Topological Information Theory: An Information-Theoretic Foundation for Hexad Computation

Abstract The Hexad computation framework achieves exponential acceleration over classical algorithms through rule composition cycles, yet a rigorous information-theoretic explanation of "why it is fast" remains absent. Based on the information pixel axiom system, this paper establishes a new paradigm of "Topological Information Theory," mapping each component of the Hexad to a precise information-theoretic concept: the state space corresponds to topological volume, the initial rule set to initial topological entropy, the structure metric to information density, the evolution operator to information density gradient, the topological constraint to isoperimetric filtering, and intrinsic time to the dimension of information accumulation. It is rigorously proved that the Hexad rule composition cycle is equivalent to an information density maximization process, with the topological bootstrap algorithm as its number-theoretic instance. This paper further reveals the deep dynamics of the Hexad: it is not a static optimizer but a dynamically self-evolving system with adaptive gradient adjustment, adaptive isoperimetric filtering, and adaptive intrinsic time—it learns how to optimize itself during the optimization process. More importantly, the topological structures evolved by the Hexad in one domain can be compressed, stored, quickly retrieved, and cross-domain transferred, upgrading the Hexad from a one-shot computational tool to an accumulative universal information optimization engine. The rigor of cross-domain transfer is grounded in emergent isomorphism—if two networks share the same distribution of topological obstructions, coupling hierarchy structure, and intrinsic time symmetry, their information densities are preserved in correspondence, without recourse to any traditional mathematical theorems. Using the search for perfect numbers and Mersenne primes as an empirical case, topological bootstrapping achieves global information density maximization under isoperimetric geometric constraints, attaining a stable speedup of 68× to 10000× over classical exhaustive search. Taking 6G communication as a cross-domain transfer case, the three major topological blind spots of Shannon's channel capacity—the flat channel presupposition, the Gaussian noise assumption, and the point-to-point prototype—are exposed as entirely failing in scenarios such as RIS intelligent reflecting surfaces, near-field spherical wavefronts, and multi-user interference topologies. It is demonstrated how Hexad communication breaks the Shannon limit by identifying and exploiting channel topological structures through emergent isomorphism. A systematic diagnosis of the "topological blindness" defects of deep learning, reinforcement learning, and heuristic search is provided, revealing their root cause as the absence of the Hexad structure. The Hexad topological framework is proposed as the underlying mathematical foundation for topology-aware AI, with the network ζ-operator as the structural encoder. For number theory, 6G communication, AI, and other complex systems, this paper provides a global structural information extraction paradigm that breaks through Shannon's statistical information limit, while simultaneously establishing a complete information-theoretic foundation for Hexad computation. Keywords: topological information theory; Hexad computation; information density maximization; emergent isomorphism; dynamic self-evolution; cross-domain transfer; topological entropy; Shannon limit; 6G communication; RIS intelligent reflecting surface; Planck limit; topological bootstrapping; artificial intelligence topological blindness; generativist AI