A Z8 Topological Symmetry Breaking Scenario and Cumulative Cosmic Folding: A Unified Framework for 7+1 Division and Logarithmic Δα/α Variation

We present a complete, unified and testable cosmological framework that integrates Z8 topological discrete symmetry breaking with Cumulative Cosmic Folding (CCF). CCF is proposed as a physical alternative to Penrose's Conformal Cyclic Cosmology (CCC) where the crossover surface Σ is not a conformal rescaling but a physical topological folding membrane preserving the topological invariant I0 = 8. Inside each aeon, a Z8-symmetric complex scalar field Φ with potential V = λv^8(|Φ̃|^8 - 1)^2 undergoes spontaneous symmetry breaking to a 7+1 split state via coupling to a secondary heavy scalar χ. The folding operation F̂: I⁺ₙ → I⁻ₙ₊₁ accumulates quantized four-form flux F⁴₍ₙ₎ = nq, with elementary membrane charge q ~ 10^-6 GeV², yielding dark energy density ρ_Λ ~ q² ~ 10^-12 GeV⁴. We introduce the Indistinguishable Degenerate Vacuum Principle (IDVP) as the physical definition of the pre-broken phase: |Φ̃| < 1 is a sea of indistinguishable quanta with V ≈ 0. The Einstein-frame dilaton φ ≡ Φ/M_Pl tracks the logarithm of accumulated flux, leading to a robust prediction Δα/α = β ln(n+1) with quantized coefficient β = [2λ/(2λ+γ)] I[F̂], I[F̂] ∈ ℤ. We provide full derivations of the 5D bulk-boundary metrics, Lanczos-Israel junction conditions, one-loop technical naturalness bound g ≤ 1.5×10^-4, chameleon screening evading MICROSCOPE constraints, and entropy compression factor 10^34. The model predicts a 5σ detection of β ~ 10^-7 with N > 150 high-resolution quasar spectra from JWST NIRSpec and ELT-ANDES.