Energy per Switching Event

Charging a capacitance $C$ to voltage $V$ through any resistance whatsoever, from a fixed supply, costs the supply $CV^2$ and leaves $\tfrac{1}{2}CV^2$ stored on the capacitor. The other half is dissipated as heat in the resistance, and the striking part is that the resistance value does not appear in the answer: charge it through a milliohm or a megohm, quickly or slowly, and exactly half the energy is lost. Only the path taken changes, never the total.

That is why the half is here and not in the whole-chip power formula. A complete cycle — charge up, discharge down — costs $CV^2$: half lost charging, half lost discharging the stored energy to ground. The dynamic power expression $\alpha C V^2 f$ counts complete cycles, so its factor of one is correct, and this expression counts one direction, so its half is correct too. Confusing the two is a factor-of-two error that is very easy to make and very hard to spot afterwards.

The escape from the "half is always lost" rule exists, and it is called adiabatic or charge-recovery logic. Charge the node slowly from a ramping supply rather than a step, and the dissipation falls in proportion to how slowly it is done; recycle the stored charge back into the supply on the way down instead of dumping it to ground. The technique is real and it works. It is also slow, needs a resonant clock, and has stayed a research curiosity because the speed sacrifice has never been worth the energy saved in mainstream logic.

For scale, put a real switching event beside the thermodynamic floor. Rolf Landauer showed in 1961 that erasing one bit of information must dissipate at least $kT\ln 2$, about $2.9\times10^{-21}$ J at room temperature — a limit that comes from thermodynamics rather than from any particular technology. A logic gate switching a few femtofarads at a volt spends around $10^{-15}$ J, some six orders of magnitude above that floor. There is, in principle, enormous room left. In practice the gap is filled by noise margin, speed, and the plain difficulty of building anything that operates a few $kT$ above thermal noise and still gives the right answer every time.