Reducing Chaos up to the Prime Barrier: An Extended N+M Filter and a Finite-Dimensional Liouville Bound

This is a finite‑dimensional model illustrating a square‑root bound. It does not imply the Riemann Hypothesis; the connection is only analogical. We extend the recently introduced N+1 filter – a finite-dimensional model of the truncated Frobenius operators of the Bost–Connes system – by adding states labelled by products of two primes. On this extended space H_{N+M} we define a discrete analogue of the Liouville function λ(n) = (-1)^{Ω(n)} and study the total weighted sum S(N) = ∑ λ(n)/√n. We prove a Barrier Lemma: for any N, |S(N)| ≤ C √dim H_{N+M}, where the constant C does not depend on N. The bound mimics the critical exponent 1/2 appearing in the Riemann Hypothesis. Moreover, we show that the barrier is universal: it persists with the same constant C if the Liouville function is twisted by any Dirichlet character. The model thus illustrates that all "reducible chaos" (composite numbers) can be controlled up to a natural threshold dictated solely by the primes, leaving the primes themselves as the ultimate arithmetic barrier.