Controlled Penumbral Inflation from Monodromic Valleys

Realizing controlled, single-clock inflation in string theory is fundamentally obstructed by the backreaction of heavy moduli. We show that in the \emph{penumbra} -- the near-boundary regime of complex-structure moduli space where asymptotic symmetries are partially broken -- this obstruction can be exactly quantified. We derive a covariant control theorem demonstrating that local branch data dictate whether a monodromic valley supports a controlled inflationary plateau, thereby isolating the first controlled penumbral inflationary window. The result turns the penumbra from a geometric regime into a dynamical filter. In the axion-saxion effective theory, a branch-displacing odd term generates a plateau when $\Delta\equiv p+2\nu-q>0$, while covariant single-clock control further requires $p<2$, or $p=2$ with $12A_pm^2/(dV_0)\gg1$ over the observational window. This splits penumbral valleys into no plateau, uncontrolled plateau, and controlled plateau before global completion is attempted. We identify a minimal analytic family with a closed-form valley and an invariant attractor equation for the full two-field dynamics, providing the first exactly solvable penumbral realization that remains predictive under the next penumbral order. The controlled corridor targets $r\sim10^{-3}$ with the correlated running $\alpha_s\simeq-r/2$ for the $d=q=1$ benchmark, providing a falsifiable target for LiteBIRD/CMB-S4.