Tetrahedral Emergent Gravity vH2: The Minimal Simplex Principle and the Quaternion Projection Paradigm

Tetrahedral Emergent Gravity (TEG vH2.0) derives the cosmological dark matter 
fraction Omega_DM = 2*ln(3/2)/3 ≈ 0.2703 from two geometric axioms and zero 
free parameters, within 0.1% of the local SH0ES measurement.

Starting from the Quaternion Vacuum Axiom (the quantum vacuum is a unit-quaternion 
field on H ≅ S³) and the Minimal Simplex Principle (the minimal convex polytope 
in R^d has z(d) = d+1 facets), the paper establishes: (i) z_fund = 4 is a theorem, 
not a hypothesis, via the stereographic projection of the 5-cell {3,3,3} onto the 
regular tetrahedron; (ii) d = 3 is the unique spatial dimension with integer 
holographic bit count N_bits = d; (iii) the ratio D_V/D_A = 3/2 is exact and 
unique to d = 3; (iv) the algebraic and stereographic projections of the 5-cell 
are physically equivalent for TEG (Theorem on projection equivalence); (v) the 
Z_2 symmetry a → -a of the S³ sigma-model derives the factor 2 in the geometric 
frustration F = 2*ln(3/2) exactly.

A new factorisation unique to d = 3 is proved constructively via the 
cuboctahedron decomposition of the FCC contact set: z_pack(R³) = N_bits * z(3) 
= 3 × 4 = 12. The Newtonian limit is resolved by a chameleon mechanism with 
analytically derived transition threshold rho/rho_c ≈ 15.9. A candidate baryon 
fraction Omega_b = D_A - S_K5 ≈ 0.0516 (5.2% from observed) is derived from the 
spectral entropy deficit of the 5-cell graph K5 with no free parameters.

Sixs open problems are stated precisely, including derivation of the vacuum 
potential from a variational principle and the joint derivation of H0 and Omega_DM 
from the same geometric axiom.

Companion to TEG v8 (Zenodo, 2026): https://doi.org/10.5281/zenodo.20423814
GitHub: https://github.com/MiguelAngelFrancoLeon/mfsu-tetraedro