London Colin,
I think I understand your question. I'm going to take a stab at it, coming from a completely different direction than the previous replies. I've wondered the same thing before and your post has caused me to dig into this a little more. I want to say up front that I have not read anywhere close to all of the info in the references I'm listing. I'm using them because they had the specific pieces of the puzzle I happened to be looking for with that particular Google search, and they both come from
http://www.bjmath.com (Archive copy), which seems to be a good source of information.
First, the SCORE is built around a 13.5% risk of ruin. According to
page 7 of this link (Archive copy), a 13.5% risk of ruin corresponds to full Kelly betting based on the initial bankroll, with no bet resizing. So I'm starting with the assumption that we are betting full Kelly.
As sagefr0g stated, "SCORE is DI squared and DI is 1000[ (winrate/hr divided by 100)/(sd/hr divided by 10)]." Another way to say this is DI = 1000 * (EV per hand) / (standard deviation per hand). I'm going to drop the "per hand" from here on, but it is assumed. Continuing, we can say that SCORE = (1,000 * EV / SD)^2 = 1,000,000 * EV^2 / SD^2. And since SD^2 = Var, we get:
SCORE = 1,000,000 * EV^2 / Var
Now,
this link (Archive copy) states that "The correct 'Kelly' bet at any level is Bankroll*ev/var, where ev is the unit gain per hand, and var is the unit variance per hand."
Since our bankroll is $10,000, our correct full Kelly bet would be 10,000 * EV / Var. For correct Kelly betting, our bet size is directly proportional to our EV for each individual bet, so our average bet size should also be proportional to our average EV. Our average per hand winrate would be our average bet times our average EV, or (10,000 * EV / Var) * EV. Our hourly winrate at 100 hands per hour would be 100 * (10,000 * EV / Var) * EV = 1,000,000 * EV^2 / Var. Since this is the same as the SCORE, we can say that the SCORE equals our hourly winrate with a $10,000 bankroll and a 13.5% risk of ruin.
More generally, hourly winrate at a 13.5% risk of ruin is equal to (hands per hour) * bankroll * EV^2 / Var. I haven't read BJA, so I don't know why SCORE was defined the way it was. I assume the formula isn't nearly so simple for hourly winrate at other risk levels, which probably makes 13.5% the only reasonable risk level to use for generating a simple to use formula. Regardless, our primary goal is to maximize our winrate for a given RoR, so creating a good way of comparing winrates is really what matters. We don't really care to understand how it affects us if our standard deviation changes. We don't care to fully grasp the impact if our %EV changes. With a predetermined acceptable RoR, those things will cause us to change our bet sizing to match our risk tolerance. We care how those factors interact to affect our hourly rate, give our risk tolerance. That's why I feel Don Schlesinger based SCORE off of the winrate formula above. It automatically rolls the EV and SD for a given situation into one number that allows direct comparison of one game or one situation to another for the purpose of determining which opportunity will give us the best winrate for a given risk level.
All that said, the constants we use for hands per hour and for bankroll are really irrelavant, although for the sake of getting numbers that have some meaning to us, it helps if we use a large enough constant to bring the formula's output up to a range we can easily think about. If the constant were one, we would be talking about how much difference there is in a SCORE of 0.000028 and 0.000032. Percentage-wise, that's no different than 28 vs. 32, and since no one will likely be playing at 13.5% RoR, the absolute numbers are not particularly useful either, but for many people it isn't as easy to think about the smaller numbers. That's one good argument for picking a large constant when defining SCORE. If we use the standard assumption of 100 hands per hour, and choose a $10k bankroll, then that gives a constant of 100 * 10,000 = 1,000,000, which puts the numbers in the "right" range and makes it easy to explain to people what the SCORE really means in meaningful (if not particularly useful) terms (i.e. winrate on a $10k bankroll with a 13.5% RoR). The fact that this makes the SCORE equal to DI^2 is an added bonus, but not really important. It is my view that the $10k bankroll assumption was probably chosen simply because it made for a convenient constant for the formula.
I have no idea if this was the thought process Don went through in developing SCORE, but it does make good sense to me. Hopefully I've answered your question, but if not, then I hope I've at least given you some food for thought.