Actually there is a way in which a field could be doubly or more occupied. If the state filling representes the coefficient of the number which divides into that number. This is nice as the product of all states labels \(k\) to the power of thier populations \(n_k\) is the number itself... \[\prod_{k}^{\infty} k^{n_k} = N\]
Such that a state \(|1,2,1,0,0...>\) would represent 1x2x2x3=12.
Intepretation. If \(k\) is a frequency and the state population is an amplitude, what does the fourier transform of the spectrum look like. Spikes lead to sharp frequencies. So numeric values would take on the form of waves which could ont be interpreted in any other way perhaps \[|1,2,1,0,0...>=1cos(t)+2cos(2t)+1cos(3t)+0cos(4t)+...\]
Might be getting this wrong... Perhaps more like a number like 9. 9 is divisible by 3 twice. So \(9=|1,0,2,0,0,0,0,0,1>\)
Doesn’t belong here: These appear to be relating to prime factors, a concept that is easily mixed up as the first few numbers are mostly prime and common divisors.
6=111001
2=110000
12=2(6)= 110000 + 111001 = 221001 = (12) - 0000000000001
So perhaps could leave the highest number off and the lowest number off... As these are always the same for each number and it allows algebraic rules to be set up.
So in this notation, 1 would not have a symbol as it does nothing to a number: Product in number space leads to an addition in divisor space, (....morphism?)
If the last digit is also taken 2 does not have a divisor representation, Could just place a \(d\).
1=\(.\) 2=\(d\) 3=\(d0\) 4=\(d01\)