Algebraic Stability and Cosmological Structure A, The Necessity of G_2 Manifolds: From Self-Referential Dynamics to Exceptional Holonomy

This paper is based on a fundamental principle: the ultimate description ofthe physical universe must be a logically self-consistent self-referential dynamicalsystem. Within this framework, by analyzing the spectral gap stability behaviorof the constraint Jacobian matrix at transfinite self-referential depths, we deriveglobal constraint conditions that lock spacetime dimensions and topological invariants. Subsequently, combining the Berger classification and singularity theory,we systematically exclude other possibilities and rigorously prove that the sevendimensional G_2 manifold is the unique, inevitable geometric carrier that realizesthe gauge symmetries of the Standard Model and the three generations of chiralfermion spectra. This framework not only naturally yields specific topological invariants (b2 = 21, b3 = 77) but also makes a series of quantitative predictionscovering particle physics and cosmological scales based on this geometric structure,thereby placing the origin of the universe’s fundamental structure on a falsifiablephysical foundation