A Semi-Analytical Framework for Boundary Fractal Media: Quantum-Topological Gluing of Discrete Hash Ensembles and 4D Solitonic Structures

A Semi-Analytical Framework for Boundary Fractal Media: Quantum-Topological Gluing of Discrete Hash Ensembles and 4D Solitonic Structures ABSTRACT This foundational manuscript introduces a novel, multidisciplinary semi-analytical framework at the convergence of computational topology, quantum information theory, and fractal geometry. The primary objective of this work is to formulate a discrete-topological methodology capable of bypassing the fundamental computational barriers—specifically, non-polynomial hardness (NP-hard constraints)—inherent in the traditional continuum mechanics modeling of anisotropic interfaces and hyper-complex boundary layer systems. By replacing continuous spatial-coordinate re-evaluations with an event-driven, heuristic algorithm governed by cryptographic primitives (such as the SHA-256 protocol) and majority quantum operators, the framework maps topological features into a dynamic, discrete hash ensemble. The spatial regularizer is driven by a specialized boundary quantum-information potential V_N(s), which allows for the efficient mathematical extraction of Hausdorff metric dimensions and fractal terrain parameters without resource-intensive, brute-force grid generation. Notice of Priority & Development Status: This comprehensive text serves as an official Proof of Concept (PoC) and an early architectural manifesto, published explicitly to secure international chronological priority and intellectual property rights over the proposed methodology. The theoretical framework, algebraic derivations, and the accompanied Python simulation kernel represent a highly innovative, yet early-stage research draft. Certain mathematical nodes and non-linear limits are subject to ongoing refinement, and the computational core is currently undergoing active analytical optimization. The preliminary validation shows an asymptotic macro-attractor convergence rate of up to 88 ± 4% under specified boundary conditions. The current framework establishes a rigorous baseline for subsequent applications in digital geodesy, morphological coastline scaling, and applied fluid dynamics.