Spectral Phase Transitions and Prime Condensation in a Serre–Hecke Mutation System

We analyse the spectral output of an effective Serre–Hecke–Pell/Fibonacci (SHP) Hamiltonian constructed on a finite mutation lattice and focus on two numerical signatures: universality-class transitions and index localisation. First, the level-spacing statistics exhibit a clear transition from Poisson to GOE and further to GUE as the mutation coupling and Hecke phase are activated. This demonstrates that the system evolves from an integrable regime to a fully chaotic regime with broken time-reversal symmetry. The transition is not imposed but emerges dynamically from the interplay between discrete mutation (Pell/Fibonacci) and continuous Serre flow. Second, we observe a statistically significant localisation of low-energy eigenstates on prime indices. The prime-weight enrichment exceeds twice the random baseline in the RG-stable sector and reaches a maximum enhancement of approximately 3.3. This effect persists under finite-size scaling and is strongest in the lowest-energy modes, indicating that it is not a numerical artifact but a structural feature of the dynamics. Taken together, the results suggest a unified interpretation: mutation-induced defects generate spectral chaos, while RG-stable modes selectively localise on irreducible indices. In this framework, prime indices emerge not as external arithmetic input but as preferred sites of dynamically stabilised eigenmodes