A Narrow-Band Prediction for Proton Decay from Geometrodynamic SO(10) with f_a = M_X/2

I present a sharp and falsifiable prediction for the proton lifetime in the channel \(p\rightarrow{e^{+}\pi^{0}}\) within an \(SO{(10)}\) grand unified setting. The only structural assumption beyond standard unification that I make is to tie the Peccei–Quinn scale to the unification scale, \(f_{a} = {M_{X}/2}\). This identification, motivated by a broader geometrodynamic \(SO{(10)}\) framework, collapses model dependence and yields a narrow lifetime band. Using two-loop unification with moderate thresholds, I find \(M_{X} \simeq {{6.8 \times 10^{15}}{GeV}}\) and \(\alpha_{unif}^{- 1} \simeq 36.0\), which imply a central estimate \({\tau_{p}{({p\rightarrow{e^{+}\pi^{0}}})}^{(0)}} \simeq {{9.05 \times 10^{34}}{yr}}\). After marginalizing over threshold and hadronic inputs I obtain a conservative \(95\%\) theory interval \({\tau_{p}{({p\rightarrow{e^{+}\pi^{0}}})}} \in {\lbrack{{6.46 \times 10^{34}}{yr}},{{1.16 \times 10^{35}}{yr}}\rbrack}\). A future bound \({\tau_{p}{({p\rightarrow{e^{+}\pi^{0}}})}} > {{1.16 \times 10^{35}}{yr}}\) would exclude this benchmark at \(95\%\) C.L. The link to the PQ sector further implies \({(m_{a},g_{a\gamma\gamma})} = {({1.68{neV}},{{\, 2.55 \times 10^{- 19}}{GeV}^{- 1}})}\), which I use only as an internal consistency check. The purpose of this paper is deliberately narrow: I isolate one testable consequence (a narrow-band prediction for \(\tau_{p}{({p\rightarrow{e^{+}\pi^{0}}})}\)), state all assumptions explicitly, and provide clear falsification criteria without relying on UV-complete details of the geometrodynamic construction. All results assume single-step SM\(\rightarrow\)SO(10) unification with dominant dimension-six gauge exchange (color-triplet scalars decoupled).