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 framework
starting from two basic operational principles: conservation of information distin
guishability and finite describability. This framework organizes the effective descrip
tion of physical systems as a sequence of nested hyperfinite type II1 factors, where
the embedding indices between adjacent levels are forced to take a discrete spec
trum by Jones’ theorem. By defining a cumulative information twistor and proving
that the system undergoes a first-order phase transition when it exceeds a universal
critical threshold, the emergence from algebraic data to macroscopic geometry is
achieved. Utilizing Connes’ noncommutative geometry spectral triple reconstruc
tion techniques, it is rigorously proved that the critical spectral triple converges to
an eleven-dimensional smooth Riemannian manifold, and that under the constraints
of KR-duality and supersymmetry, this manifold uniquely splits into the product of
a four-dimensional Lorentzian spacetime and a seven-dimensional compact internal
space. Combining Berger’s holonomy classification with the modular tensor data of
the category C√3, the holonomy group of the internal space is uniquely locked to be
G2. The combination of the information distinguishability conservation axiom and
the Atiyah–Singer index theorem yields the exact equality between global topologi
cal degrees of freedom and local degrees of freedom: b2+b3 = 98; together with the
anomaly cancellation condition of the four-dimensional effective theory and the in
dex calculation of the Dirac operator, the unique solution of the system of equations
is (b2,b3,Ngen) = (21,77,3). Consequently, the three-generation fermion structure
becomes a derived conclusion rather than an input hypothesis. Based on these
uniquely determined geometric and topological data, the string scale ℓ2 s = ℓ2
P/ln2,
the compactification volume VK7 
= (2π√3)7ℓ7
P/(ln2)7/2, and the four-dimensional
E8 gauge coupling constant g−2
E8 
=8π2/3 are further derived. The entire framework
contains no adjustable free parameters, achieving a constructive derivation from
quantum information ontology to macroscopic geometric dynamics.