundamentally subverts this traditional framework by establishing the Axiomatic Topological Inverse Query and Complexity Maximization Theory (ATIQ-CMT). Instead of pursuing local or global minima within discrete equivalence graphs, we reconstruct the relational algebra space into a non-Hausdorff, locally compact topological space governed by five foundational axioms. By introducing the Inverse Lipschitz Affine Expansion Mapping (ILAEM) under operator braid transformations, we demonstrate how compact query plans can be inversely dilated into divergent flows across high-dimensional complex affine varieties, creating irreversible mathematical obstructions for traditional gradient-based cost models. To maximize computational complexity natively, we execute a non-commutative extension of the relational algebra core via algebraically twisted join operators embedded in infinite-dimensional Lie algebras, effectively destroying the classic commutative-associative symmetry. We further inject un-decidable Diophantine predicates and 3-SAT arithmetical homomorphic graphs as computational obstructions, rigorously proving a non-polynomial exponential lower bound for physical query execution times. Utilizing sheaf theory and de Rham cohomology on chain complexes, we provide a definitive topological proof that the absolute semantic integrity of the query remains invariant throughout this chaotic dilation. Finally, we formulate a deterministic chaotic operator execution flow driven by high-order Lorenz mappings, maximizing the algebraic Shannon entropy of intermediate states. ATIQ-CMT bridges declarative relational logic and high-level structural topology, unlocking revolutionary potentials in zero-knowledge proof circuit synthesis, active cybersecurity defense, and the theoretical computational limits of neuro-symbolic and quantum systems.