Physicists Prove Surprising Rule of Threes

I haven't argued that a physical constant which terminates in base 10 implies that 10 is more special than other bases. I've argued to support the idea that a physical constant terminating in base 10 is surprising. It is, superlatively so. In the words of the OP, the question you need to answer, if you think you have a constant exactly equal to 227/10, is "why is the denominator 10?". It could be that 10 is a holy number, or it could be that the 10 represents some other physical quantity that you weren't paying attention to before. But assuming it's a coincidence is a strange thing to do, and in fact it is not natural to think that if you just looked at enough numbers of course you'd eventually see a rational one. Unless you're choosing numbers by a process that biases you heavily towards getting rationals, you are not likely to see a rational until long after you've chosen an infinite number of irrationals.

The whole genesis of string theory was, first, observing that a sequence generated by a badly-understood phenomenon was the same sequence given by a mathematical result about strings; and then, second, concluding "aha! There must be strings involved in this!" Plenty of people argue that string theory isn't going anywhere, but nobody says the conclusion "there must be some strings around here somewhere" was unjustified. The "it's a coincidence" theory persuades no one.