Gap-Induced Non-Markovianity and Oscillatory Quantum Capacity in Dephasing Channels

This preprint (Pillar 11 of the Gapped Quantum Systems Framework) presents an exact information-theoretic analysis of pure-dephasing channels coupled to a bath with a hard spectral gap. Core Findings:Using the exact dephasing function derived in our previous work, we rigorously prove that the spectral gap induces strictly positive non-Markovianity (information backflow) and causes the exact quantum capacity to exhibit damped oscillations around its asymptotic plateau. We also derive the exact quantum Fisher information and a novel identity linking the Rényi-2 entropy to the concurrence. Methodology & Rigor:The derivation relies entirely on exact analytical methods and established quantum information theorems (e.g., BDS quantum capacity theorem), without any fitting, phenomenological parameters, or circular reasoning. Reproducibility:The upload includes the PDF, LaTeX source, high-resolution figures, and self-contained Python scripts (test_pillar11.py, plot_pillar11.py) that validate every analytical claim to machine precision and generate the figures.