Algebraic Stability and Cosmological Structure C, Information-Algebraic Dynamics and the Emergent Uniqueness of G2 Manifold Compactification

This paper constructs a discrete, hierarchical information-dynamical frameworkstarting from two basic operational principles: conservation of information distinguishability and finite describability. This framework organizes the effective description of physical systems as a sequence of nested hyperfinite type II1 factors, wherethe embedding indices between adjacent levels are forced to take a discrete spectrum by Jones’ theorem. By defining a cumulative information twistor and provingthat the system undergoes a first-order phase transition when it exceeds a universalcritical threshold, the emergence from algebraic data to macroscopic geometry isachieved. Utilizing Connes’ noncommutative geometry spectral triple reconstruction techniques, it is rigorously proved that the critical spectral triple converges toan eleven-dimensional smooth Riemannian manifold, and that under the constraintsof KR-duality and supersymmetry, this manifold uniquely splits into the product ofa four-dimensional Lorentzian spacetime and a seven-dimensional compact internalspace. Combining Berger’s holonomy classification with the modular tensor data ofthe category C√3, the holonomy group of the internal space is uniquely locked to beG2. The combination of the information distinguishability conservation axiom andthe Atiyah–Singer index theorem yields the exact equality between global topological degrees of freedom and local degrees of freedom: b2+b3 = 98; together with theanomaly cancellation condition of the four-dimensional effective theory and the index calculation of the Dirac operator, the unique solution of the system of equationsis (b2,b3,Ngen) = (21,77,3). Consequently, the three-generation fermion structurebecomes a derived conclusion rather than an input hypothesis. Based on theseuniquely determined geometric and topological data, the string scale ℓ2 s = ℓ2P/ln2,the compactification volume VK7 = (2π√3)7ℓ7P/(ln2)7/2, and the four-dimensionalE8 gauge coupling constant g−2E8 =8π2/3 are further derived. The entire frameworkcontains no adjustable free parameters, achieving a constructive derivation fromquantum information ontology to macroscopic geometric dynamics.