Dual Big Bang from Quantum-Cosmological Uncertainty: From ∆E∆p Planck Bound to Genus-0 Freeze-Out and the No-Go Triangle

We present a unified quantum-cosmological framework combining two previouslyseparate results into a Dual Big Bang model. First, we rigorously derive the rela-tivistic phase-space bounds ∆E∆x ≥ c/2 and the corrected ∆E∆p ≥ c4 /(4G)at the Planck scale, showing the vacuum cannot be confined to lp without gener-ating 1 Ep and 1 mp , with Schwarzschild radius rs = lp . Second, the competition22Γp /Hp10−21 is shown to enforce primordial freeze-out: expansion time−44 s is two orders faster than annihilation t−42 s, pre-texp = 1/Hp ∼ 10ann ∼ 10venting matter-antimatter annihilation and allowing Di domains. The underlyingdeformation is encoded in a Lorentz-covariant cubic-root Generalized UncertaintyPrinciple (GUP) derived from a genus-0 rational curve C : y 2 F 2 + 2F − 2y = 0with incomplete elliptic period Ω(0.5) = 0.484. This yields NCMB = 67.33 e-foldsand a second bang ∆N = 3.17, total NΛ = 70.5 = NCMB , proving a multi-stageinflation No-Go theorem. The framework is UV-finite in 4D (δm ≈ 0.00079 m) andpredicts (i)mν ≈ 0.0911 eV, (ii) r ≈ 0.00088, ns ≈ 0.9673, (iii) flat anti-heliumratio RHe ∼ 1.2 × 10−10 and PBH dark matter MPBH ∼ 5.6 × 1022 kg. Equilibriumand infinite density never existed.