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 OmegaDM = 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 Rᵈ has z (d) = d+1 facets), the paper establishes: (i) zfund = 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 Nbits = d; (iii) the ratio DV/DA = 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 Z₂ 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: zₚack (R³) = Nbits * z (3) = 3 × 4 = 12. The Newtonian limit is resolved by a chameleon mechanism with analytically derived transition threshold rho/rhoc ≈ 15. 9. A candidate baryon fraction Omegab = DA - SK5 ≈ 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 OmegaDM from the same geometric axiom. Companion to TEG v8 (Zenodo, 2026): https: //doi. org/10. 5281/zenodo. 20423814GitHub: https: //github. com/MiguelAngelFrancoLeon/mfsu-tetraedro