Defect-Induced Channel Shift and Energy Conversion in a Modular Spectral System

We show that channel compatibility in the modular spectral system is naturally described by a module extension structure, where coupling between adjacent channels corresponds to a nontrivial gluing of modules rather than a simple superposition. At a distinguished modular point, one channel loses its base component and survives only as a first-order response. This activates a unique extension between neighboring channels, forcing a one-step shift and coupling a near-degenerate pair. channel mixing is equivalent to forming a non-split extension of modules, forward and backward transitions correspond to compositions of extensions, the resulting asymmetry produces a geometric phase that cannot be removed. The arithmetic input organizes this structure:Pell-, Fibonacci-, and Ramanujan-type sequences define compatible projection data that determine how channels are glued and how energy is redistributed. Because the coupled channels are nearly degenerate, the extension-induced interaction is strongly amplified, leading to a large and directed energy conversion localized at the defect. Thus, the system realizes a minimal mechanism in which arithmetic structure enforces module-level compatibility, and this compatibility directly drives physical energy transfer.