Tetrahedral Emergent Gravity (TEG) v8: Deriving D_eff = ln8, r_J = 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 D_eff = ln 8, the holographic codimension ∂ = 3 - ln 8, and a predictive relation for the Jeans radius r_J from a single geometric axiom: the quantum vacuum in R^3 selects tetrahedral network coordination (z_fund = 4) by maximising holographic entropy density among all Platonic solids. The derivation chain is parameter-free at every algebraic step:z_fund = 4 => D_eff = ln 8 => sigma_UV = 0.3263 => N_bits = 3 (exact) => sigma_eff = 0.1088. Applied to 171 real SPARC galaxies with sigma_eff = 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 sigma_eff(z) is derived independently at each z. The Jeans radius r_J ≈ 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), sigma_eff all derived from the tetrahedral axiom. For H_0 = 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 d_s in [2.0,2.2]) and Causal Dynamical Triangulations (D_eff 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 & Gamonal (2026) on EPRL edge orientations is reported. Eight open problems are documented with complete transparency, including the formal derivation of the UV-IR equilibrium (Op. 7), the cosmological extension connecting ∂ to the Hubble tension (Op. 5), and the physical origin of the mass threshold log M_b ≈ 10.5 (Op. 8). Note on scope and objectives: The central goal of TEG is to test whether a single geometric axiom — tetrahedral vacuum coordination selected by holographic entropy maximisation — can generate a complete derivation chain to observable galactic parameters. The current RMSE (0.152 dex) reflects open problems in the M_vac(r) profile beyond r_J that are identified as well-defined mathematical targets, not failures of the axiom. TEG does not claim to compete with MOND or LambdaCDM in fitting precision at this stage; it claims to offer what neither provides: a geometric derivation of its characteristic parameter (sigma_eff = 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: r_J = ( l_Pl^σ_eff × R_H^(1−σ_eff) ) / ( ∂ × √π ) where:- σ_eff = 0.1088 from holographic equipartition under S_3-symmetry,- ∂ = 3 − ln 8 is the holographic codimension,- √π arises from the solid angle of a tetrahedral face on the unit sphere. For H_0 = 70 km s^−1 Mpc^−1, this gives r_J ≈ 0.619 kpc, agreeing with the SPARC empirical median to 7%. A reproducible Jupyter notebook (TEG_verificacion_tensorial.ipynb) is provided as supplementary material. It verifies the representation-theoretic foundations: dim(H_int^(4)) = 2, S_vN = ln 2, the S_3-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. ---