Hardy–Littlewood prime-pair asymptotic valid pointwise when the shift and

search length are both \(P\) would give

For a primorial,

so (8) predicts

[C: Hardy–Littlewood heuristic; not a theorem]

The exact missing lemma on this route is a positive pointwise lower bound

for (7), uniformly at \(h=x=P_n\). Ordinary two-dimensional sieve gives

upper bounds but not a positive lower bound for two simultaneous primes:

this is the parity obstruction.

Qualitative Schinzel H applied after fixing a single \(P\) predicts

infinitely many primes \(p,p+P\), but it gives no uniform guarantee that

one occurs before \(p=P\). Thus the qualitative fixed-polynomial statement

does not, by itself, prove the upper bound required in #779. **[A: logical

quantifier observation]**

Likewise, bounded-gap theorems locate some recurring bounded gap; they do

not force the specified growing gap \(P_n\). [A]

5. Reproducible computation

The standalone verifier is:

It uses only the Python standard library and the installed PARI/GP

executable. It does not download or trust the OEIS table at runtime:

candidate offsets are embedded, the first 4602 primes are rebuilt with an

Eratosthenes sieve, and every asserted arithmetic relation is recomputed.

Run:

python runs/erdos779_wave6l_verify.py

PARI/GP version in the successful run: 2.15.4.

5.1 Exact finite block \(2\leq n\leq100\)

For all 99 values \(2\leq n\leq100\), the verifier checks:

• the offset \(p\) is prime;
• \(p_n
• isprime(P_n+p,2) == 1.

PARI documents flag 2 as the deterministic APRCL primality test, not BPSW.

No external certificate file is emitted, so I conservatively classify this

finite block as [D] computational-only, albeit with a proof-grade

primality algorithm rather than a probable-prime test.

Selected rows:

| \(n\) | \(p_n\) | witness \(p\) | digits of \(P_n\) | result |

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

| 2 | 3 | 5 | 1 | APRCL prime \(P+p=11\) |

| 3 | 5 | 7 | 2 | APRCL prime \(P+p=37\) |

| 4 | 7 | 13 | 3 | APRCL prime \(P+p=223\) |

| 5 | 11 | 23 | 4 | APRCL prime \(P+p=2333\) |

| 10 | 29 | 61 | 10 | APRCL prime |

| 25 | 97 | 103 | 37 | APRCL prime |

| 50 | 229 | 293 | 92 | APRCL prime |

| 75 | 379 | 409 | 154 | APRCL prime |

| 100 | 541 | 641 | 220 | APRCL prime |

The complete 99-row transcript is deterministically represented by

small_table_sha256 =
a2bfc46bfeea17cb9f9b674167f506261dcd019fe8244250db1f2746e87a164d

The largest offset in this block is \(1381\), at \(n=70\), and all full

integer rows are generated by the verifier. [D]

5.2 Exhaustive short-window identity

For each \(2\leq n\leq15\), the script enumerates both sides of (3) for

\(H=p_{n+1}^2\). All tested integers are below \(2^{64}\); it uses the

standard deterministic seven-base Miller–Rabin theorem for this range.

The two sets agree in every case. **[D, rigorous modulo the deterministic

64-bit Miller–Rabin base theorem]**

The counts are

[(2, 6), (3, 10), (4, 19), (5, 23), (6, 29), (7, 25),
 (8, 38), (9, 42), (10, 35), (11, 56), (12, 54),
 (13, 45), (14, 60), (15, 67)]

and the enumerated-set transcript hash is

251e0911a815011d7bbd920ff11211e975bdbf5515c5675f07990a151f1378ee

This computation tests the implementation; Proposition 4 itself was proved

for all \(n\) independently of the computation.

5.3 A deliberately non-rigorous checkpoint at \(n=3001\)

The current A005235 b-file gives the candidate offset \(p=33211\).

The verifier independently recomputes:

n             = 3001
p_n           = 27457
digits(P_n)   = 11833
p             = 33211
isprime(p,2)  = 1
p_n < p < P_n
ispseudoprime(P_n+p) = 1   # PARI BPSW
P_n mod 1000000007 = 149874951

The exact integer \(P_n+p\), represented without printing all 11833 digits,

has

sha256(decimal(P_n+p)) =
79f309b87fca624b205ea1090716f8ccd3502fc61e4e95e384c225f04b31e777
first 32 digits = 51602038179433115646345337470223
last  32 digits = 18921544259369476798319951327881

This is [D] computational-only. BPSW passing does not prove that

\(P_{3001}+33211\) is prime, and I do not call it prime in this report.

