No engine beats 1 − Tc/Th

Two reservoirs, each the open chain of d sites, a hot one at Th and a cold one at Tc, start in their thermal states and evolve by a joint unitary U. Proved in Lean: the work W = Q_h − Q_c never exceeds (1 − T_c/T_h) Q_h, and a single reservoir gives no work at all (Kelvin–Planck). Every point you add is one unitary; none can enter the orange region.

sites d
1.50
0.50

Work W against heat Qh from the hot chain (tap a point)

unitaries W = (1 − T_c/T_h) Q_h, Lean forbidden

Efficiency η = W / Qh of the engines found

–
Carnot 1 − T_c/T_h
–
best η found
–
selected: W, Q_h
–
one bath: lowest ΔE
–

Units with k_B = 1; each reservoir is the open chain of d sites, levels −2 cos(kπ/(d+1)). The unitaries permute the joint energy levels (one to three swaps): every state stays diagonal and the numbers are exact. An engine takes heat from the hot chain (Q_h > 0) and gives out work (W > 0). With T_h ≤ T_c the bound is W ≤ 0: no engine exists. "One bath" applies every unitary to a single chain at T_h; its energy never goes down. Lean: MState.carnot_temperature, carnot_entropy, inner_gibbsState_le_uConj — source.