Actually I spoke too soon. The method works fine for s(4,5) and also for s(3,7). However, for s(7,8) it gives a bound of s=8/3. This is certainly a bound, but it's not achievable: however, it's not necessary to drop down to 7/3, since the intermediate value of 5/2 is achievable (again, with a quite easy but fiddly proof).
So I must make a more modest claim: the method gives a reliable bound b, which is usually achievable. When it isn't achievable, m/3 may be the best achievable, but there may be some other intermediate value that will work.
As you can see, I've simply relaxed one of the two claims I made without any good reason. I'm still making one claim without proof (that m/3 is always achievable, because there is so much floppiness in the dissection).
For s(4,9) my method gives a bound of s<=9/5, which is another clearly not achievable case, so my new weakened claim is that 4/3 <= s < 9/5. As I haven't found the value yet this is an interesting test case ... |