We present an exact algebraic mechanism by which non-abelian holonomy in the modular group PSL(2,ℤ) generates a rigid 5:1 channel decomposition of the spectrum. Assigning the generators to a ℤ⁴ lattice, we compute six square holonomies and show that five belong to a Fibonacci class while a unique one forms a Pell class, yielding an invariant Frobenius ratio ηₚ = 24/43. The induced 1+5 block structure leads, via Schur reduction, to a resolvent-type secular equation whose pole structure enforces avoided crossings and strict interlacing of eigenvalues. In the perturbative regime, we prove a universal shift relation in which the displacement of the distinguished Pell mode is fixed by the dimensionality of the Fibonacci sector, independent of the coupling strength. Crucially, we show that the statistical properties of the spectrum are not determined by the modular holonomy data alone. Instead, the level statistics depend sensitively on the spectral position of the Pell channel relative to the Fibonacci band. As this position is varied, the system exhibits a continuous crossover from Poisson-like clustering to strong level repulsion, governed by the minimum channel separation. The previously reported Brody exponent β ≃ 0.53 is recovered only for a specific choice of channel position and is not a universal signature of PSL(2,ℤ). The modular structure fixes the channel decomposition and algebraic constraints, but the statistical regime of the spectrum emerges from geometric placement rather than arithmetic invariants.
Jeong Min Yeon (Mon,) studied this question.