Positive lower density of large prime gaps

Source and verification

This article explains family 026 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Gap size and gap frequency

Let $p_n$ be the $n$th prime and $d_n=p_{n+1}-p_n$. The prime number theorem makes $\log p_n$ the average scale of a gap. Unbounded values of $d_n/\log p_n$ establish the existence of unusually large gaps, but say little about their frequency.

For a set of indices $A\subseteq \mathbb{N}$, its lower natural density is

$$\underset{N\to \infty }{\mathrm{liminf}}\frac{\#(A\cap [1,N])}{N}.$$

Claimed positive gap frequency

For every fixed $C>0$, the manuscript asserts constants $c(C)>0$ and $N_0(C)$ such that

$$\begin{array}{c}\#\{n\le N:d_n>C\log p_n\}\\ \ge c(C)N,\\ N\ge N_0(C).\end{array}$$

The threshold $C$ is fixed before the limit. This is an ordinary lower-density result, not a limiting gap distribution or a bound uniform for growing $C$. (OpenAI, 2026)

There is no conflict with the mean gap: the allowed positive proportion $c(C)$ may become very small as $C$ grows.

🏷️ Why empty intervals do not suffice

Consider integer-start intervals $(m,m+h]$. A prime gap of length $d$ contributes roughly $(d-h{)}_{+}$ values of $m$ for which such an interval is empty. Therefore counting empty intervals weights long gaps by their length.

A small number of exceptionally long gaps could account for many empty intervals. To prove a positive fraction of prime indices, one needs a way to bound how often the same gap is counted.

The source uses adjacent intervals:

$$\begin{array}{c}(m,m+h]\\ \text{and}(m+h,m+2h].\end{array}$$

It requires the first to contain a prime and the second to contain none. Let $p$ be the last prime in the first interval. The next prime is beyond $m+2h$, so its gap from $p$ is greater than $h$.

Moreover, a fixed $p$ can be selected by at most $h$ integer starts, because $m<p\le m+h$ forces $p-h\le m<p$. This bound is independent of the length of the following gap.

Thus a positive proportion of successful starts in an interval of length $X$ gives $\gg X/h$ distinct prime gaps. With $h\asymp \log X$, this is the prime-counting scale.

🏷️ Positive weights with asymmetric prime behavior

The manuscript constructs a nonnegative weight $w(m)$ for which the first interval has substantial weighted probability of containing a prime, while the expected number of primes in the second interval is arbitrarily small.

Two uses of the second moment have different roles. A prime-count second moment detects occupancy of the first interval. A weight second moment transfers positive weighted mass to many ordinary starts.

For the latter, if $E$ is the event of success, Cauchy–Schwarz yields

$${\left(\sum _{m\in E}w(m)\right)}^2\le \#E\sum _{m}w(m{)}^2.$$

A lower bound for the weighted mass and an upper bound for $\sum w^2$ therefore give a lower bound for $\#E$. A large weighted mass without this control could be concentrated on very few starts.

🏷️ Cancellation between neighboring dimensions

The weights are squares of signed combinations of smooth divisor sums attached to subsets of shifts in the second interval. Squaring keeps the final weight nonnegative while allowing cancellations inside it.

The test functions make distinct shift sets orthogonal in the leading unmarked second moment. If a shifted value is prime, the support of the divisor sum forces the associated divisor coordinate to be one. Removing that coordinate couples terms whose shift sets differ in size by one.

An alternating choice of coefficients makes these adjacent-dimensional terms nearly cancel. The unmarked average stays positive, while the weighted prime mass in the second interval becomes small.

The first-interval detector has a constant independent of the size of the alternating family. That separation of constants permits suppression in the second interval to be chosen strong enough without losing the first-interval signal.

The analytic inputs are smooth divisor-sum correlations, Bombieri–Vinogradov distribution, and an averaged prime-tuple singular series. The latter is an average of local congruence-density factors, not an assumed Hardy–Littlewood prime-tuple asymptotic.

All dimensions and test functions are fixed before $X$ grows. Uniform errors over the shifts are needed because their number increases with $h$.

🏷️ Passing to index density

Choose $h=\lfloor \lambda \log X\rfloor$, with $\lambda$ larger than the desired fixed threshold. The adjacent-interval construction yields many gaps at prime locations of size comparable to $X$, where $\log p=\log X+O(1)$. Hence $d>C\log p$ for sufficiently large $X$.

