= Solution
The relevant claim concerns a machine that repeats its complete operating cycle while delivering a continuing net output of <work> without a corresponding supply of <energy>. \b[A machine cannot supply net work in a cycle without taking in energy or depleting an energy store.] This follows from <conservation of energy>, an experimentally supported physical law rather than a theorem of pure logic.
Let $E$ include the machine's <internal energy>, <kinetic energy> and <potential energy>. With $Q$ the net <heat> received and $W$ the net work delivered, the <first law of thermodynamics> gives
$$
\Delta E=Q-W.
$$
For a complete cycle all stores return to their initial values, so $\Delta E=0$ and $W=Q$. If there is neither heat input nor another energy input, $W=0$. A descending <mass>, unwinding <spring>, consumed fuel or discharged <battery> supplies energy, but it cannot be omitted from the accounting while calling the result <perpetual motion>.
Changing an overbalanced wheel's geometry does not evade the argument. A <conservative force> gives zero net work around a closed configuration cycle, while <friction> dissipates usable mechanical energy. Resetting a mechanism or returning a raised weight costs work that must be included.
Indefinitely continuing motion without useful output is a different possibility: an ideal frictionless oscillator or freely moving body need not stop. Connecting a load removes energy and changes that situation. A cyclic machine supplied with heat from only one reservoir and converting all that heat into work is excluded instead by the <Kelvin-Planck statement> of the <Second law of thermodynamics>. The first-law and second-law objections address different proposed energy balances.
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