Matrix Representation of Prime Gaps via a Telescopic Difference Operator

This paper completes the spectral bridge by introducing a novel matrix operator framework for the distribution of individual prime gaps. Starting from the spectral mean gap vector ⃗¯ G derived from the bounded divisor operator T, we construct a discrete telescopic difference operator D that rigorously maps the cumulative mean gap sequence to the pointwise vector of individual prime gaps ⃗ G . We prove that D is a bounded Toeplitz matrix acting stably on the sequence space of logarithmic growth, matching the natural asymptotic behavior of the primes. Furthermore, we analyze its spectral properties, demonstrating that since the unilateral shift operator generates a closed unit disk, the spectrum of D is explicitly confined within the shifted complex domain |z − 2| ≤ 1. This framework establishes a well-posed, continuous linear transformation in the space of gap sequences, bridging the algebraic structure of the divisor network with the deterministic pointwise distribution of prime numbers without relying on unproved arithmetic hypotheses.