The OP does bring up a valid point- when we rock the bike left to right, we are transfering ARM work to the pedals. Someday somebody will create a handlebar powermeter to measure work done by the arms. When that happens we will see that indeed some portion of the arm power is wasted on overcoming the gyroscopic forces of the wheels.
However for all intents and purposes everyone is limited by their legs tiring out. Nobody is particularly limited by our arms giving out. So how much power is wasted by your arms is mostly academic (to be fair, arm power also has to be replenished with food, so it is not completely academic)

Yup. That's what we found. When predicting power for speed (or speed for power) you can either include or leave the MOI out and compare the two results. With the current resolution of power and speed measurements, you'll be hard-pressed to spot a consistent difference--so we left it out in favor of parsimony.

https://pubmed.ncbi.nlm.nih.gov/28121252/

To be fair, we often can get away with assuming that Crr and CdA are fixed with speed because over *most* of the speeds and surfaces that bicycles travel, that's *almost* so. The Reynolds number changes with speed which affects CdA, and at high speeds you start to run into limitations with the "spring rate" of pneumatic tires. But for the speeds that most of us go, we can usually get away with treating them as if they were constant and our predictions will still work pretty well. That's actually a clue: when the predictions start to not match with observations, then we usually sigh and add the extra damn variables back in.

Or they could find that the gyroscopic resistance actually helps with the leverage exerted on the pedals. The only force that matters is what goes through the pedals, however you push and pull on the bars. The bike only rocks because of the forces you are applying and any resistance to that rocking motion does not necessarily reduce your power output through the pedals.

The rocking motion increases the force applied to the pedals. The point I'm making is that of the force applied to the pedals only some of it (most of it, granted) goes to moving the bike forward. Some of it goes to overcoming aerodynamic drag, rolling resistance, friction/ heat loss, and overcoming gyroscopic inertia.

Unlike aero drag and rolling resistance, the gyroscopic force is not even opposing forward motion of the bike. It just makes the bike slightly more stable and even that is questionable as it is not even required for balance. So I'm not sure where you are going with this line of thought?

It's not directly opposing forward motion, but overcoming gyroscopic force requires energy that could otherwise go toward increasing forward motion.

I'm dubious that there's anything to your hypothesis that gyroscopic force materially affects sprinting efficiency, but if it does, it would be that it provides a useful opposition to pedaling effort. Think of the classic paradox of the irresistible force meeting the immovable object.

More simply, think of the difference between running on asphalt and running on loose sand. (The gyroscopic effect, such as it is, would be analogous to the asphalt, providing a force to push against.)

The rocking motion only occurs to maintain dynamic balance when exerting high pedal forces. If anything the trivial gyro force helps with the balance.

Take a look at the dedicated sprint trainer in this video. It has massive lateral support arms and dampers to simulate the rocking resistance. Does the force exerted on those lateral arms reduce the power produced at the crank or does it aid in generating it?

https://m.youtube.com/watch?v=y0Bw5OnHQrM

The sand is harder to run on because, like the wheel, it moves.

And that equates to your speculation regarding the effect of gyroscopic force; its absence would be analogous to the sand, because it would similarly permit movement that would dissipate force (running force in the one case, pedaling force in the other).

If no motion happens, no work happens. If motion, then work. When running in sand, work is done to move the sand that does not contribute to moving the runner forward: that's an inefficiency. When rocking a spinning wheel, work is done to overcome the gyroscopic force and move the wheel side-to-side. That work does not contribute to moving the bike and rider forward, and it is an inefficiency.

But you are only rocking the bike to maintain a dynamic balance as you exert force on the pedals. If the gyroscopic force was higher then the bike would simply rock less. It might actually be more efficient in this regard. But to put it into context we are talking about transient forces in the order of 10-20N. They are inconsequential in balancing the bike.

Put a bike on a repair stand by the top tube with wheels off the ground. Rotate the front wheel forward as fast as you can. Try to turn the handlebars. This will tell you how much force is involved. Also, try tilting the bike laterally by pushing the crank away from you and pulling toward you, also noting whether the front wheel steers as a result; This part I especially wonder about, if the direction the steering goes, matches road bikers as they climb a hill standing on the pedals; Some turn the wheel with each pedal stroke, I tend to keep the steering straight. Now put the bike in high gear and crank forward to get the rear wheel going wicked fast, then again, try to tilt the bike laterally by the (stationary) crank; noted the forces involved, but again, this is very fast wheel rotation, versus slow climbing speed.

Let us know the answers, whomever does this, thanks in advance. I'd do it but I have no repair stand.

"One valid test is worth a thousand expert opinions." - Engineering Proverb