The axion solution to the strong CP problem requires the Peccei–Quinn (PQ) symmetry to be of extremely high quality, so that the vacuum QCD angle satisfies \(|\bar{\theta }|\lesssim 10^{-10}\) . In many extra-dimensional axion models, this quality is protected because explicit PQ breaking is effectively nonlocal: it is communicated through the bulk only via effects that probe the whole compact dimension, leading to exponential suppression. There is a similar but different type of axion model, in which PQ symmetry is broken explicitly and locally on the UV brane, yet a parametrically light pseudoscalar mode can appear in the KK spectrum due to warped localization. This invites a nontrivial question: does a light KK mode necessarily imply that the PQ symmetry is of high quality in the vacuum relevant for QCD? We address this by formulating the vacuum problem in terms of boundary angular variables. Instead of focusing solely on a single propagating KK eigenmode near a chosen background, we compute the axion-dependent vacuum energy over the full \(2\pi \) range by treating the UV and IR boundary phases as two compact angular fields. Bulk dynamics then induces a monodromy-like “stiffness” that energetically correlates the two boundary angles, while QCD provides an additional periodic potential for the IR angle. This two-angle description provides a direct and systematic criterion to assess PQ quality in the strong-CP vacuum, and clarifies how the global potential structure can differ from the intuition based only on the quadratic KK spectrum.
