Grand Unification and Proton Stability from Quantum Rigidity N 43

Here, we demonstrate that grand unification emerges from quantum geometric effects encoded in a scalar field $\Pi$ with rigidity parameter $N \approx 43$. The $\Pi$ field modifies renormalization group running, driving gauge coupling unification. Initial predictions give $M_{\text{GUT}} \sim 6 \times 10^{15}$ GeV, but proton decay constraints require $M_{\text{GUT}} > 1.6 \times 10^{16}$ GeV. We resolve this tension by discovering that $N$ predicts the GUT scale through $M_{\text{GUT}} = M_{\text{Pl}} \exp(-\sqrt{N}) = 1.73 \times 10^{16}$ GeV, giving proton lifetime $\tau_p \sim 1.6 \times 10^{34}$ years, consistent with Super-Kamiokande limits. This exponential suppression arises from instanton effects with action $S \sim \sqrt{N}$, providing a non-perturbative origin for the GUT scale. Hyper-Kamiokande will test this prediction in the coming decade.