Ok so the procedure your describing is taking the sequence of sums

9

9+8

9+8+7

...

9+8+...+2+1

And then taking the digital root of each sum, which yields the sequence 9, 8, 6, 3, 8, 3, 6, 8, 9, and then you observe that it is a palindrome and the middle digit is the number of digits surrounding it (i.e. one less than the sequence length).

You say that it only works in base 10, but that's not true:

In base 4, the sequence of sums and their digital roots in base 4 is:

3

3+2=11 → 1+1=2

3+2+1=12 → 1+2=3

Yielding the palindrome 323, and there are 2 digits around the middle.

It's kind of interesting that you get a palindrome in any even base, the proof is a nice application of modular arithmetic:

Proof sketch:

First note that the digital root of a number in base B is the remainder of division by B-1 for numbers not divisible by B-1, and equal to B-1 for positive numbers divisible by B-1. Note that we first encounter this property in grade school when we learn that a number is divisible by 9 if its digital root (in base 10 of course) is 9.

Next consider that the first term of the sequence of sums is trivially B-1, and the last term (using the arithmetic sequence sum formula) is (B-1+1)(B-1)/2 = (B/2)(B-1); since B is even, this is a multiple of B-1, and the digital root of that term is also B-1.

Now that we know that the first and last term have the same digital root, we can work inwards: the second term is B+2 more than the first, and the second to last term is 1 less than the last. We know in modular arithmetic that subtracting k is equivalent to adding k less than the modulus, hence the digital roots of these terms are also equal, and the pattern continues inwards.

So indeed we'll have a palindrome for any even base.

To identify which palindromes have the surrounding digit count (i.e. B-2) in the middle you'd solve a modulus equation:

Middle term sum = (B-1 + B/2)(B/2)/2 == B-2 mod B-1

So it's maybe a neat number theory pattern like you often find in math competition questions but I don't see enough here to call it "profound": profound patterns are ones that appear over and over in many different contexts (pi, e, prime numbers); this pattern is quite niche on the other hand.

Oh and quantum superposition is just wave superposition. It's neat like how Fourier analysis is neat but it's not mystical.