Spectral Analysis of the Divisor Network on the Prime-Exponent Space: A Bounded Generalized Toeplitz-Catalan Operator

This paper establishes a novel algebraic and spectral framework linking thediscrete divisor network to the spectral geometry of the prime-exponent space.We demonstrate that the total divisor network function Ntotal(n) vanishesidentically at prime numbers, assigning a unique algebraic weight to composite numbers based on their prime factorization. By mapping every naturalnumber to a vector of prime exponents in a countable Hilbert space, we definea generalized Toeplitz-Catalan operator whose elements are directly weightedby the algebraic weight W(v(n)). Using a rigorous diagonal-by-diagonal expansion and the Bounded Extension Theorem, we prove the Hilbert-Schmidtcompactness, boundedness, and self-adjointness of the proposed operator T.This operator reflects the algebraic fabric of the divisor network, providing arigorous bridge between arithmetic geometry and operator theory.