Instead of sieving by divisibility, we sieve by the size of the gaps between originally consecutive primes.

We process gap sizes in strict ascending order. To be eliminated, two primes must currently be next to each other in the surviving list AND they must have been consecutive in the original sequence.  

Here is an example of how the sieve works in the first 50 primes:

We start with an unbroken chain of the first 50 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229.  

Step 1: Processing Gap 1

We start with the absolute smallest gap between any two primes: a gap of 1.

Scanning: We look for any original neighbors separated by 1. There is only one such pair in existence: (2, 3). Eliminated: 2 and 3.

Surviving List: 5, 7, 11, 13, 17, 19, 23... (all primes from 5 to 229).

Step 2: Processing Gap 2 (The Twin Prime Cleansing)

Next, we scan left-to-right for original neighbors separated by a gap of 2 (Twin Primes).

Matches Found: (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199), (227, 229). Eliminated: All of the twin primes listed above.

Surviving List: 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223. 

Step 3: Processing Gap 4 (Cousin Primes)

 We look for originally consecutive primes separated by 4, but only if both primes are still alive.

Checking (19, 23): 19 was eliminated in Step 2. The pair is broken. 23 is shielded and survives.

Checking (37, 41): 41 was eliminated in Step 2. 37 is shielded and survives.

Matches Found: Scanning our surviving list, we find exactly three pairs where both members survived Step 2 and were original neighbors separated by 4: (79, 83), (127, 131) and (163, 167)Eliminated: 79, 83, 127, 131, 163, 167.

Surviving List: 23, 37, 47, 53, 67, 89, 97, 113, 157, 173, 211, 223.

Step 4: Processing Gap 6 (Sexy Primes)

We scan the remaining primes for originally consecutive pairs separated by 6.

Checking (23, 29): 29 is gone. 23 is shielded.

Matches Found: We scan the surviving list left-to-right. We find two intact pairs: (47, 53) and (211, 223). Eliminated: 47, 53, 211, 223.

Surviving List: 23, 37, 67, 89, 97, 113, 157, 173.

Step 5: Processing Gap 8

We scan the remaining 8 primes for original pairs separated by 8.

Match Found: There is exactly one pair left that fits this criteria: (89, 97). Eliminated: 89, 97.

Surviving List: 23, 37, 67, 113, 157, 173.

Step 6: Processing Gaps 10+

At this point, we check for gaps of 10, 12, 14, and so on. However, if we look at our surviving primes, 23, 37, 67, 113, 157 and 173, none of them have their original right-side neighbor remaining.

Because all of their original neighbors have been eliminated by smaller gaps, these remaining primes are permanently shielded. No larger gap size can ever eliminate them. 

So, the final survivors are: 23, 37, 67, 113, 157, 173.  

We are interested in the density of the remaining primes. We have computed the following results for the first k primes:

10^2 primes -> 14 survivors

10^3 primes -> 134 survivors

10^4 primes -> 1374 survivors

10^5 primes -> 13494 survivors

10^6 primes -> 135530 survivors

10^7 primes -> 1358010 survivors

We see that the density for the first 10^7 primes is 0.135801.

Conjecture:

We conjecture that when k goes to infinity, the density approaches a constant and this constant is 1/e^2 = 0.1353352832366...

Q1. Can you verify our calculations?
Q2. Can you extend our calculations beyond 10^7?

Q3. Can you prove the above conjecture?