Paper CLXXXVIII: The Frozen e₇ Lattice as the Origin of the Cosmological Constant: Two Zero-Free-Parameter Derivations of ρΛ, the Metric Signature, and w = −1

We derive the cosmological constant ρΛ from first principles via two independent, zero-free- parameter routes grounded in the One-Octonion Brane-Bulk framework. The starting point is that the matter–antimatter asymmetry δM/M = 6.10×10 ¹ (Paper CLX) produces exactly one ⁻ ⁰ unpaired e₇ lattice node per baryon. Each e₇ node is frozen — left-handed, color-singlet, unable to annihilate — because [e₄,e₅,e₆] = −2e₇ algebraically forbids a partner. These frozen nodes constitute a spatially isotropic, Lorentzian lattice whose causal geometry derives the metric signature (−,+,+,+) and whose isotropy forces w = −1, both without free parameters. Route A (top-down, Papers XLII + G₂ mode selection): The Fano zero-point vacuum energy is ρΛ(XLII) = ħH₀²/(ℓ_P²c). Of the 14 G₂ generators, only e₇ is frozen; the other 13 cancel through H /H pairing. This gives ρΛ = ħH₀²/(14ℓ_P²c) = 4.589×10 ¹ J/m³, which is 87.4% of the ⁺ ⁻ ⁻ ⁰ observed value. Route B (bottom-up, e₇ node counting): Each frozen node carries vacuum energy Eγ = 19.51 GeV (Paper CLXXIX, G₂-curved transit). The node density is n_b = δM/M × n_γ = 0.2505 m ³. ⁻ The spatial Lorentzian fraction 3/4 and sump frozen fraction f_sump = exp(−t/T_sump) = 0.827 give ρΛ = (3/4) × n_b × f_sump × Eγ = 4.859×10 ¹ J/m³, which is 92.5% of the observed value. ⁻ ⁰ The two routes agree with each other to 5.6% and with observation to 7.5–12.6%. The residual gap is within the theoretical uncertainty of the framework parameters. Both routes bracket the observed ρΛ = 5.251×10 ¹ J/m³ from below with zero free parameters. ⁻ ⁰

Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.