Again, if it’s guaranteed to be mathematically exactly fair, then by the maths I posted earlier, claiming you have better than 1/6 chance of getting the next one right is mathematically impossible, by definition.

To be clear: you’re defining a situation whereby you are guaranteed to only have a 1/6 chance of getting the next number correct, whichever you pick, and then saying “isn’t it better to stick with the number that came up before?”. Simply, no, it’s not, because of the way you defined the system.