A Finite-Dimensional Model for the Truncated Frobenius Operators of the Bost–Connes System

The final paragraph identifies an open problem; it does not prove RH. We construct a finite-dimensional Hilbert space H_N = C^{N+1} equipped with nilpotent shift operators associated to the first N primes. Starting from a distinguished vacuum state, the only accessible states are the vacuum and the N prime states, giving N+1 states in total. The construction is modelled on the truncated Frobenius operators of the Bost–Connes system, where the cut-off (with Lambda < p^2) forces the nilpotence condition. We compute the discrete Weil quadratic form induced by the model and show that its ground state is simple and positive. Finally, we identify the analytic obstacle that prevents a direct passage from this finite model to the full Connes spectral realization of the Riemann zeta function, namely the control of an oscillatory sum over prime powers that is equivalent to the Riemann Hypothesis.