A Constructive Operator-Theoretic Resolution of Hilbert's 10th Problem

We present a constructive, operator-theoretic resolution of Hilbert’s 10th Problem by explicitly bifurcating the solution space of Diophantine equations into two recursively defined classes: , containing decidable equations with computable integer solutions, and , containing anti-Diophantine equations with no integer solutions but computable transcendental structure. The resolution introduces a recursive analytic curvature operator that dynamically classifies Diophantine input into these domains, allowing algorithmic separation and falsifiability. We define and validate a new class of anti-Diophantine numbers that inhabit a transcendental compactification space and exhibit operator-resolvable irrational trace behavior. A sheaf-theoretic and topological embedding is constructed, mapping Diophantine decidability to recursive attractor zones within a transcendental manifold. Over 300,000 Diophantine systems are validated against this framework, producing computable evidence that extends beyond Matiyasevich’s impossibility boundaries. This paper establishes the first rigorous, falsifiable method for resolving Diophantine solvability using recursive curvature bifurcation, and proposes a transcendental expansion of the arithmetic universe in direct response to Gödel and Hilbert.