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\newtheorem{theorem}{Theorem}
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\title[A Type I/II estimate to smooth moduli]{A sharper Type I/II estimate for primes
in arithmetic progressions to smooth moduli, and \texorpdfstring{$H_m\ll e^{3.8043\,m}$}{Hm << exp(3.8043m)}}
\author{David Niedbala Giraudin}
\address{ORCID 0009-0009-1526-1178}
\date{3 November 2026}

\begin{document}

\begin{abstract}
In Stadlmann's Type I/II equidistribution estimate for $x^\delta$-smooth moduli, one of the three
conditions, $64\varpi+20\delta+2\sigma<1$, can be replaced by $48\varpi+18\delta+2\sigma<1$. This
weaker condition follows from the second one, $48\varpi+16\delta+4\sigma<1$, whenever
$\delta\le\sigma$. The proof follows Stadlmann's argument and makes two changes: the length of a
block in one Taylor expansion depends on the pair of shifts, and the bounds are summed over the
pairs instead of being multiplied by their number. Combined with Stadlmann's general Harman-sieve
construction, this gives a minorant for the primes with exponent of distribution $1/2+2\varpi$ to
smooth moduli for every $\varpi<7/544$, instead of $\varpi=0.01265$. As a consequence,
$\liminf_{n\to\infty}(p_{n+m}-p_n)\ll_c e^{cm}$ for every $c>3.80428$, and for every $c>3.80423$
if a numerically computed value of the sieve deficit is used. This improves Stadlmann's exponent
$3.8075$.
\end{abstract}

\maketitle

\section{Statement}

We use the notation of Stadlmann~\cite{Sta23}, \S3. Let $\varpi,\delta,\sigma>0$. Let $\alpha,\beta$
be coefficient sequences at scales $M,N$ with $MN\asymp x$ and $N=x^\gamma$, where $\beta$ has the
Siegel--Walfisz property. The moduli are the $q\le x^{1/2+2\varpi}$ dividing $P(x^\delta)$.
Stadlmann proved \cite[Prop.~1]{Sta23} that the equidistribution estimate \cite[(3.28)]{Sta23}
holds for all $\gamma\in[1/2-\sigma,1/2]$ whenever
\begin{equation}\label{eq:S}
72\varpi+24\delta<1,\qquad 48\varpi+16\delta+4\sigma<1,\qquad 64\varpi+20\delta+2\sigma<1.
\end{equation}

\begin{theorem}\label{thm:main}
The conclusion of \cite[Prop.~1]{Sta23} holds when the third condition in \eqref{eq:S} is replaced
by $48\varpi+18\delta+2\sigma<1$. More precisely, in \cite[Lemma~17]{Sta23} the lower bound
$\gamma\ge32\varpi+10\delta+400\eps$ may be replaced by $\gamma\ge24\varpi+9\delta+400\eps$.
\end{theorem}

The new condition follows from the second condition of \eqref{eq:S} whenever $\delta\le\sigma$,
and this is the case in all the applications below.

\begin{remark}\label{rem:scope}
The theorem does not improve the exponent of distribution $1/2+1/40$ of the primes themselves
\cite[Thm.~1]{Sta23}. That deduction goes through the Heath-Brown identity and
\cite[Lemma~1]{Sta23}, which require $\sigma>1/10$. With the second condition of \eqref{eq:S},
$\sigma>1/10$ already gives $80\varpi+\tfrac{80}{3}\delta<1$ \cite[Cor.~2]{Sta23}, so the third
condition is not the binding one there. It is the binding one in the Harman-sieve application:
\cite[Lemma~18(II)]{Sta23} needs $0.40481>32\varpi$. OpenAI extended \cite[Prop.~1]{Sta23}, with the
same three conditions, to triply densely divisible moduli \cite[Prop.~2.18 and \S A.4]{OAI}. Their
proof follows the same lemmas with the same block length, and we expect the same modification to
apply. We have not checked the details and claim nothing for those moduli.
\end{remark}

\section{Proof of Theorem~\ref{thm:main}}

All references in this section are to \cite[\S3]{Sta23}, whose notation we keep:
$z\in Z_5(x)$, $\lambda,\lt\sim\Lambda$, $w_1,w_2$, $m$, $q_0$, $\Delta_1$, $H$, $T(z)$ and the
classes $W_6,W_7,W_8$. We write $s_2=(w_2,m)$ and $\lambda^*=(\lambda,\lt)$. Since $\lambda\sim\Lambda$
we have $|\lambda|\le2|\Lambda|$. We go through Lemmas~7--17 and indicate every change. Everything not
mentioned is used unchanged.

