Operate the most efficient heat engine that can possibly exist. Drag the T–S rectangle — its top edge is the hot reservoir, its bottom edge the cold, its width the gas you let expand. Every drag re-solves the four corners of the P–V loop on the left, whose enclosed area is the net work. The efficiency rises toward a wall it can never cross — η = 1 − Tc/Th — and when you try to beat it, you feel the wall.
Every cycle that runs between the same two reservoirs loses to Carnot — that is the Second Law, and on this bench you can feel the wall. Reshape the loop and the gold work shrinks below the dashed ceiling, a red lost-work wedge opens between your loop and the Carnot rectangle, and if you let heat leak across a finite temperature gap the universe's entropy ticks irreversibly positive. You cannot win, and you cannot break even.
Two independent derivations agree to machine precision and are re-run live in the
self-test badge above: the work as the loop area (∮P dV, from-scratch Simpson quadrature) equals
the heat bookkeeping (Q_h − Q_c via ∫T dS), and no reshaped cycle ever beats
1 − T_c/T_h. · The microscopic floor this engine is made of lives one wing over, in the
Cavern's Maxwell–Boltzmann gas.
And the one line this bench takes on faith — pressure(n,T,V)=n·R·T/V — is
derived next door from wall collisions →