Nor did I re-establish that 33211 is the least offset; minimality is

irrelevant for its use as a #779 candidate.

The final complete verifier run took 49.66 wall-seconds. It also

reproduced the \(n=1001\) BPSW checkpoint from Section 6.1 and the

\(n=4601\) size and heuristic inputs from Section 6.2.

5.4 Core verification code

The full executable code is in the standalone file. Its decisive PARI

calls are:

# Exact finite block:
for n in range(2, 101):
    assert p_n < witness[n] < P_n
    assert isprime(witness[n], 2)       # APRCL
    assert isprime(P_n + witness[n], 2) # APRCL

# Large checkpoint, intentionally not promoted to a proof:
assert ispseudoprime(P_1001 + 8831)     # BPSW only
assert ispseudoprime(P_3001 + 33211)    # BPSW only

The actual file constructs the primes and primorials from scratch, invokes

GP in one checked subprocess, parses all 99 certificates, performs the

independent 64-bit bijection enumeration, and hashes the transcripts.

6. Bounded failed computations and realistic cost

6.1 Proof attempt at \(n=1001\)

For the candidate \(p=8831\), \(P_{1001}+8831\) has 3397 decimal digits.

A BPSW check took about one second, but this is not a proof. PARI APRCL

required repeatedly expanding its stack; the final bounded attempt used

93.17 CPU-seconds and reached about 4.8 GB resident memory before I

interrupted it at an 8 GB configured stack. Pocklington–Lehmer mode also

failed to finish within two CPU-minutes. No exact primality claim is made.

6.2 Attempt to extend the b-file to \(n=4601\)

Here

p_4601        = 44203
digits(P_4601)= 19113

PARI nextprime(P+2) (which searches for a BPSW pseudoprime, not a proven

prime) was stopped at the hard cap of 239.94 CPU-seconds without returning

a candidate. Therefore no \(n=4601\) value is claimed.

A separate timing probe of the first prime offset \(p=44207\) found

\(P_{4601}+44207\) BPSW-composite in 22.62 seconds (PARI reported

22,620 ms). This can be repeated with

--skip-large --benchmark-4601. [D]

Under the prime-pair heuristic, a prime offset has conditional success

probability about

or roughly 2310 expensive survivors per success. [C] At the measured

22.7 seconds per composite test, a straightforward serial extension is

therefore approximately 15–30 core-hours, allowing for the slower successful

BPSW test and search overhead. [C: cost estimate] At a representative

\$0.05–\$0.15 per vCPU-hour, that is roughly \$1–\$5; parallel execution

would reduce wall time but not core-hours. This is why I did not run it on

the present few-minute budget.

A proof-grade ECPP certificate for an 11,833- or 19,113-digit general-form

candidate is a much larger job than the BPSW search. Extrapolating published

general-purpose ECPP/APRCL timings gives at least hundreds, plausibly

thousands, of core-hours, with substantial input-to-input variance.

[C: engineering estimate, not a theorem]

7. What has and has not been established

• The lower inequality \(p_n[A]
• Every prime value \(P+m\) with \(1

offset \(m\). [A]

• Equation (5) is an exact reduction: the remaining alternatives are a

Cramér-scale gap at the primorial or a positive prescribed-gap prime-pair

count in the tail. [A]

• Explicit #779 witnesses for \(2\leq n\leq100\) were independently

rechecked with APRCL. [D]

• The literature reports computational verification of the stronger

Fortune conjecture through \(n=3000\), and the live OEIS b-file currently

has 4600 candidate rows. [D]

• The \(n=3001\) shifted primorial passed BPSW in this run, but has not been

proved prime here. [D]

• Neither the required uniform Cramér-scale prime-gap lemma nor the required

pointwise lower-bound prime-pair lemma is presently available. Hence

there is no uniformity/finiteness step closing #779.

PARTIAL: proved the exact Cramér-window/tail decomposition, APRCL-reverified witnesses for 2<=n<=100, and BPSW-checked (not proved) the n=3001 candidate; the uniform prime-gap/prime-pair step remains open.