Then (a)

OEIS A078515 lists starts where \(\ell(n)\) sets a strict new record; it equals the list of distinct locations on the frontier \(R(k)\), but it need not list \(R(k)\) once for every \(k\) if a record jumps by more than one.

Streaming algorithm and completeness certificate

(a) Immediately before processing gap \(d_r\), maintain the largest interval

\([L,r-1]\) whose gaps are pairwise distinct, together with the most recent index of every gap value. If the previous occurrence of \(d_r\) is \(q\ge L\), then:

• every start \(i=L,\ldots,q\) encounters its first duplicate at \(r\);
• hence its exact local length is \(\ell(i)=r-i\);
• the largest length in this finalized batch is \(r-L\), at start \(L\);
• set \(L=q+1\) and continue.

Thus record detection needs constant work per collision, not a quadratic scan. Once \(L\) exceeds a requested start limit \(K\), every start \(n\le K\) has been finalized even though its run was allowed to extend past \(K\). This is the stopping certificate needed to determine \(h(x)\) exactly for every \(x\le K+1\).

The supplied program generates primes by an odd-only segmented Eratosthenes sieve; it imports no prime list and no number-theory library.

Verified record frontier through 105,000,000 starts

The following table is (d). Each row is a strict record event; \(p_n\) is included as a compact witness locator.

| start \(n\) | \(\ell(n)\) | \(p_n\) |

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

| 1 | 2 | 2 |

| 7 | 3 | 17 |

| 23 | 4 | 83 |

| 30 | 5 | 113 |

| 94 | 6 | 491 |

| 219 | 7 | 1367 |

| 279 | 8 | 1801 |

| 773 | 9 | 5869 |

| 1856 | 10 | 15919 |

| 3724 | 11 | 34883 |

| 6999 | 12 | 70639 |

| 7000 | 13 | 70657 |

| 19205 | 15 | 214867 |

| 184163 | 16 | 2515871 |

| 280103 | 17 | 3952733 |

| 849876 | 18 | 13010143 |

| 1870722 | 19 | 30220163 |

| 3570761 | 20 | 60155567 |

| 4114341 | 21 | 69931991 |

| 11271072 | 22 | 203674907 |

| 55282774 | 23 | 1092101119 |

| 68256040 | 24 | 1363592621 |

| 68256041 | 25 | 1363592677 |

| 104011359 | 26 | 2124140323 |

The jump row has the explicit gap word

whose 15 entries are distinct; the next gap is \(10\), its first duplicate. No earlier start has length \(14\), so

Consequently \(h\) jumps from \(13\) to \(15\) as \(x\) passes \(19205\), and it never takes the value \(14\). This is why A078515 is not literally a one-value-at-a-time inverse.

The terminal record in the scanned regime is

with next gap \(6\).

Some exact values, all (d), are

| \(x\) | \(h(x)\) |

|---:|---:|

| 10 | 3 |

| 100 | 6 |

| 1,000 | 9 |

| 10,000 | 13 |

| 100,000 | 15 |

| 1,000,000 | 18 |

| 10,000,000 | 21 |

| 100,000,000 | 25 |

| 105,000,001 | 26 |

Together with the record table and the strict condition \(n

Independent checks performed

The full run:

python3 runs/erdos852_wave6n_reverify.py \
  --start-limit 105000000 --small-crosscheck 20000

did the following.

1. Generated all required primes from scratch and stopped only after every start \(1,\ldots,105{,}000{,}000\) was finalized.

2. Scanned 105,000,003 primes / 105,000,002 gaps through prime 2,145,390,563.

3. Recomputed all 24 record events above.

4. Independently recomputed starts \(1,\ldots,20{,}000\) using a monolithic sieve and a direct per-start set scan. This independently catches and confirms the \(13\to15\) jump.

5. For every record witness, used deterministic 64-bit Miller–Rabin to check every endpoint prime and every intervening integer composite. Hence the displayed differences really are consecutive prime gaps, independently of the segmented sieve flags.

6. Rechecked the elementary finite-sum formulas and every rational exponent used in Sections 2–3.

Two full measured runs on this VM took 93.46–100.84 wall seconds (93.39–100.76 user CPU seconds) and at most 20.7 MB peak RSS. The final post-edit run ended PASS. The default 12,000,000-start check takes about 10.4 seconds and proves the frontier through length 22.

5. What remains, and the exact wall

The first power-of-log question is answered by the repaired lower-bound argument, but the requested estimate and the question

remain open in the sources checked.

Why the standard tools here do not decide it

• (a/b) The fixed five-prime collision count reaches exactly the

\(H\asymp(\log N)^{1/3}\) scale. The exponent balance fails above it unless one uses the “no intermediate primes” constraints or another source of saving.

• (b) The global mean-square method cannot get even a subpolynomial upper bound; with the conjecturally optimal second moment it still stops near \(x^{1/3}\).
• (c) The iid and singular-series-pressure models reported in the discussion suggest that \(h(x)\) may instead have linear-log extreme scale. They are not proofs and are not used in any boxed conclusion here.

To disprove \(h(x)=o(\log x)\) along the conditional route, a sufficient missing lemma would be a power-saving, lower-bound form of Hardy–Littlewood uniform for growing patterns of \(k=O(\log N)\) primes in a window \(O((\log N)^2)\), together with a uniform Bonferroni minorant enforcing that all unselected offsets are composite and an averaged control of the resulting singular-series pressure. Existing fixed-\(k\) upper-bound sieves do not provide this; lower-bounding prime patterns runs into the parity barrier, and even the fixed twin-prime case is unresolved.

To prove \(h(x)=o(\log x)\), one would instead need a uniform collision theorem forcing a repeated gap in every block of \(\varepsilon\log x\) consecutive prime gaps for every fixed \(\varepsilon>0\). None of the global moment or fixed-pattern estimates above supplies such an extreme, every-block statement.

Cost of extending the finite scan

The remaining published A078515 starts are about \(1.01\times10^9\), \(1.31\times10^9\), and \(7.89\times10^9\). At the measured Python rate, reproducing through the last of these would cost about \(7.0\times10^3\) CPU seconds, i.e. roughly 2 core-hours, and sieve primes to about \(2\times10^{11}\). That exceeds the permitted few-minute run, so it was not attempted. An optimized C/C++ primesieve implementation would be substantially faster, but finding a new record after \(7.89\times10^9\) has no known finite stopping bound and in any case would not settle an asymptotic statement.

6. Re-verification artifact

Standalone checker:

runs/erdos852_wave6n_reverify.py

It is self-contained Python 3, has a fast 12-million default, and accepts

--start-limit 105000000 to reproduce every computational claim in this report.

PARTIAL: Verified the repaired \(h(x)\gg(\log x)^{1/3}\) and \(h(x)\ll_\varepsilon x^{0.41+\varepsilon}\) reductions, computed the exact frontier through 105,000,000 starts, and found \(R(14)=R(15)=19,205\); the \(o(\log x)\) question remains beyond current fixed-tuple sieve and global-moment methods.