Algebraic Stability and Cosmological Structure F: The η16 Index Rigidity Criterion

Within the cross-disciplinary domain of Jones subfactor theory, G2 manifoldtopology, and the Atiyah–Singer index theorem, this paper establishes an algebraicstability framework derived from first mathematical principles and containing nofree parameters. We prove that the total dimension of the chiral fermion zeromode space in the four-dimensional effective theory is uniquely locked to dmatter =16. This number is an inevitable consequence of three independent mathematicalstructures: (i) the fusion rules of the Jones discrete spectrum {4cos2(π/n)} and thequantum SU(2) representation category, which force the tensor product structure ofthe Fibonacci category and the SU(2)3 category, whose global quantum dimension’sinteger part exactly equals the dimension of the G2 Lie algebra; (ii) the topologicalinvariants (b2,b3) = (21,77) of G2 manifolds, uniquely determined by informationgeometric duality and Diophantine equations; (iii) the generalized index theorem,which requires that the η-invariant contribution of two E8 singularities be 40, forcingthe Dirac operator index to the integer value 16. The parameters d0 = 22 anddgeom = 6 are independently fixed by the spectral gap stability convergence pointand the minimality of moduli stabilization, respectively, and satisfy the informationconservation identity d0 = dgeom+dmatter. The entire parameter system constitutesan overdetermined closed structure with no adjustable parameters. The frameworkoutputs twelve rigid quantitative predictions covering frontier testable signals suchas the proton lifetime, axion-like dark matter, neutrino mass spectrum, primordialgravitational wave background, Hubble constant tension, and new gauge bosonresonances; experimental verification is the sole means of theoretical adjudication.