If that failed to be stable, all that would prove is that particular design was unstable, not that there is no design with those features that is stable. This is a problem of finding a general rule, not just a particular design that is stable or unstable.
The issue is that no one has found a way to characterize all possible designs (within certain constraints, such as two wheels each attached to a rigid frame, the frames joined by a hinge) which are self-stable. They know some conditions that are necessary, such as at least one factor linking lean to steering and the design needing a steering force applied to turn in a steady turn. They have not yet characterized what conditions are sufficient for a stable design.
What you want, to say that you fully undertsand how a bicycle works, is a set of conditions which are both necessary and sufficient for a bicycle to be self-stable. If you build a bicycle which meets those conditions, then it will be self-stable (at some speed; certain designs may be self-stable over a wider range of speeds while some may only be self-stable at a narrow range of speeds); if you build a bicycle which does not meet those conditions, it will not be self-stable at any speed.
We've gotten closer over the last century; initially it was believed that gyroscopic force was necessary, but that was disproved. Later it was believed that trail was necessary (or either gyroscopic force or trail was necessary; I haven't read the older paper), but that has now been disproved. We now know a couple of necessary conditions (listed above), but they are somewhat weak necessary conditions, and we don't yet have (as far as I know) a set of sufficient conditions (conditions which, if they hold true, will guarantee that the bicycle will be stable, regardless of other changes to the design), beyond a few known designs which are demonstrably stable.
In fact, if you follow from the paper in Science to the "Supplementary Online Materials" (which is actually the full-length paper; what's published in Science is really an extended abstract), you will see that they prove that "no combination of positive gyroscopic action, positive trail, or positive steer axis tilt are either necessary or sufficient for self-stability over at least a small range of speeds." They construct models of bicycles that lack each of these things but are stable, and have all of these things but are unstable.
The issue is that no one has found a way to characterize all possible designs (within certain constraints, such as two wheels each attached to a rigid frame, the frames joined by a hinge) which are self-stable. They know some conditions that are necessary, such as at least one factor linking lean to steering and the design needing a steering force applied to turn in a steady turn. They have not yet characterized what conditions are sufficient for a stable design.
What you want, to say that you fully undertsand how a bicycle works, is a set of conditions which are both necessary and sufficient for a bicycle to be self-stable. If you build a bicycle which meets those conditions, then it will be self-stable (at some speed; certain designs may be self-stable over a wider range of speeds while some may only be self-stable at a narrow range of speeds); if you build a bicycle which does not meet those conditions, it will not be self-stable at any speed.
We've gotten closer over the last century; initially it was believed that gyroscopic force was necessary, but that was disproved. Later it was believed that trail was necessary (or either gyroscopic force or trail was necessary; I haven't read the older paper), but that has now been disproved. We now know a couple of necessary conditions (listed above), but they are somewhat weak necessary conditions, and we don't yet have (as far as I know) a set of sufficient conditions (conditions which, if they hold true, will guarantee that the bicycle will be stable, regardless of other changes to the design), beyond a few known designs which are demonstrably stable.
In fact, if you follow from the paper in Science to the "Supplementary Online Materials" (which is actually the full-length paper; what's published in Science is really an extended abstract), you will see that they prove that "no combination of positive gyroscopic action, positive trail, or positive steer axis tilt are either necessary or sufficient for self-stability over at least a small range of speeds." They construct models of bicycles that lack each of these things but are stable, and have all of these things but are unstable.
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