Tetrahedral Emergent Gravity (TEG) v7: Deriving D_eff = ln8, r_J = 0.62 kpc, and ∂ = 3 - ln8 from a Single Geometric Axiom — Validation on 171 SPARC Galaxies

After 40 years, MOND's acceleration scale a0 still has no derivation from first principles — it is an empirical constant inserted by hand. Newtonian Fractional-Dimension Gravity (NFDG) identifies an effective spacetime dimension D_eff ≈ 2.0–2.2 in galaxy outer regions, but cannot derive it from first principles either. We present Tetrahedral Emergent Gravity (TEG), which derives both from a single geometric axiom.

The axiom: the quantum vacuum in R³ selects tetrahedral network coordination (z_fund = 4) by maximising holographic entropy density among all Platonic solids. The derivation chain is algebraic and free of adjustable parameters at every step:

z_fund = 4 → D_eff = ln8 ≈ 2.079 → σ_UV = 0.3263 → N_bits = 3 (exact) → σ_eff = 0.1088

Applied to 171 real SPARC galaxies with σ_eff = 0.1088 (zero fitting), TEG achieves RMSE = 0.138 ± 0.076 dex, subject to three conventional scales (r_ref, M_J,ref, N0) documented in Table 1. Performance degrades on gas-dominated dwarf galaxies (log M_b < 9.5), identified as the primary target for improvement in v8.

The Jeans radius r_J ≈ 0.62 kpc is derived from the LQG volume operator for 4-valent nodes, coarse-grained with D_eff = ln8, agreeing with the empirical SPARC median to 7%. The universal holographic bit D_V - D_A = ln2 is an algebraic identity verified to 16 significant figures. The holographic codimension ∂ = 3 - ln8 ≈ 0.921 is derived algebraically from the tetrahedral axiom and yields H0_TEG ≈ 70.3 km/s/Mpc as a strong conjecture (Open Problem 5).

Six open problems are documented with full honesty: the spin-foam derivation of the orientation factor (Op.1), G_eff normalization (Op.2), the effective GR action (Op.3), the Lorentzian spectral dimension (Op.4), the cosmological extension (Op.5), and the derivation of external scales (Op.6). D_eff = ln8 is consistent with LQG spectral dimensions and CDT de Sitter phase results.

Reproducibility code runs in under 5 minutes on the public SPARC dataset. TEG's claim is a complete algebraic derivation chain from a single vacuum geometry axiom to confirmed effective parameters, with all assumptions and limitations explicitly documented.

TEG v7 — 

**Tetrahedral Emergent Gravity (TEG)** derives algebraically the effective vacuum dimension \(D_{\rm eff} = \ln 8\), the Jeans radius, and the holographic codimension from a single geometric axiom: the quantum vacuum in \(\mathbb{R}^3\) selects tetrahedral network coordination (\(z_{\rm fund} = 4\)) by maximising holographic entropy density among all Platonic solids.

Applied to 171 real SPARC galaxies with zero free parameters, TEG reproduces the observed rotation curves with RMSE = 0.152 dex using the cubic vacuum profile (independently verified on public SPARC data). The framework includes full validation, an empirical z-sweep confirming \(z = 4\), explicit connections to Loop Quantum Gravity (LQG) and Causal Dynamical Triangulations (CDT), and eight open problems documented with complete transparency.

All model parameters are derived from the single tetrahedral axiom. TEG shows an independent a-posteriori convergence with the recent results on causal rigidity and edge orientations in EPRL vertices by Bianchi, Chen & Gamonal (2026), interpreted as a mutual cross-check rather than a derivation.

**Note on the scope and objectives:**  
The primary goal of TEG is not to compete with MOND or ΛCDM in fitting precision at this stage (current RMSE = 0.152 dex). Instead, the central objective is to test the hypothesis that the quantum vacuum selects tetrahedral coordination (z = 4) as the structure that maximizes holographic entropy density, leading to a fractal-like emergence of spacetime properties.

Parameters such as σ_eff = 0.1088 are derived algebraically from this single geometric axiom, with zero fitting.

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