Regular Simplex Hierarchical Gravity Part III: Hierarchical Renormalization and the 600-cell as a Separation Device

Parts I and II established the static geometry and the dynamics of Regular Simplex Hierarchical Gravity (RSHG): from Axiom Zero and the ineliminable tetrahedral packing deficit δ = 2π − 5 arccos(1/3) ≈ 7.36° (Niven's theorem), the 600-cell arises as the unique four-dimensional closure, and its projection onto the three-dimensional brane yields gravity (Ω_local ≈ 100, reproducing G to 1.1%) together with the Light-Speed Resource Allocation Principle c² = v² + τ². Part III develops the thermodynamics of this structure. We assign the 600-cell a second role, as a separation device that, at every hierarchy boundary, partitions the incoming frustration into three mutually exclusive channels—structural stress (rigidity transmitted upward), computational entropy (heat dissipated downward, the arrow of time), and unresolved four-dimensional residue (a dark-matter candidate)—under the conservation of frustration Φ_in = Φ_A + Φ_B + Φ_C. We then establish the central identity of the paper: time-averaging the vibrating force chains over a time scale τ is equivalent to spatial coarse-graining over a length cτ, which is exactly a renormalization operation, so that the observation time scale itself selects the observed hierarchy. Iterating this renormalization across six hierarchies—each a jamming transition forced by the capacity limit of frustration, with a scale ratio of 10^6 and a per-step rigidity transmission ε_n ≈ 10^−19.2—produces a cumulative suppression ε_total ≈ 10^−115.2 that approaches the 122-order cosmological-constant discrepancy as an arithmetic consequence, while φ ≈ 0.62, Ω_local ≈ 100, and δ ≈ 7.36° remain invariant at every scale. Finally, we analyze the dynamic stability of that operating point, where gravity (structural stress) and Pauli repulsion (computational entropy) balance just inside the packing limit: the universe never reaches φ_c, and the residual structural vacancy φ_c − φ ≈ 0.02 simultaneously averts asymptotic freezing, sustains the renormalization dynamics, and serves as the leak path through which force reaches three dimensions. A system poised at this boundary between structural stress and computational entropy offers a physical perspective on the boundary conditions that may sustain complex systems such as life. We state explicitly which quantities are fixed with zero free parameters and which—notably the first-principles value of ε_n—remain open.