Awesome, thanks for that!
This is why I don't like the naming of the normal distribution. What I wanted to say about the chi squared test is that it only tests whether the frequencies of individual outcomes match those of the theoretical distribution, which says nothing about the order in which those outcomes occur. That's great if the sampling process is drawing independent and identically distributed observations, but not so much if it's a stochastic process (or if its time-series data, if you prefer). When you're considering streaks (which is what Larian claim to be doing with the karmic dice), the order in which observations happen matters.
As an aside, from this stochastic processes perspective, it's interesting to look out for critical values that are 'too good', which may suggest that your sampling process isn't IID (independent and identically distributed). See here for graphs
https://imgur.com/a/pjNf9HRYou can see the intuition behind this by imagining a coin flip - you'll get a much better fit to a uniform distribution by first flipping your coin with 2 heads on it, then flipping your coin with 2 tails on it, etc, etc. In fact on any even number of flips your empirical distribution is going to be exactly the distribution you 'expect', and on odd flips it won't be far off. If you were to perform Pearson's chi squared test, the test statistic would be 0 (exactly, for an even number of flips), but you'd be super skeptical about it. You might argue that this sequence isn't even a random variable, but I can flip an actual fair coin to decide which of my two rigged coins to start with, and now it's a genuine random variable. The marginal distribution of heads/tails outcomes is genuinely uniform, but the joint distribution of the nth flip given the outcome of the n-1th is either [1,0] or [0,1], and never [0.5,0.5]. This example is pretty extreme, so you get literal 'suspiciously good' values of the Chi squared statistic, but in more realistic scenarios you'll never realize that the system is rigged in this way from a Chi squared test. You can see from my graph that my rigged dice simulation does better on Chi squared tests than the IID uniform one, but not so much so that you'd notice.
What you want to look at is the autocorrelation, shown in the second plot in my link. Here it's very obvious that my rigged dice aren't IID uniform, and this is what we would LIKE to see from the karmic dice. You can formally test for autocorrelation using the Durbin-Watson test, but at this point it's a little academic. The karmic dice data doesn't even have the right mean, so who cares if it's autocorrelated or not...
Also, be careful with multiple testing. I see in one of those threads you posted "BUT! If we combine your unweighted chart with @Saberem's unweighted data, we get a Chi^2 of 32.26 which is GREATER than the 95% confidence value of 31.14. Thus, we can conclude that Larian's unweighted dice rng is NOT random." - you need to correct for multiple comparisons. If you test enough times, you'll eventually get a test statistic greater than your critical value by chance (unless your sampling method actually IS biased). Applying Bonferroni correction and assuming you performed only two tests, the critical value would be 34.17, and since you got a Chi^2 of 32.26 you would fail to reject the hypothesis that the data followed a uniform distribution.
The non-karmic dice data looks good. If you wanted to scrutinize it I'd be looking for autocorrelation - you can see from my graph that for samples of size 300, autocorrelations stronger than about 0.2 in magnitude would be suspicious. I'd be very surprised if there was anything wrong with the basic dice rolling implementation though. Good random sequences are precisely the ones that look bad to the layman - if the players didn't think it was a bad implementation, that would be a strong indicator that it actually was. That is exactly why Larian have tried to implement the karmic dice, they want to make a version of dice rolling that the players think is fair, rather than one that actually is fair, and those two things are mutually exclusive. It's a shame they seem to have done such a bad job of it...
Edit: I should say, there's no real reason the system wouldn't be 'fair' in the sense of justice, but it wouldn't be 'fair' in the technical sense of 'fair dice'. In some ways it's more fair that low rolls tend to be followed by high ones, and vice versa. Whether it's more fun or not is another question. Fewer people would experience the feeling of good or bad luck. Personally I enjoy that, but I've also spent most of my life studying probability theory so I'm about as biased as a karmic dice...
