Interference Valley and Operator-Level Phase Locking in Neutrinoless Double Beta Decay within a Left--Right Symmetric Model

Neutrinoless double beta decay ($0\nu\beta\beta$) is a key probe of lepton-number violation and the Majorana nature of neutrinos. While the standard interpretation is based on light Majorana neutrino exchange, many well-motivated extensions of the Standard Model generate additional higher-dimensional contributions that interfere coherently with the dimension-5 amplitude. In this work, we investigate multi-operator interference within a left--right symmetric model (LRSM) matched onto the low-energy effective theory. Performing a two-dimensional scan in $(\log_{10} v_R, \alpha_\phi)$ space, we identify a pronounced interference valley in which the total $0\nu\beta\beta$ amplitude is strongly suppressed. A refined scan confirms that the suppression is not a grid artifact, but a genuine dynamical feature. Decomposing the amplitude into dimension-5 and dimension-7 components, we demonstrate that the valley originates from robust anti-phase alignment between the two contributions. The cancellation strength exhibits a sharp minimum, and the relative phase remains locked near $\pi$ across the valley region. Along the main-valley trajectory, the dimension-7 amplitude follows a clean power-law scaling in $v_R$, indicating control by a coherent operator chain rather than accidental tuning. Operator-level diagnostics show that a specific subset of dimension-6 structures governs both the phase and scaling behavior of the dimension-7 contribution. Our results reveal a structurally stable interference mechanism in multi-operator $0\nu\beta\beta$, highlighting the importance of phase correlations and operator alignment in interpreting future experimental constraints.