Why should the frequency of heads in a coin toss converge to anything at all? Suppose we have any kind of coin. Why should the relative frequency of getting a heads converge to any value at all?
One answer is that this is simply what we empirically observe this to be the case, and I think this is a valid answer.
However, my question is, is there some simple set of principles/axioms that apply to coin tossing from which we can derive this fact?

The set of principles that applies to coin tossing from which we can derive that, are indeed axioms. That means they’re open to question only inasmuch as the whole idea.
I see a hugely important secondary question as whether this is about prediction, measurement or explanation.
Whichever matters most, how is this not a sophisticated version of the standard high-school query, has a single toss the same odds as one in a sequence and then, which sequence?
After User7344, Ben asks why there should be a single probability that applies to all coin tosses. Rather, how could there not be if “all coin tosses” are equal?
How is it not axiomatic, in- or outside chaos theory and the nature of randomness that from a choice of two outcomes with all things equal, the likelihood of either is 1/2?
At risk of being pedantic, the actual wording of the Question negates much of its value. That looks like a linguistic niggle but consider the detailed discussion that’s developed from it! I hope we all saw the same meaning but how could I prove that. In reality, there can be neither frequency nor convergence in “a coin toss”; only in a series of tosses. Take that same ambiguity back from linguistics to probability, and what certainty remains?
In my view “what we empirically observe” is a starting point, often to be explained but rarely needful of justification.
One thing I suggest most people won’t accept until they’ve tried it, is that whether convergence is what we “empirically observe” depends on how patient we are.
I built a simple roulette simulator and until someone shows how they don’t, I suggest red/black spins follow the same probabilities as heads/tails tosses.
Will you take the time to guess how long an uninterrupted string of either outcome was not uncommon?
I thought that might be four or five; perhaps six but in fact after millions of runs, nothing less than 14 turned out to be uncommon. I’m told that both the result, and my amazement at it, are frequently seen in studies of advanced maths.
If any sequence is equally likely and all are combined, how could the results not converge? How would that not describe the idea of an “average”?
If any sequence is equally likely and they are not combined, how could 13-in-a-row not skew the empirical view of whichever observer saw it?
