For single leg training purposes how does a counterweight system differ from a fixed wheel ?

A fixed wheel offers the same resistance across the pedal stroke, and becomes a balanced flywheel. If you stop pedalling and pull your feet of the pedals, the pedals would spin a long time due to the flywheel.

A counterweight on a pedal would offer more resistance when the weight is at the bottom (you have to lift it), and less when the counterweight is past top dead center, because gravity would pull it down. If you stop pedalling, the crank would stop with the weight at the bottom.

A weight attached to the BB would offer the same as a fixed wheel: a balanced flywheel.

Lifting the weight requires you to push down more. Leg extensions and squats, basically. More stomp, less spin.

Lifting the weight requires more energy for half the pedal stroke, but you get it back when the weight drops and pulls the chain and turns the wheel for you. You don't have to use your "lift" muscles.

I am only interested in the effect on the upstroke of the working single leg where pulling up is sometimes done during two legged pedalling. Your last sentence above demonstrates that during two legged pedalling, when you know how to correctly unweight the rising pedal without any conscious pulling up, you can remove most of that wasted energy required to push the idling leg up and gain the advantage of the weight of that leg on its down stroke.

I agree with the whole statement except "remove".

You need to expend energy to lift the "dead" leg. If the leg isn't lifted 30-40 cm, it cannot push down on the pedal. Either using that leg's hip flexors (and other muscles) or the other leg's quads (and other muscles.) The leg needs to go up against gravity. You will spend the energy. You cannot remove the work requirement unless you remove gravity. The work needed is the same, no matter which muscles you use. The only potential difference is metabolic cost.

You could, in theory, use a string and your arm muscles to lift your dead leg, but you must lift your dead leg, and that work requirement is determined by leg weight and gravity and physics.

To answer your original question: you store the work needed in the crank counterweight using your quads when pushing your only leg down, and lifting the crank weight. You don't remove the energy required to lift your leg. You spend it half a pedal stroke in advance.

Unless you only have one leg, why would you cycle with just one?

Unless you only have one leg, why would you cycle with just one?

For single leg training purposes how does a counterweight system differ from a fixed wheel ?

You need to expend energy to lift the "dead" leg. If the leg isn't lifted 30-40 cm, it cannot push down on the pedal. Either using that leg's hip flexors (and other muscles) or the other leg's quads (and other muscles.) The leg needs to go up against gravity. You will spend the energy. You cannot remove the work requirement unless you remove gravity. The work needed is the same, no matter which muscles you use. The only potential difference is metabolic cost.

I don't believe the work needed is the same. When you unweight the rising pedal you get 100% return from the muscles used, when you use pushing down to raise the idling leg it will take more effort because of the tangential effect reduction in the force you are applying to raise that leg.

you sure it isn't more like operating on 2 cylinder in a four cylinder engine? was not aware that there were valves that shut off and reduced blood flow to one half of the body. even though the other leg isn't doing any work aside from hanging there and keeping out of the way of pedals/cranks, it is still receiving its blood supply. or are you saying blood pressure changes from one half of the body to the other through some means if only one leg is cycling? I cant see what mechanism is shipping more o2 to the used leg. not saying you are wrong, I just don't understand

Isn't this irrelevant anyway? You have one less leg extracting O2 from the blood as it makes its journey through the body. That leaves more for the working leg.

Is reduced oxygenation of blood ever a limiting factor?

That you're asking this question is illuminating.

But not surprising.