Tetrahedral Emergent Gravity: A Geometric Information Framework for Galactic Rotation Curves

TEG presents a phenomenological framework for galactic rotation curves grounded in a single axiom: the quantum vacuum selects tetrahedral network coordination (z=4) via holographic entropy density maximisation. All principal model constants derive from this axiom.

Derivation chain:

1. z=4: Derived from Platonic solid entropy maximisation.

2. Information Scales: DA=ln4, DV=ln8. The holographic bit ln2=DV−DA is exact and universal.

3. Geometric Frustration: σUV=0.3263, derived from the mismatch between the vacuum coordination (z=4) and matter packing (z=12).

4. Spatial Dimensions: Nbits=log2(2z)=log28=3 (exact integer representing three spatial dimensions).

5. Global Roughness (σeff): σeff=σUV/Nbits=0.326/3=0.1088. This theoretical value agrees with the empirical SPARC fit of 0.108±0.005 to within 0.72%.

Key caveat on σeff: The derivation assumes isotropic equipartition of UV roughness across the 3 degrees of freedom. While physically motivated, a first-principles proof from LQG (Loop Quantum Gravity) or CDT (Causal Dynamical Triangulations) remains an open challenge. It is important to note that σeff=0.1088 is a derived-and-confirmed parameter, not a freely fitted one.

Empirical results: Applied to 171 real SPARC galaxies using the derived σeff=0.1088 (zero fitting), TEG achieves a mean RMSE of 0.1617±0.081 dex. Including the vacuum entropy term Mvac(r)=ln2⋅Mb,tot⋅(r/rmax)3 improves the fit to 0.1455±0.078 dex (a 10.4% improvement in 127/171 galaxies with zero new parameters). The current r3 scaling represents the uniform-density limit of the free vacuum network; a linear profile Mvac∝r for exact flatness is anticipated for version 1.2. A "z-sweep" analysis confirms that z=4 is empirically preferred over other configurations (z=3 or z=5).

Three falsifiable predictions:

1. Euclid Mission: 5% shear enhancement at θ<1′.

2. Cosmic Voids: 8% density deficit compared to ΛCDM.

3. Statistical Robustness: z=4 consistently outperforms other coordination numbers on the full SPARC catalog (reproducible via the provided Python code in Appendix B).

Keywords: galactic rotation curves, holographic entropy, emergent gravity, tetrahedral vacuum, SPARC, geometric information theory, zero free parameters.