Proof that all positive integers will appear in A360706.

Required knowledge for this poof:

Theorem 1: 
 We know that if we have an infinite sequence S which may be any permutation of natural numbers, 
 then any finite subset of bits in its binary representation will reach eventually all possible combinatorial
 states of zeros and ones, if only the range of numbers that will be observed is big enough.
 This will still hold if we exclude any set of number from S, if the set of excluded numbers is finite in size.

Example:
 S = 2,7,22,9,1,51,47,88,11,13,5,17,12,3,...   this shall be part of some randomly chosen permutation
 We observe three randomly chosen bits of the binary representation of S:
 2^0: 0,1,0,1,1,1,1,0,1,1,1,1,0,1,...
 2^2: 0,1,1,0,0,0,1,0,0,1,1,0,1,0,...
 2^3: 0,0,0,1,0,0,1,1,1,1,0,0,1,0,...


Theorem 2:
 Let V be an infinite sequence that is a subset of some permutation of natural numbers. ( see 1. )
 Consider W as the binary bitwise XOR operation over consecutive terms of V. For each finite subset of bits in the 
 binary representation of W we will observe all possible combination of one bits in the numbers from within some interval 
 of consecutive terms in W, if the size of this interval is big enough. 

Example:
 We observe the bits 2^0, 2^2 and 2^3 of the cumulative XOR over S:
 2^0: 0,1,1,0,1,0,1,1,0,1,0,1,1,0,...
 2^2: 0,1,0,0,0,0,1,1,1,0,1,1,0,0,...
 2^3: 0,0,0,1,1,1,0,1,0,1,1,1,0,0,...

Proof: 
 Let us assume this sequence would be a permutation of the positive numbers excluding the number x,
 this means x shall not be part of this sequence. This would be only then the case if the count of usage
 for the 1-bits in the binary representation of x reaches never equal parity. Not reaching equal parity for some 
 finite set of bits would violate Theorem 2, as excluding the number X is excluding a finite subset of the permutation.
 This shows that if a set of numbers is missing in A360706 that this set of numbers must be countable infinite.
 If a such a set of numbers exists, which is not part of A360706, then each member of this now excluded set depends
 on a further infinite set of numbers which are required to be missing in A360706, as for each such member the count of usage 
 for the 1-bits in the binary representation may never reach equal parity. This will leads in all cases to a contradiction 
 as by iterating through this process we will exclude eventually all numbers from the sequence.