= Solution
For a truly isolated <Hamiltonian> bunch, <Liouville's theorem> preserves the fine-grained volume in its full $6N$-dimensional <phase space>. <Hamiltonian> manipulation cannot increase its full phase-space density by compressing all canonical position and <momentum> directions simultaneously. Merely focusing the beam to a smaller transverse spot generally increases angular or <momentum> <divergence>; it does not by itself reduce <beam emittance>.
The distinction between full volume and projected spreads matters. Coupling transverse and longitudinal degrees of freedom can exchange emittance or move spread into other coordinates and correlations, so a projected transverse quantity may decrease without reducing the complete phase volume. That is redistribution, not cooling of every degree of freedom. In linear uncoupled optics, preservation of each transverse canonical area makes the spot-size/<divergence> tradeoff particularly direct.
Actual cooling allows the beam to exchange <energy> and entropy with other systems. In <electron cooling>, a colder comoving electron population receives the protons' random <kinetic energy> and is subsequently renewed or cooled. In <stochastic cooling>, a pickup measures deviations and a downstream kicker applies feedback corrections; the electronics, measurement noise and environment belong to the enlarged system. CERN's description identifies exactly this pickup-and-kicker arrangement.<Stochastic cooling by CERN>[Feedback and corrections: pickup measurement of sample motion and downstream electric-field correction]{ref}
Thus these methods relax the idealization that the proton bunch alone is isolated. Including the cooling apparatus does not require the complete closed dynamics to violate <Liouville's theorem>. For protons, direct radiative damping is much weaker than for electrons, so stochastic or electron cooling is a more relevant practical suggestion than relying on strong synchrotron damping.
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