We derive a microscopic mechanism for spectral energy redistribution in a modular Pell–Fibonacci system by choosing the off-diagonal interaction sector from the same quasimodular origin as the diagonal spectral data. Concretely, the couplings are built from the Ramanujan–Eisenstein ring generated by P, Q, RP, Q, RP, Q, R, which is closed under the Ramanujan differential system. As a result, taking ∂τ_∂τ does not leave the model space: derivative channels, Berry-curvature numerators, and sideband amplitudes all remain inside the same arithmetic function ring. This removes the arbitrariness of the mixing sector and makes the model internally closed under modular deformation. Expanding the coupling matrix around the CM point τ=i=iτ=i, we find that R (i) =0R (i) =0R (i) =0 while R′ (i) =−πi Q (i) 2≠0R' (i) =- i\, Q (i) ² 0R′ (i) =−πiQ (i) 2=0. This singles out channel 3 as a pure first-order Fourier source: unlike the other channels, it has no zeroth-order background term and therefore generates an isolated m→m+1m m+1m→m+1 sideband. The sideband couples the near-degenerate mode pair (2, 3) (2, 3) (2, 3), giving a leading coupling scale β23∼πε∣Q (i) ∣2λ3−λ2, ₂₃ |Q (i) |²₃-₂, β23∼λ3−λ2πε∣Q (i) ∣2, up to eigenvector-overlap factors. Thus the CM arithmetic constraint does not merely select a special point; it actively creates a symmetry-breaking coupling channel. The same closure structure propagates to the energetic level. Because the Berry response is concentrated in the dipolar pair (2, 3) (2, 3) (2, 3), the induced channel-shift energy obeys ΔEshift∼πε∣Q (i) ∣2 EGeom (λ3−λ2) 2, Eₒ₇₈₅ₓ |Q (i) |²\, E₆₄₎₌ (₃-₂) ², ΔEshift∼ (λ3−λ2) 2πε∣Q (i) ∣2EGeom, showing that geometric energy is amplified by the inverse square of the near-degeneracy gap. The mechanism is therefore two-layered: CM arithmetic fixes the sideband source through R (i) =0R (i) =0R (i) =0, while spectral near-degeneracy magnifies its physical effect through the small denominator. Despite this strong amplification, the defect-local conversion efficiency remains low, ηabs≈0. 03₀₁ₒ 0. 03ηabs≈0. 03, because the Berry response is dipolar rather than monopolar. In other words, the CM defect is highly effective at generating localized geometric response, but only a small fraction of the active statistical energy is converted into net channel-resolved transfer at small loop radius. This distinction between large local amplification and low net efficiency is a central physical feature of the model. Taken together, these results establish a closed analytical chain CM arithmetic → quasimodular sideband source → Berry dipole geometry → channel-resolved energy transfer, CM arithmetic \;\; quasimodular sideband source \;\; Berry dipole geometry \;\; channel-resolved energy transfer, CM arithmetic→quasimodular sideband source→Berry dipole geometry→channel-resolved energy transfer, all within a single Ramanujan-closed function ring. In this sense, the model is not an ad hoc mixture of arithmetic and physics, but a self-consistent modular effective theory in which the same PQRPQRPQR structure governs coupling variation, geometric curvature, and energy redistribution. .
Jeong Min Yeon (Mon,) studied this question.