The prime number theorem converts $\gg X/\log X$ qualifying gaps into a positive fraction of prime indices. The source makes this conversion uniformly over sufficiently large initial segments.

A concrete corollary concerns increases in $p_n/n$. Algebra gives

$$\frac{p_{n+1}}{n+1}>\frac{p_n}{n}⟺d_n>\frac{p_n}{n}.$$

Since $p_n/n\sim \log p_n$, the claimed theorem with $C=2$ gives a positive lower density of such increases.

🏷️ Key steps and proof outlines

The proof has two distinct tasks: construct a probability weight that strongly disfavors primes in one interval, and remove that weight without losing a positive proportion of starting points. The essential formulas are in “A square with a small marked moment” and “Cancellation between adjacent dimensions”; “From two adjacent intervals to gap counts” completes the counting argument (OpenAI, 2026).

Prime deletion and neighboring dimensions

Put $L=\log X$, $h=\lfloor \lambda L\rfloor$, and let $J=\{1,\dots ,h\}$ and $I=\{h+1,\dots ,2h\}$. Define $V_B(m)=L^{-1}{\sum }_{b\in B}(\log (m+b))1_{m+b\text{ prime}}$. Thus $V_B>0$ detects a prime, and then $V_B\ge 1$. A smooth divisor sum attached to a shift set $S=\{b_1,\dots ,b_j\}$ is

$$D_{F_j}(m;S)=\sum _{d_i\mid m+b_i}\mu (d_1)\cdots \mu (d_j)F_j\left(\frac{\log d_1}{L},\dots ,\frac{\log d_j}{L}\right).$$

Here $\mu$ is the Möbius function. Start with symmetric smooth functions $f_j$ supported in $t_i>0$, ${\sum }_i{t}_i<1/8$, and let $F_j$ be their coordinatewise tail integrals. Then $(-1{)}^j{\partial }_1\cdots {\partial }_j{F}_j=f_j$. The signed sum and its square are

$$Z(m)=\sum _{j=0}^k\sum _{S\subset I,|S|=j}{D}_{F_j}(m;S),W_X(m)=Z(m{)}^2/w.$$

Why do adjacent dimensions interact? If $m+a$ is prime for $a\in I$, every contributing divisor in that coordinate must equal one: the alternative divisor $m+a$ lies outside the support. Removing $a$ from a subset replaces $F_j$ by $F_j+F_{j+1}(\cdot ,0)$. Taking mixed derivatives replaces $f_j$ by $f_j+T{f}_{j+1}$, where

$$(T{f}_{j+1})(\mathbf{t})={\int }_0^{\infty }{f}_{j+1}(\mathbf{t},u)du.$$

This is an exact deletion identity, with no extra factor of $j+1$. It explains the source’s limiting quadratic forms, with ${\alpha }_j={\lambda }^j/j!$:

$$\begin{array}{rl}w& =\sum _{j=0}^k{\alpha }_j\Vert f_j{\Vert }_2^2,\\ v& =\lambda \sum _{j=0}^k{\alpha }_j\Vert f_j+T{f}_{j+1}{\Vert }_2^2.\end{array}$$

The moment theorem gives ${\mathbb{E}}_X{Z}^2\to w$, ${\mathbb{E}}_X(Z^2{V}_I)\to v$, and ${\mathbb{E}}_X(Z^2{V}_J)\to \lambda w$, where ${\mathbb{E}}_X$ averages $X<m\le 2X$. Its arithmetic proof expands divisor tuples, uses the Chinese remainder theorem, and evaluates compatible local densities. In the limiting quadratic forms, unmatched shifts force a complete mixed derivative onto a coordinate face; the interior support makes that term vanish. Matching subsets survive and produce the displayed squared norms. The support restriction keeps the moduli within the prime-distribution estimates used in the source.

Alternating signs create a small marked moment

Choose $g\ge 0$ smooth and compactly supported in $(0,\infty )$ with

$$\int g^2=1,\int g=\sqrt{\lambda },{\mu }_g:=\int ug(u{)}^{2}du<\beta <\gamma <1/8.$$

The small first moment is compatible with the prescribed integral. To see the mechanism, take a smooth profile approximately $1/u$ on $[1,R]$, normalize its $L^2$ norm, and rescale its argument. The resulting first moment is $O(\lambda /\log R)$, which can be made small before any dimension is chosen.

