S-Matrix Stability and Quark Confinement Potential via the Non-Markovian K-Kernel Framework

In Quantum Chromodynamics (QCD), the confinement of quarks is a fundamental phenomenon where color-charged particles cannot exist in isolation. The interaction between quarks is typically modeled by a short-range Yukawa potential, supplemented by a phenomenological linear term to explain confinement. In this paper, we propose a novel, deterministic derivation of the confinement potential using the non-Markovian K-Kernel framework. By introducing a spacetime memory term into the effective potential, we demonstrate that a linear confinement regime naturally emerges at large distances due to the historical accumulation of the strong force field. Furthermore, we calculate the S-Matrix for elastic scattering and show that the unitarity magnitude |S| remains perfectly equal to 1.0. This proves that the K-Kernel modification preserves probability conservation, offering a mathematically rigorous alternative to the phenomenological string models.