"The Cosmological Constant from K₈ Topology: Λeff = Vₘᵢₙ × e-96π." Paper IV in MEF Series

We derive the effective four-dimensional cosmological constant Λeff from the topology of the twelve-dimensional manifold M₁₂ = (S¹Sol × S³) × K₈, where K₈ = [(CP² × S²) ×w (T²/Z₂)]Spinᶜ is the internal compactification manifold of the Master Equation framework. The derivation proceeds without free parameters. Two factors combine to produce the observed value: (i) Topological suppression. The Euler characteristic χ(K₈) = 12 fixes the quantum coherence coupling constant ζ = 12π via tadpole cancellation. Dimensional reduction from 12D to 4D introduces a warp factor exponent $n = 8$ (six from the fibre volume, two from Einstein-frame Weyl rescaling), giving: e-8ζ = e-96π ≈ 10⁻¹³¹. (ii) Geometric prefactor. The unwarped fibre volume Vol(CP² × S²) = 2π³ τ₂² Rc⁶ and the Planck-to-compactification scale ratio contribute a prefactor Vₘᵢₙ ≈ 10⁹ MPl⁴, where τ₂ ≈ 1.86 is the orbifold aspect ratio derived from sin²θW = 0.224 via τ₂² = (1 - sin²θW)/sin²θW. The product yields: Λeff = Vₘᵢₙ × e-96π ≈ 10⁹ × 10⁻¹³¹ = 10⁻¹²² MPl⁴, in precise agreement with the observed value Λobs ≈ 10⁻¹²² MPl⁴. The positive sign (de Sitter) is a structural prediction: the Spinᶜ gauge flux energy on the T²/Z₂ orbifold stabilises the Quantum Coherence Dilaton and provides geometric uplift from the Anti-de Sitter flux vacuum. Additionally, the same Spinᶜ holonomy that stabilises the vacuum generates a broken mirror symmetry between visible and dark sectors: the ψ-cycle holonomy phase φ_ψ = 5π/12—derived from SU(3) centre symmetry and the Euler characteristic—shifts the effective Weinberg angle at the AQ-wall to sin²θW(AQ) ≈ 0.948, producing a ~ 4.2× enhancement of dark electromagnetic coupling and providing a geometric mechanism for the observed ΩDM/Ωb ≈ 5 ratio. The result is overconstrained: the same topological integer χ(K₈) = 12 simultaneously determines three fermion generations, the electroweak mixing angle, the Planck–weak hierarchy, the cosmological constant, and the dark-to-visible matter coupling ratio. Altering Λ would cascade into violations of all these observables.