We study a five-channel coupling model governed by Ramanujan–Eisenstein dynamics and show how a hidden defect at a special point drives a new type of channel interaction. The key mechanism appears at the complex multiplication (CM) point τ=i = iτ=i, where R (i) =0R (i) = 0R (i) =0 while ∂τR (i) ≠0_ R (i) 0∂τR (i) =0. This removes the zeroth-order contribution of one channel and leaves behind a pure first-order oscillation. As a result, this channel becomes a unique source of coupling, inducing a precise shift between neighboring modes m→m+1m m+1m→m+1 with amplitude β23∼πε∣Q (i) ∣2λ3−λ2. ₂₃ |Q (i) |²₃ - ₂. β23∼λ3−λ2πε∣Q (i) ∣2. This defect-driven process converts geometric (Berry) energy into spectral redistribution. The effect is strongly enhanced when nearby modes are nearly degenerate, leading to ΔEshift∼πε∣Q (i) ∣2Egeom (λ3−λ2) 2. Eₒ₇₈₅ₓ |Q (i) |² E₆₄₎₌ (₃ - ₂) ². ΔEshift∼ (λ3−λ2) 2πε∣Q (i) ∣2Egeom. We further show that the Ramanujan system defines a local flow in channel space, which governs how these interactions build up. While this flow does not directly represent global holonomy, its accumulation along modular loops gives rise to the observed sideband coupling, with the associated Berry phase encoded in the eigenstructure of the system. Together, these results reveal a minimal mechanism by which a single hidden defect can drive channel mixing and energy transfer in a modular spectral system.
Jeong Min Yeon (Mon,) studied this question.