Geometrization of Manifold k; A Meta-Analysis of String Theory under the Boundary of Realizability

This paper, within the theoretical framework of the geometrization of renormalization group (RG) flow, introduces a ”boundary of realizability” based on fundamental principles of information theory and computational complexity, serving as aprior principle for screening physical theories. This framework posits that the physical legitimacy of a mathematical structure not only requires internal self-consistencybut must also satisfy fundamental limits set by the holographic information capacity upper bound and the feasibility of computational processes. By performing a”geometric survey” of the theory space, this study rigorously distinguishes between”mathematical self-consistency” and ”physical realizability”. The analysis indicatesthat the core value of the string theory framework lies in its role as a ”metaframework” or ”dynamical engine” for generating and exploring the vast ”theorylandscape”, rather than being a stable ”physical continent” uniquely locked by itsown topological fingerprint. Through a systematic screening of the exceptional Liegroup hierarchy, the study reveals that highly complex symmetry structures such asE8×E8, whose description may require information capacities exceeding the cosmicholographic bound and whose derivation of low-energy physics is computationallyinherently infeasible, are deemed unrealizable in principle. In contrast, schemesexemplified by compactification on G2 holonomy manifolds, with topological fingerprints (e.g., (b2,b3) = (21,77)) possessing extremely low information densityand clear, computationally feasible paths for deriving low-energy physics, becomeparadigms within the boundary of realizability. This research aims to provide astructured, quantitatively evaluable meta-theoretical coordinate system and a future research program for the exploration of quantum gravity and unified theories,based on the principle of realizability.