Tetrahedral Emergent Gravity (TEG) v8: Deriving Dₑff = ln8, rJ = 0. 62 kpc, and ∂ = 3 - ln8 from a Single Geometric Axiom — Validation on 171 SPARC Galaxies

Tetrahedral Emergent Gravity (TEG) derives algebraically the effective vacuum dimension Dₑff = ln 8, the holographic codimension ∂ = 3 - ln 8, and a predictive relation for the Jeans radius rJ from a single geometric axiom: the quantum vacuum in R³ selects tetrahedral network coordination (zfund = 4) by maximising holographic entropy density among all Platonic solids. The derivation chain is parameter-free at every algebraic step: zfund = 4 => Dₑff = ln 8 => sigmaUV = 0. 3263 => Nbits = 3 (exact) => sigmaₑff = 0. 1088. Applied to 171 real SPARC galaxies with sigmaₑff = 0. 1088 (zero fitting), TEG reproduces observed rotation curves with RMSE = 0. 152 dex using the cubic vacuum profile (independently verified on public SPARC data). An empirical z-sweep on the same dataset confirms z = 4 as the coordination that minimises error when sigmaₑff (z) is derived independently at each z. The Jeans radius rJ ≈ 0. 62 kpc emerges from a UV-IR equilibrium condition conjectured in this work (Appendix L), combining the Planck length, the Hubble radius, and factors ∂, sqrt (pi), sigmaₑff all derived from the tetrahedral axiom. For H₀ = 70 km/s/Mpc the prediction agrees with the SPARC empirical median to 7%. This equilibrium condition awaits formal derivation from the EPRL tensor-network renormalization (Open Problem 7). TEG establishes explicit consistency with Loop Quantum Gravity (4-valent nodes, intertwiner dimension 2, spectral dimension dₛ in 2. 0, 2. 2) and Causal Dynamical Triangulations (Dₑff in 2. 0, 2. 5), interpreted as mutual cross-checks rather than derivations. An independent a-posteriori convergence with the causal rigidity results of Bianchi, Chen it claims to offer what neither provides: a geometric derivation of its characteristic parameter (sigmaₑff = 0. 1088) from a Platonic-solid axiom with zero empirical input. Supplementary Note – TEG Volume 8 This work presents an analytical UV–IR equilibrium condition for the galactic Jeans radius within the Tetrahedral Emergent Gravity (TEG) framework: rJ = (lPl^σₑff × RH^ (1−σₑff) ) / (∂ × √π) where: - σₑff = 0. 1088 from holographic equipartition under S₃-symmetry, - ∂ = 3 − ln 8 is the holographic codimension, - √π arises from the solid angle of a tetrahedral face on the unit sphere. For H₀ = 70 km s^−1 Mpc^−1, this gives rJ ≈ 0. 619 kpc, agreeing with the SPARC empirical median to 7%. A reproducible Jupyter notebook (TEGᵥerificacionₜensorial. ipynb) is provided as supplementary material. It verifies the representation-theoretic foundations: dim (Hᵢnt^ (4) ) = 2, SᵥN = ln 2, the S₃-invariant equipartition theorem, and the tetrahedral closure condition with exactly two global solutions. This result is presented as a motivated conjecture (Open Problem 7) awaiting full tensor-network renormalization derivation. ---