Set $Q_j(\mathbf{t})=k^{j/2}{\prod }_{i}g(k{t}_i)$. Then $\Vert Q_j{\Vert }_2=1$ and $T{Q}_{j+1}=\sqrt{\lambda /k}{Q}_j$. On the $r=\lfloor \sqrt{k}\rfloor$ levels nearest $k$, choose

$$f_j(\mathbf{t})=\frac{(-1{)}^j}{\sqrt{r{\alpha }_j}}{Q}_j(\mathbf{t})\chi \left(\sum _i{t}_i\right),$$

where $\chi$ equals one up to $\beta$ and vanishes beyond $\gamma$; set the other levels to zero. Away from the cutoff, neighboring levels cancel with residual factor $1-\sqrt{(j+1)/k}$. It is at most $r/k$ in magnitude.

The cutoff costs little because under density $Q_j^2$, the sum $S_j={\sum }_i{t}_i$ has mean at most ${\mu }_g$ and variance at most $s_g^2/k$, where $s_g^2$ is the variance under $g^2$. Chebyshev’s inequality controls the region where the cutoff changes. Writing $d=\beta -{\mu }_g>0$, the source obtains

$$1-\frac{s_g^2}{k{d}^2}\le w\le 1,\frac{v}{\lambda }\le \frac{2}{r}+{\left(\frac{r}{k}\right)}^2+\frac{4s_g^2}{k{d}^2}.$$

If $B$ bounds the support of $g$, choose $k\ge 2B/d$. On $S_j\le \beta -B/k$, every cutoff $\chi (S_j+u/k)$ equals one for all $u$ in that support. Thus Chebyshev under $Q_j^2$ controls the deletion integral against $g(u)du$ through a pointwise support bound, without identifying these two measures.

The $2/r$ term is the cost of the two endpoints of the active range of dimensions. Thus $w\to 1$ and $v\to 0$. Choose one finite family with $v/w\le \epsilon$, freeze it, and only then let $X\to \infty$. No estimate uniform in an increasing dimension is needed.

Two different uses of Cauchy–Schwarz

The remaining moment estimates give

$$\mathrm{limsup}{\mathbb{E}}_X(W_X{V}_J^2)\le K_{\lambda },{\mathbb{E}}_X{W}_X^2\le M$$

for large $X$. The detector constant $K_{\lambda }$ is independent of $\epsilon$; $M$ may depend on it. Let $A=\{V_J>0\}$ and $B=\{V_I>0\}$. Weighted Cauchy–Schwarz first gives

$${\mathbb{E}}_X(W_X{1}_A)\ge \frac{({\mathbb{E}}_X{W}_X{V}_J{)}^2}{{\mathbb{E}}_X(W_X{V}_J^2)}.$$

Since $1_B\le V_I$, choose $\epsilon ={\lambda }^2/(2K_{\lambda })$ to leave limiting weighted mass at least $\Delta ={\lambda }^2/(2K_{\lambda })>0$ on $A\setminus B$. Ordinary Cauchy–Schwarz then removes the weight:

$$\frac{|A\setminus B|}{X}\ge \frac{({\mathbb{E}}_X{W}_X{1}_{A\setminus B}{)}^2}{{\mathbb{E}}_X{W}_X^2}\ge \frac{{\Delta }^2}{4M}>0.$$

These inequalities solve different concentration problems. The first prevents prime detection from resting on too little weighted mass; the second prevents that mass from resting on too few integers.

From starts to gaps and prime indices

For each successful start $m$, select the last prime $p$ in $(m,m+h]$. The empty interval $(m+h,m+2h]$ forces its successor beyond $m+2h$, so the gap exceeds $h$. A fixed $p$ is selected at most $h$ times because $p-h\le m<p$. Therefore a positive proportion of successful starts yields $\gg X/\log X$ distinct gaps. Choose $\lambda >C$ so $h>C\log p$ for large $X$.

Finally, to obtain a lower density among the first $N$ prime indices rather than merely a subsequence of dyadic counts, take $X=\lfloor p_N/3\rfloor$. All selected starts lie below $p_N$, and the prime number theorem gives $X/\log X\sim N/3$. This closes the quantifier needed for positive lower natural density.

🏷️ Connections

Bounded prime gaps develops sieve methods for prime clusters. Here the target is a prime next to a vacant interval. Jacobsthal’s function studies the opposite covering question: how long an interval can remain without an integer coprime to a prescribed modulus.

🏷️ Manuscripts and verification scope

• Positive lower density of large prime gaps — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The selected formal statement concerns positive lower density of increases in p_n/n, the corollary rather than the full all-threshold large-gap theorem. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Positive lower density of large prime gaps.