Exact Quantum Thermodynamics of Hard-Gap Dephasing: Residual Ergotropy and Entropy Production without Lindblad

This manuscript (Pillar 13 of the unified gapped-bath framework) presents an exact derivation of the quantum thermodynamic properties of a central spin undergoing pure dephasing in a hard-gap bosonic bath. By strictly avoiding the Lindblad master equation and any phenomenological approximations, we prove that the spectral gap acts as a thermodynamic resource. The gap protects a finite residual ergotropy at long times, E∞=(ℏω0/2)exp⁡(−J0/Δ)E∞=(ℏω0/2)exp(−J0/Δ), and caps entropy production strictly below the classical maximum. At the exact critical coupling J0=4Δ/π2J0=4Δ/π2, the residual ergotropy ratio is the pure mathematical number e−4/π2≈0.666787e−4/π2≈0.666787. We also demonstrate that pure dephasing induces zero heat exchange (Q=0Q=0), with the entropy saturating at exactly 0.9000460.900046 bits for J0=ΔJ0=Δ. The paper includes rigorous proofs, five falsifiable predictions, a comparison with standard frameworks (Lindblad, Caldeira-Leggett), and a fully reproducible Python validation script.