What prime-gap statistics would imply

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Put $d_n=p_{n+1}-p_n$. These are author-recorded comparisons of the target statements. They do not establish the conjectured prime-gap laws.

Support is stronger than existence of a distribution

Problem 234 asks for a continuous limiting distribution of $d_n/\log n$. If such a distribution $F$ gives positive mass to every open interval in $(0,\infty)$ and to every interval $(0,b)$, then every $c\ge0$ is a limit point, as requested by Problem 5. For $c>0$, take arbitrarily small intervals about $c$ with positive mass; convergence of their frequencies gives infinitely many indices in each interval. Choose an increasing sequence of those indices. For $c=0$, use $(0,1/j)$ instead. Since $\log p_n/\log n\to1$, the normalizations have the same finite limit points.

Continuity alone does not supply full support. The conditional exponential distribution reported on Problem 234 has it, but its prime-tuple hypothesis remains unproved. The unconditional positive measure and bounded-gap properties of the set of limit points do not give full support either. For example, $\bigcup_{j\ge0}[j,j+1/2]$ is a proper closed subset of $[0,\infty)$, has measure at least $T/2$ in $[0,T]$, and meets every interval of length one. This example tests only those two consequences, not all the stronger combinatorial assertions in Merikoski's paper. Its Theorem 1 and Corollaries 2--3 were inspected at statement depth. The next test for a proposed route is positive limiting mass in an arbitrary prescribed interval. Stop if the available conclusion only bounds total measure or gaps in the set of limit points.

Two half-density inequalities do not prove infinitely many equalities

For Problem 218, let $A_N$ count $n\le N$ with $d_{n+1}\ge d_n$, let $B_N$ count those with $d_{n+1}\le d_n$, and let $E_N$ count equalities. Then $A_N+B_N=N+E_N$. Thus $A_N/N\to1/2$ and $B_N/N\to1/2$ together imply $E_N/N\to0$, not $E_N\to\infty$. The numerical sequence alternating between $2$ and $4$ has both inequality densities $1/2$ and no equality; this is a logical example, not a prime-gap construction. Marginal distributions of individual gaps likewise do not determine adjacent pairs. [[../library/primes/banks_2023_ratios_consecutive_prime_gaps/_index|Banks's ratio formula]] is presented as a conjecture motivated by a heuristic, not an unconditional adjacent-gap theorem. A route must supply both the needed joint distribution and a separate reason for infinitely many exact equalities.

A second moment still leaves a power-sized distinct run

Let $M_2(N)=\sum_{n\le N}d_n^2$. Cauchy--Schwarz and the prime number theorem give $M_2(N)\gg N(\log N)^2$, so Problem 233 asks for an upper bound of the correct order. The inspected Theorem 1 of Stadlmann is instead $\sum_{p_n\le X}d_n^2\ll_\varepsilon X^{1.23+\varepsilon}$. Changing from prime value $X$ to index $N$ does not remove its excess power.

Even the desired bound would only give a power estimate for the length $H$ of a run of pairwise distinct gaps starting at index $n<x$, the quantity in Problem 852. Distinct positive integers have sum at least $H(H+1)/2$. The telescoping sum of these gaps is at most $p_{n+H}\ll(x+H)\log(x+H)$. For large $x$ this first forces $H<x$, and then $H\ll\sqrt{x\log x}$. The run therefore ends before $2x$. Its squared gaps have sum at least $\sum_{j=1}^Hj^2\gg H^3$, while the hypothetical bound for Problem 233 gives $M_2(2x)\ll x(\log x)^2$. Consequently

$$H\ll\bigl(x(\log x)^2\bigr)^{1/3}.$$

This is far above $o(\log x)$. It is a limitation of this direct moment argument, not a proof that Problem 233 could never help by another method. A useful next test must exploit local repetition or dependence beyond a global second moment; stop the bare sum-of-squares route at the displayed bound.

First occurrences and Problem 853

For a positive even integer $t$, let $\tau(t)=\min\{n:d_n=t\}$, with $\tau(t)=\infty$ if no such index exists. For real $y\ge2$ put $T(y)=\max_{2\le t\le y,\ 2\mid t}\tau(t)$. In the notation of Problem 853, $r(x)$ is the least missing positive even gap among indices $n\le x$. The positivity convention matters: including zero among candidate gap sizes would make $r(x)$ identically zero. Exactly

$$r(x)>y\quad\Longleftrightarrow\quad T(y)\le x.\tag{1}$$

It follows that $r(x)\to\infty$ is equivalent to every positive even gap occurring at least once. Infinitely many occurrences of each gap would be stronger than needed. The stronger requested conclusion is equivalent to

$$T(y)<\infty\text{ for every }y,\qquad \log T(y)=o(y).\tag{2}$$

Indeed, if $r(x)/\log x\to\infty$, take $x=\exp(\varepsilon y)$. Eventually $r(x)>y$, so $T(y)\le\exp(\varepsilon y)$ for every fixed $\varepsilon>0$. Conversely, fix $K>0$ and use (2) with $\varepsilon<1/K$. For $y=K\log x$ and all sufficiently large $x$, $T(y)\le\exp(\varepsilon y)<x$, so (1) gives $r(x)>K\log x$. Integer rounding changes neither conclusion. This exact interface exposes the hard step: a subexponential upper bound for the worst first occurrence, uniform over all even $t\le y$. A fixed-pattern existence theorem with uncontrolled constants does not supply (2). Stop a proposed deduction if its threshold still depends ineffectively on each individual $t$.

Adjacent near-record gaps and Problem 1137

For integer $N\ge2$, define

$$D_N=\max_{1\le n\le N}d_n,\quad M_N=\max_{2\le n\le N}d_nd_{n-1},\quad B_N=\max_{2\le n\le N}\min(d_n,d_{n-1}).$$

Both factors in every product are bounded by $D_N$, hence

$$B_N^2\le M_N\le D_NB_N.$$

Thus Problem 1137 is equivalent to $B_N/D_N\to0$. Equivalently, for every $\varepsilon>0$, all sufficiently large prefixes contain no adjacent pair both at least $\varepsilon D_N$. The omitted initial undefined product and changing $n<x$ to an integer prefix have no asymptotic effect. This is a relative isolation assertion. A lower bound for one large gap, such as Theorem 1 of [[../library/primes/maynard_2016_large_gaps_between_primes/_index|Maynard's large-gap paper]], gives no such upper bound for adjacent products. A route must control the smaller member of every adjacent pair relative to the running maximum. Stop before inferring isolation from a single-gap construction, even if that construction substantially increases $D_N$.