Recursive Quantum Numbers (RQN): Global Limit Theorem Toward a Controlled Euler-Product Approximation

This paper presents a rigorous global limit theorem connecting a novel cosmologically-motivated number system; the Recursive Quantum Numbers (RQN) framework, to the classical Riemann zeta function via an explicit, controlled Euler product bridge.
The RQN framework defines a time-indexed family of finite non-Archimedean rings R_p(t), each isomorphic to Z/p^(N(t)+1)Z, where the recursion depth N(t) is determined by a decaying cosmological energy function. As cosmic time approaches zero; corresponding to the early universe where energy was effectively unlimited,  the depth grows without bound and the system recovers the classical p-adic integers.

The central result shows that the RQN-Euler approximant Z(s,t), a finite product of energy-truncated Euler factors built from RQN fibers, converges uniformly on compact subsets of Re(s) > 1 to the Riemann zeta function ζ(s) as t → 0. The convergence is fully controlled: explicit error bounds are derived in terms of the prime cutoff P(t) and per-prime recursion depths N_p(t), both of which are linked to cosmological parameters in the RQN model. A numerical illustration confirms the approximation at s = 2 + i with quantified error.
This establishes a mathematically legitimate and physically motivated bridge between RQN truncations and classical analytic number theory. Future directions include analytic continuation into the critical strip and numerical investigation of zero movement as depth increases; a potential path toward connections with the classical critical line.