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That definitely sounds wrong - if you're leaving unburnt fuel in your exhaust, that is wasted fuel, you could burn it further and get more energy out of it - they seem exactly in line with eachother as goals. Do you have any source for that statement?

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It's not that there is unburnt fuel in your exhaust - unburnt fuel, while a pollutant, is not the problem.

The issue is that higher temperature combustion, while more efficient, produces much more NOx. So you get more energy per unit fuel at higher temperatures, but you also get more pollutants. Lower temperatures (less oxygen) result in "cleaner" combustion (essentially, more CO2 and N2, less NO and NO2), but produce much less energy per unit fuel.

This is only true in some regions. The creation of Ozone in the troposphere is dependent on availability of NOx, and availablity of hydrocarbons. In the southeast US, forests produce large amounts of volatile hydrocarbons, so the system is NOx-limited. In California's Central Valley, excess NOx from nitrogen fertilizers makes ozone production hydrocarbon-limited (and also basic rain from the NH3). So from a pollution management standpoint, either could be important, but usually only one.

I had forgotten about the NOx at higher temperatures. But running lean is also more efficient and can result in incomplete combustion:

https://en.wikipedia.org/wiki/Lean-burn

There really is a lot at play, but I stand by the statement that "clean" and "efficient" are directly at odds with each other.

"directly at odds" is going too far.

For example, electronic fuel injection is cleaner and also much more efficient than using a carburetor.

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