We present a novel operator-theoretic framework to settle the long-standing problem concerning the arithmetic nature of the Riemann zeta function at odd positive integers, $zeta(k)$ for $k > 1$. By factoring the structural core of the Hurwitz zeta function into a regularized infinite product of commuting, self-adjoint second-order differential operators $mathcal{Z} = prod_{m=1}^{infty} mathcal{Z}_m$, we map the evaluation of $zeta(k)$ to an infinite-dimensional matrix expectation value. We demonstrate that truncating this operator product at a finite depth $N$ generates a sequence of rational approximations satisfying an absolute three-term Ap'{e}ry-type recurrence relation. By systematically extracting a single integer node $m$ from the operator chain, we derive a rigid functional identity connecting the approximation errors of $zeta(k)$ and $zeta(k+2)$. An asymptotic analysis of the resulting diophantine linear forms reveals that a rational decoupling of these states is algebraically forbidden by the underlying spectral growth. This establishes that if $zeta(k)$ is irrational for an odd integer $k>1$, then $zeta(k+2)$ is also irrational, providing a complete inductive proof of irrationality for all odd zeta values.Finally, while this method of proof has been extended to Dirichlet-$beta$ function to establish the irrationality of Catalan's constant,we failed to adapt it for the purpose of uncovering the arithmetic nature of Euler-Mascheroni constant.