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

Tetrahedral Emergent Gravity (TEG v8) derives galactic rotation curves from tetrahedral coordination zfund = 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 ℝᵈ has z (d) = d+1 facets — yields five theorems: (i) zfund = 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 Nbits = d; (iv) the ratio DV/DA = 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: zₚack (ℝ³) - z (3) = 12 - 4 = 8 = 2Nbits (3) This identity is a corollary of the factorisation zₚack = Nbits · 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: ΩDM = 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/ρc ≳ 15. 9. A candidate for the baryon fraction is derived from the spectral entropy deficit of the 5-cell graph K₅: Ωb = DA - SK₅ ≈ 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: αTEG = ln 8 / 4 = 3 ln 2 / 4, Dₑff = 4αTEG = 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 Vₜilde (a, ρ) = V (a, ρ) - DA with boundary condition Vₜilde (0) = 0. The fractal correction to the scalar equation of motion exists but is IR-suppressed by (H₀/HPl) ² ≈ 10^-122, leaving all main-text results unchanged. The holographic codimension ∂ = 3 - ln 8 acquires a first-principles geometric interpretation as ∂ = Dₛpatial - Dₑff. A conjectural Hubble constant H₀TEG = HCMB / √∂ = 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 fσ₈ (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 Ωₘ: TEG predicts ΩₘTEG = 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 zₚack (ℝ³) = 12. The derivation of ΩDM 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 = HCMB / √∂) 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 Mᵥac (r) from Vₜilde (a, ρ), and boundary condition Vₜilde (0) = 0 from a symmetry or renormalisation-group principle (Open Problem 3) ; joint derivation of H₀ and ΩDM 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 zₚack (ℝᵈ) = d · 2^ (d-1), with d=3 as the unique solution (Open Problem 5). Companion to TEG v8 (Zenodo, 2026): https: //doi. org/10. 5281/zenodo. 20423814GitHub: https: //github. com/MiguelAngelFrancoLeon/mfsu-tetraedro