Problem 218: prime-scale growth does not turn oscillation into density
1) OBJECTIVE AND SOURCE REVIEW
Attempt dated 2026-09-23. For consecutive prime gaps $d_n$, the target concerns density $1/2$ for each of $d_{n+1}\ge d_n$ and $d_{n+1}\le d_n$. I read
gpt_pro_5.4/218.tex completely. It invokes Pintz to obtain arbitrarily large and small consecutive-gap ratios. I test whether those oscillations together with the first-order total gap scale can imply the requested frequencies.
2) WORK
Here is an explicit obstruction to that inference. Define positive even model gaps by \[
g_n=2\lceil\log(n+2)\rceil\quad\text{unless }n=2^j\ (j\ge2),
\qquad g_{2^j}=2j^4.
\] The baseline has total $2N\log N+O(N)$, by comparison of $\sum\log(n+2)$ with its integral and bounded rounding errors. Modifications at powers of two contribute $O((\log N)^5)$ up to $N$. Hence \[
\sum_{n\le N}g_n\sim2N\log N.
\] The baseline changes only $O(\log N)$ times up to $N$, and the modifications affect only $O(\log N)$ consecutive comparisons. Thus \[
\#\{n\le N:g_{n+1}\ne g_n\}=O(\log N).
\] Both weak-comparison sets therefore have density 1, while both strict-comparison sets have density zero. Nevertheless, immediately before $2^j$ the ratio is of order $j^3$, and immediately after $2^j$ it is of order $j^{-3}$. Therefore \[
\limsup g_{n+1}/g_n=\infty,\qquad\liminf g_{n+1}/g_n=0.
\] The model has positivity, evenness, prime-scale total growth, and frequent-scale extreme ratios, but completely different comparison densities. These elementary properties cannot settle the prime question.
3) ADVERSARIAL VERIFICATION
This is explicitly a model of gaps, not a sequence of actual prime gaps or a counterexample to the conjecture. The total-growth coefficient is immaterial to the logical obstruction. The equality set matters: for any sequence the two weak-comparison indicators add to one plus the equality indicator. Accordingly their two proposed densities of $1/2$ require the equality set to have density zero, a fact not supplied by oscillation alone.
4) FINAL
UNRESOLVED. Actual prime-specific frequency estimates, including control of equal consecutive gaps, are still missing. The additional result is an explicit limitation on what the supplied qualitative conclusions can imply.
5) SOURCES CHECKED
Complete supplied file; J. Pintz,
On the ratio of consecutive gaps between primes,
https://arxiv.org/abs/1406.2658, checked 2026-09-23 for the qualitative extreme-ratio theorem. The model construction is proved here.
6) COMPLETION ESTIMATE
Subjective progress toward the density assertions.
COMPLETION: 10%