\(3\leq t\leq T\). In any representation \(p-t=q-r\), one has

\(r\leq t\leq T\) and \(p-T

for \(r\) per prime \(q\) in that terminal interval, so counting possible

pairs proves (4). (a)

By the prime number theorem, (4) forces at least

\((\tfrac12+o(1))\log T\) primes in that terminal interval as \(T\to\infty\).

This is the prime-clustering mechanism used by the cited upper-bound papers.

(b), modulo the prime number theorem

3. Exact finite computation

The complete standalone source is

erdos17_wave5e_verify.py. It uses only the

Python standard library.

The verification layers are:

1. an Eratosthenes sieve, with every entry through \(20{,}000\) independently

compared to trial division; (d)

2. a full enumeration using the transition lemma; (d)

3. a separately written full enumeration in the original difference

variables, without calling the \(t\)-criterion routine; (d)

4. a third, literal global difference-set construction through \(10^5\),

adding every difference \(p-q\) whenever a new prime \(p\) arrives; (d)

5. direct exhaustion of every possible smaller prime for each of the first

thirty noncluster witness certificates. (d)

The two full enumerators agreed at every prime through \(10^8\), and the

literal global method agreed at every prime through \(10^5\). (d)

3.1 Verified decade table

Here \(2\) is excluded, following the convention in the 1999 paper.

| \(x\) | cluster primes \(\leq x\) | noncluster odd primes \(\leq x\) |

|---:|---:|---:|

| \(10^2\) | 23 | 1 |

| \(10^3\) | 99 | 68 |

| \(10^4\) | 420 | 808 |

| \(10^5\) | 1,807 | 7,784 |

| \(10^6\) | 8,287 | 70,210 |

| \(10^7\) | 40,017 | 624,561 |

| \(10^8\) | 202,208 | 5,559,246 |

Every entry is recomputed from scratch, not read from OEIS. They reproduce

the corresponding Blecksmith--Erdős--Selfridge table rows; this run does not

claim a range record beyond their published \(10^{13}\) computation. (d)

3.2 Exact list fingerprint and extrema through \(10^8\)

• The ascending list of all 202,208 cluster primes through \(10^8\), encoded

as consecutive little-endian unsigned 64-bit integers, has SHA-256

41c46f29d4a26a62101597e6e9be03fca7bb3383017be871d9aac0053f6a6e9d.

Both full enumerators independently produced this digest. (d)

• The last cluster prime at most \(10^8\) is \(99{,}999{,}551\). (d)
• Among consecutive cluster primes whose upper endpoint is at most \(10^8\),

the largest value gap is

\(95{,}573{,}461-95{,}561{,}341=12{,}120\). Exactly 629 noncluster primes

lie between these two cluster primes. (d)

3.3 First explicit failure certificates

The second column is an even \(n\leq p-3\) which the direct checker proves is

not \(q_1-q_2\) for any primes \(q_1,q_2\leq p\).

| noncluster \(p\) | missing difference \(n\) |

|---:|---:|

| 97 | 88 |

| 127 | 118 |

| 149 | 140 |

| 191 | 182 |

| 211 | 202 |

| 223 | 214 |

| 227 | 202 |

| 229 | 202 |

| 251 | 242 |

| 257 | 242 |

These are finite exhaustive certificates, not probabilistic tests. (d)

3.4 Reproduction

The default dual enumeration through \(10^7\), plus the literal \(10^5\)

audit, takes about 14 seconds on this VM:

python3 runs/erdos17_wave5e_verify.py

The exact \(10^8\) dual run used for the table and extrema was:

python3 runs/erdos17_wave5e_verify.py \
  --limit 100000000 \
  --cross-limit 100000000 \
  --brute-limit 100000

It completed with all_assertions_passed: true in 130.25 CPU-seconds and

used 213,660 KiB maximum resident memory on this VM. (d)

The core transition check is:

def transition_failure_t(previous_cluster, candidate, is_prime):
    largest_t = candidate - previous_cluster + 1
    for t in range(9, largest_t + 1, 2):
        if is_prime[t]:
            continue
        for r in range(3, t + 1, 2):
            if is_prime[r] and is_prime[candidate - t + r]:
                break
        else:
            return t
    return None

The full source contains the independent enumerators, hashes, published-row

regressions, primality audit, witness exhaustion, and command-line interface.

4. What remains, and the exact wall

4.1 Two hard consequences of a positive answer

If infinitely many cluster primes exist, then every positive even integer is

a difference of two primes: given even \(n\), choose a cluster prime

\(p\geq n+3\). Thus #17 implies Maillet's still-open conjecture. (a)

For any sufficiently large cluster prime \(p\), apply the definition to

\(p-9\). The smaller prime in a representation can only be \(3,5,\) or \(7\);

therefore at least one of \(p-6,p-4,p-2\) is prime. Consequently infinitely

many cluster primes would imply infinitely many prime gaps of size at most

\(6\), i.e. \(H_1\leq6\). (a)

The current unconditional bounded-gap theorem only gives \(H_1\leq246\);

Polymath8b gives \(H_1\leq6\) only under generalized

Elliott--Halberstam. See

[*Variants of the Selberg sieve, and bounded intervals containing many

primes*](https://arxiv.org/abs/1407.4897). (b)

Even the conditional \(H_1\leq6\) handles only the first transition

constraint \(t=9\); it supplies none of the simultaneously growing

constraints for \(t=15,21,25,\ldots\). (a)

4.2 The precise missing extension lemma

By the transition lemma, infinitude is equivalent to the following extension

statement:

> For every cluster prime \(C\), there is a prime \(p>C\) such that, for every

> odd composite \(t\leq p-C+1\), some prime \(r\leq t\) makes \(p-t+r\)

> prime.

One direction recursively produces infinitely many cluster primes; the other

chooses the next larger cluster prime and invokes the transition lemma.

(a)

No searched theorem gives this extension. The obstruction already begins at

\(t=9\), where a lower bound would imply \(H_1\leq6\); the full statement then

requires a growing family of correlated prime-pair alternatives. The Brun and

large-sieve machinery in the two direct papers is one-sided here: it proves

strong upper bounds on how often the required clustering can occur, but

does not give a positive lower bound. (b)

The \(\Delta^*_{721}\)/syndetic prime-difference results do not bridge the

gap: they guarantee that some differences occur frequently in every bounded

combinatorial configuration, whereas #17 requires all even differences up

to a moving endpoint, with their representing primes also below that same

endpoint. (a), using the cited theorem's statement

A finite computation, regardless of range, cannot supply the missing

uniform infinitude step. The present result is therefore an exact reduction

plus a reproducible finite audit, not a solution or a claimed closure. (a)

PARTIAL: proved an exact transition/frontier reduction and independently verified all cluster primes through 10^8 (202,208 total), but infinitude remains blocked already by the unproved H_1<=6/Maillet-level lower-bound barrier and the growing transition constraints.