Dark Energy Equation of State from the Critical Dimension of Parallelizable Spheres

A parameter-free prediction for dark energy, derived from a 1962 theorem in pure mathematics. Adams' theorem proves that only spheres of dimension 0, 1, 3, and 7 are parallelizable — a deep topological constraint tied to the four normed division algebras (real numbers, complex numbers, quaternions, octonions). We show that the maximum parallelizable dimension, dc = 7, sets the recursion depth of holographic vacuum encoding. The fraction 1/7 of vacuum degrees of freedom constituting the encoding structure itself cannot be holographically encoded; these modes contribute as pressureless matter (w = 0), shifting the dark energy equation of state from w = −1 to: w₀ = −1 + 1/7 = −6/7 ≈ −0. 857 This prediction has zero continuous free parameters. Comparison with data: DESI DR2 (2025) measures w₀ = −0. 838 ± 0. 055. Our prediction deviates by only 0. 35σ. Bayesian model selection yields a Bayes factor of 941× over the two-parameter CPL model and 72× over ΛCDM. Key results: - Parameter-free prediction: w₀ = −6/7, wₐ = 0 - Explicit Lagrangian via assisted quintessence with 7 scalar fields from G₂-holonomy compactification - Cross-domain validation: the same dc = 7 independently predicts 8 fermion mixing parameters (χ² = 2. 63) in a companion paper - Falsifiable: DESI DR5 (~2029) will distinguish this from ΛCDM at >4σ Companion paper: "Fermion Mixing Angles from Three Algebraic Parameters" (Paper 1 in this series) — https: //doi. org/10. 5281/zenodo. 18675985 Correspondence: johnie5waddell@outlook. com