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This is going to be a weird question. I am an academic researcher. In my field there has been a key topic whose "accepted" solution has been known not to work since 1962. Nobody had been able to work out why. I did. There was a subtle but fundamental math error in the original scientific work. In fact, the accepted solution can be shown to be completely uncorrelated with the actual solution. I replaced the solution with a mathematically valid solution and empirically tested against the entire historical data set the field has. It works.

My problem is that there are 651 software and process patents built upon the invalid methodology. I am concerned the patents will interfere with correct scientific development in the field. Are patent claims valid if they are built on something which is mathematically or scientifically impossible? The math error was that it violated the rules of general summation. The accepted process is wrong in the same sense that 2+2=77 is wrong. These are heavily used processes. They don't work, but nobody had been able to figure out a better solution.

My second question is "is there an easy way to get these patents declared invalid without bearing tremendous expense?"

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For your second question: No.

Now to the first one, as this is harder. If the claims state something similar to after adding two to two we get 4 but 4 is sufficiently close to 77 so whatever or we add 2 and 2 and procede, then this should not in any way forbid you to use your invention. BUT: And this is actually what you asked:

Are patent claims valid if they are built on something which is mathematically or scientifically impossible?

Yes, if the patent is granted and the claim is sufficiently broad, it can be used against you. And to add another one, it might even be that you are not able to get that patent invalidated, depending very much on the exact problem and the wording of the patent.

Edit: And as a comment said, impossible patents would/should not be granted, so if there are many granted patents on this topic, either it is not totally impossible or they are directed to things only in the vecinity of your method.

What can you do?

You can patent it yourself if you think it is worth some money. From there on, if your way is really as much better as you state, you do have a good bargaining position. Anyone who uses the process would have to use yours afterwards (?), making your patent potentially stronger. And if it really comes down to it, you can probably invalidate a patent used against you.

But helping you with the strategic part is really just guessing here, so let me know if you have any more specific questions.

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    I think the answer addresses your actual concerns. But in terms of jargon, there could be some scope for confusion. A patent cannot be granted on an impossible device (no perpetual motion machines). But theoretically flawed analysis may be fine. If the claims were directed to a system that took four dollars (and nothing else) and made it 77 dollars, that patent should be invalid (although invalidating it might be expensive). But if, as the original response says, the patent is theoretically unsound but the claims nonetheless describe a vaguely useful system, then they may well hold up.
    – Ben Kleinman
    Commented Nov 30, 2016 at 7:45

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