On a Class of Infinite Hermitised Toeplitz-Catalan Operators and Their Bounded Spectral Properties

This paper establishes a novel algebraic and spectral framework bridging combinatorial discrete structures and the spectral geometry of the Riemann zeta function. We introduce a brand-new class of infinite dense convolution Toeplitz operators, embedded with Catalan numbers and regulated by an absolute geometric weight 4^{-|i-j|}. By employing a rigorous diagonal-by-diagonal expansion and the Bounded Extension Theorem, we prove the Hilbert-Schmidt compactness, finiteness, and self-adjointness of the proposed operator T over the Hilbert space \ell^2(\mathbb{N}). Crucially, the paper develops a rigorous pathway for analyzing the critical line \Re(s) = 1/2 via a local complex extension and a spectral trace minimization argument. By exploiting the exact closed-form solution of the universal Hadamard trace formula, we show that any hypothetical complex deviation from the critical line imposes a pointwise spectral rigidity constraint that forces a strict energy deficit, thereby contradicting the structural completeness and statistical count required by the Riemann-von Mangoldt density function. Consequently, we demonstrate that all non-trivial zeros are spectral invariants of the closed system and must be uniquely accommodated on the critical line. This framework offers an innovative "Universal Spectral Model" and an original methodology for spectral exclusion via uniqueness in operator theory.