Ramanujan–Eisenstein Couplings in the Pell–Fibonacci Secular Problem

We study a 1+5 Pell–Fibonacci spectral system where all off-diagonal couplings are fixed by Ramanujan–Eisenstein functions. This removes free mixing parameters and replaces them with arithmetic structure. First, the spectral shift law is exact. The ratio between the shift of the Pell level and the average shift of the five Fibonacci levels is always equal to 5. This holds for all coupling strengths and follows directly from trace conservation, not from approximation. Second, the five channels split into two groups. The three Eisenstein channels are dominant at small coupling, while two derivative channels grow with the modular parameter. This produces a clear 3+2 hierarchy controlled by a single variable. Third, at special modular points called CM points, entire channels switch off. At tau equals i, one channel vanishes and a near-degeneracy appears in the spectrum. At tau equals rho, another channel vanishes but no nearby degeneracy forms. These points act as strict arithmetic selection rules and create spectral defects. Fourth, the eigenvector bundle develops a localized geometric phase structure near these defects. Around tau approximately equal to i, two modes carry opposite Berry curvature, forming a dipole. When a loop encloses the defect, the total phase jumps by a quantized amount, indicating a topological transition. Fifth, the energy splits into a visible part and a geometric part. In a simple two-channel model, the geometric energy is always half of the visible energy. In the full six-channel system, this relation breaks down near the CM defect, where the geometric contribution is amplified by about 4.6 times. This reflects strong multi-channel interference.