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

Tetrahedral Emergent Gravity (TEG v8) derives galactic rotation curves from tetrahedral coordination z_fund = 4 in ℝ³ with zero free parameters. That coordination was a hypothesis. We identify it as a theorem by two independent routes, both anchored in two physical inputs stated explicitly as axioms. Physical inputs (axioms):(i) The quantum vacuum is fundamentally described by the algebra of unit quaternions ℍ (topologically S³); all observed physics in ℝ³ emerges as the natural projection π: ℍ → ℝ³ (Axiom 2.1).(ii) The 5-cell {3,3,3} uniquely maximises holographic entropy density among all regular 4-polytopes at R=1 (Proposition 3.1, established by direct computation on all six regular convex 4-polytopes; not derived from a dynamical principle — see Open Problem 2). Derived results:Under these two axioms, the Minimal Simplex Principle — the combinatorial fact that the minimal convex polytope in ℝ^d has z(d) = d+1 facets — yields five theorems:(i) z_fund = z(3) = 4 is a theorem, not a hypothesis;(ii) the holographic bit ΔS = ln 2 is universal;(iii) d=3 is the unique dimension with integer bit count N_bits = d;(iv) the ratio D_V/D_A = 3/2 is exact and unique to d=3;(v) projection reduces coordination by exactly one. A new identity unique to d=3 connects three independent results:z_pack(ℝ³) - z(3) = 12 - 4 = 8 = 2^[N_bits(3)] This identity is a corollary of the factorisation z_pack = N_bits · z(3) = 3 × 4 (Proposition 5.2) combined with Theorem 4.4. The ℤ₂ symmetry a ↦ -a of the S³ sigma-model (Theorem 5.5) derives the factor 2 in the geometric frustration exactly. Together with the equipartition theorem of TEG v8, this yields the cosmological dark matter fraction without free parameters:Ω_DM = 2 ln(3/2) / 3 ≈ 0.2703 This value converges within 0.1% of the local SH0ES measurement and 4.4% of Planck 2018. The Newtonian limit is resolved by a chameleon mechanism with analytically derived transition threshold ρ/ρ_c ≳ 15.9. A candidate for the baryon fraction is derived from the spectral entropy deficit of the 5-cell graph K_5: Ω_b = D_A - S_K_5 ≈ 0.0516 (5.2% from observed). Covariant extension (Appendix F):A covariant action for TEG is constructed by integrating the multi-fractional spacetime formalism of Calcagni (2012) with the tetrahedral axiom. The single free parameter of Calcagni's family is fixed exactly by the geometric axiom:α_TEG = ln 8 / 4 = 3 ln 2 / 4, D_eff = 4α_TEG = ln 8 The resulting field equations reduce to standard Einstein equations sourced by matter and the scalar field a(x); the multi-fractional measure cancels from the gravitational sector. General Relativity is recovered exactly in the high-density limit via the renormalised potential V_tilde(a,ρ) = V(a,ρ) - D_A with boundary condition V_tilde(0) = 0. The fractal correction to the scalar equation of motion exists but is IR-suppressed by (H₀/H_Pl)² ≈ 10^-122, leaving all main-text results unchanged. The holographic codimension ∂ = 3 - ln 8 acquires a first-principles geometric interpretation as ∂ = D_spatial - D_eff. A conjectural Hubble constant H₀^TEG = H_CMB / √∂ = 70.25 km s^-1 Mpc^-1 (Route A, Conjecture 3) is within 0.3σ of CCHP 2025 and 1.5σ of DESI 2024; an alternative structural route gives 61.8 km s^-1 Mpc^-1 (Route B), and reconciling the two constitutes the primary open sub-problem of the covariant extension. Observational tests (Appendix D):Systematic tests against current public data are reported with honest assessments of discriminating power: the GWTC-3 second law (459 BBH events, 99.0% compliance); EHT shadow sizes for M87* and Sgr A* (consistent with the high-density GR limit); cluster lensing convergence for Abell 2744 (χ²/N = 3.91 vs. NFW 0.52, limited by 2011-era angular resolution); and the growth-rate parameter fσ_8(z) from DESI DR1 (χ²/N = 1.24, zero free parameters, vs. Planck 0.95 with two parameters). The decisive near-term test is a direct Euclid Full Mission measurement of Ω_m: TEG predicts Ω_m^TEG = 0.3193, separable from Planck ΛCDM at ~8σ with Euclid precision. Scope of the derivation:All results from Section 4 onward follow by proved theorems from Axiom 2.1, Proposition 3.1, and the classical kissing number z_pack(ℝ³) = 12. The derivation of Ω_DM contains no additional conjectures beyond these three inputs. Proposition 3.1 is established within TEG by direct computation but is not derived from a variational or dynamical principle; its deeper justification is identified as Open Problem 2. One open conjecture (Conjecture 3: H₀ = H_CMB / √∂) and five open problems are stated precisely: derivation of the baryon fraction from the causal asymmetry of the 5-cell (Open Problem 1); derivation of V(a,ρ) from a single variational principle in the S³ sigma-model (Open Problem 2); completion of the covariant formulation — Friedmann equation in FRW, derivation of M_vac(r) from V_tilde(a,ρ), and boundary condition V_tilde(0) = 0 from a symmetry or renormalisation-group principle (Open Problem 3); joint derivation of H₀ and Ω_DM from the same geometric axiom to resolve the Planck–SH0ES discrepancy (Open Problem 4); and characterisation of all dimensions where the optimal-packing contact set decomposes as z_pack(ℝ^d) = d · 2^(d-1), with d=3 as the unique solution (Open Problem 5). Erratum (vH3.1). A previous version of this manuscript (sub. June 2026) contained an arithmetic error in Eq. (4) and its "clarification" in Appendix E.4: a spurious factor of DV/DA=3/2 was introduced into the definition of the geometric frustration F(R^3) and did not cancel as claimed. The correct expression is F(R^3)=2ln(z_pack/(z_pack-z_fund))=2ln(3/2), with no DV/DA factor. This was already the value used consistently in Table 6 and in the final numerical result Omega_DM=2ln(3/2)/3 approx 0.27031 (Theorem 5.9), so no numerical prediction, table, or figure in this paper is affected; only the symbolic derivation shown in Eq. (4) and Appendix E.4 has been corrected. See Sections 5.3 and E.4 for the corrected derivation. Companion to TEG v8 (Zenodo, 2026): https://doi.org/10.5281/zenodo.20423814GitHub: https://github.com/MiguelAngelFrancoLeon/mfsu-tetraedro