Primes are not rigidly set, the integer size is

We make a consistently spaced x axis, and that makes the primes what they are. We can change scale as we go along, and spread or shrink the primes as we want. Let the integer one be 13/17 the size of integer two. Then integer one including all groups in multiples 2,3. The next integer, being 17, includes groups made of 5,3; and so one.

The question is why are they rigidly set with respect to a constant multiplicative identity. The answer is counting between integers, the fractions. A constant multiplicative identity has a perfect inverse in the fractions.

So make two series, count up and down, the prime pairings have to add up to the current prime plus one:

1   2   3    4  5 6 7 8 9 10 11 12 13
13 10 11 10 9 8 7 6 5   4  3   2   1

Primes within a prime. Find the pattern below. I have counted primes by k, and look for:

 Pk +1 = Pj+Pi, i,j < k, skipping trivials

1
2 
3  2,2
5  3,3
7   5,3
11 7,5
13 11,3 |  7,7
17 13,5 | 11,7
19 17,3 | 13,7
23 19,5 | 13,11
29 23,7 | 19,11 | 17,13
31 29,3 | 19,13
37 31,7 | 19,19
41 37,5 | 23,19
43 41,3 | 37,7   | 31,13 | 23,11
47 43,5 | 41,7   | 37,11 | 31,17 | 29,19
53 47,7 | 43,11 | 41,13 | 37,17 | 31,23
59 53,7 | 47,13 | 43,17 | 41,19 | 37,23 | 31,29
61 59,3 | 43,19 | 31,31
67 61,7 | .....................