\subsection*{Step 1: a block length depending on the pair (Lemma~10)}
In the proof of Lemma~10, the variable $d$, of length $\Delta_1$, is cut by a smooth partition of
unity into blocks of length $\Delta^*=\min\{N/(|\Lambda|x^{5\eps}),\Delta_1\}$, which does not depend
on $(\lambda,\lt)$. For a fixed pair we use instead
\[
\Delta^*_\lambda:=\min\Bigl\{\frac{\lambda^*}{|\lambda|}\,\frac{N}{x^{5\eps}},\ \Delta_1\Bigr\},
\qquad N_\lambda:=\frac{\lambda^*}{|\lambda|}\,N .
\]
Since $\lambda^*\ge1$ and $|\lambda|\le2|\Lambda|$, we have $\Delta^*\le2\Delta^*_\lambda$. The block length
is used in three places, and each still holds with $\Delta^*_\lambda$:
\begin{enumerate}
\item[(a)] the number of blocks is $\ll\Delta_1/\Delta^*_\lambda$, and the cut-off functions
$\tau_m$ satisfy the same derivative bounds, since $\Delta^*_\lambda\le\Delta_1$;
\item[(b)] the Taylor expansion (3.13) of $\psi^*_N((\lambda/\lambda^*)n_1+F_kd)$ in $d$ needs
$|F_k(d-x_0)|/N\ll x^{-5\eps}$ for $|d-x_0|\ll\Delta^*_\lambda/z_1$. Since $0<F_k\le|\lambda|/\lambda^*$,
\[
\frac{|F_k(d-x_0)|}{N}\le\frac{|\lambda|}{\lambda^*}\cdot\frac{\Delta^*_\lambda}{z_1N}\le\frac{x^{-5\eps}}{z_1};
\]
\item[(c)] in Step~2 of the proof of Lemma~11, the expansion (3.15) needs
\[
\frac{H_kf_4\,\Delta^*_\lambda/(z_1f^*)}{(\lambda^*/|\lambda|)N}\ll x^{-5\eps}.
\]
Since $H_kf_4\le[f_2/f_2^*,f_3/f_3^*]\le w_1$, $f^*\ge1$ and $w_1\mid z_1$, the left side is at
most $(w_1/z_1)x^{-5\eps}\le x^{-5\eps}$.
\end{enumerate}
Thus Lemma~10 holds with $\Delta^*_\lambda$ in place of $\Delta^*$, and Lemma~11 holds with $W_8$ defined
by $\Delta_2\le\Delta^*_\lambda$ instead of $\Delta_2\le\Delta^*$.

\subsection*{Step 2: the length of the \texorpdfstring{$n_1$}{n1}-sum (Lemmas~11 and~14)}
After the substitution $n=(\lambda/\lambda^*)n_1+F_kd$ in Lemma~10, the variable $n_1$ runs over a
coefficient sequence at scale $(\lambda^*/\lambda)N$. In Lemma~11 it becomes $n_3$, at scale
$(\lambda^*/(\lambda w_3))N\le N_\lambda$. The definition of $W_8$ records only $N_2\le N$; we keep
$N_2\le N_\lambda$. Lemma~13 applied as in the proof of Lemma~14 then gives, for this pair,
\begin{equation}\label{eq:L14}
\Sigma_6(z,w_6,k)\ll_\eps\Bigl(\bigl(\tfrac{z_1}{w_1}\bigr)k+(\lambda-\lt)B,m\Bigr)\frac{x^{3\eps}}{q_0}
\,F_\lambda,\qquad
F_\lambda:=\frac{\Delta_1}{\Delta^*_\lambda}\Bigl(\frac{N_\lambda}{m^{1/2}}+m^{1/2}\Bigr)
\Bigl(\frac{\Delta^*_\lambda}{m^{1/2}}+m^{1/2}\Bigr).
\end{equation}
Lemmas~15 and~16 do not involve the block length. The range of $k$, the condition
$(\cdot,m)\le T(z)$ and the sequence $(\xi_k)$ are the same as in \cite{Sta23}. They give
$\Sigma_6(z,w_6,k)\ll_\eps\xi_k\,x^{4\eps}s_2T(z)F_\lambda/q_0$ with $(\xi_k)\in\Xi(z)$. The proof
of Lemma~9 is linear in the target bound: if $\Sigma_6\ll\xi_kI$ for all relevant $k$, then
$\Sigma_5(z)\ll q_0(q_0,\ell)I$. Hence
\begin{equation}\label{eq:S5}
\Sigma_5(z)\ll_\eps(q_0,\ell)\,x^{4\eps}\,s_2\,T(z)\,F_\lambda .
\end{equation}
Expanding $F_\lambda$ gives four terms:
\[
F_\lambda=\frac{\Delta_1N_\lambda}{m}+\frac{\Delta_1}{\Delta^*_\lambda}N_\lambda+\Delta_1
+\frac{\Delta_1}{\Delta^*_\lambda}m .
\]
By $N_\lambda\le N$ and $\Delta^*\le2\Delta^*_\lambda$, the first and the last are at most once and twice
the corresponding terms $\Delta_1N/m$ and $(\Delta_1/\Delta^*)m$ of \cite{Sta23}. For the middle two,
consider the two possible values of $\Delta^*_\lambda$. If $\Delta^*_\lambda=N_\lambda x^{-5\eps}$ then
$(\Delta_1/\Delta^*_\lambda)N_\lambda=x^{5\eps}\Delta_1$, and otherwise
$(\Delta_1/\Delta^*_\lambda)N_\lambda=N_\lambda$. Hence
\begin{equation}\label{eq:mid}
\frac{\Delta_1}{\Delta^*_\lambda}N_\lambda+\Delta_1\le2x^{5\eps}\Delta_1+N_\lambda .
\end{equation}

\subsection*{Step 3: summing over the pairs (Lemma~7)}
In the proof of Lemma~7, $\Sigma_3\ll_\eps\Upsilon_T+\sup_qx^\eps\Upsilon_S(q)$, and the bound
$\Upsilon_S(q)\ll_\eps x^{-48\eps}N^2$, which includes the diagonal terms, gives the term
$O(x^{-47\eps}N^2)$. Here $\Upsilon_T$ is a sum over pairs $(y,\widetilde y)$ of the inner sums
$\Sigma_4(z_1,z_2,z_3,(y,\widetilde y))$. Stadlmann bounds it by the number of pairs,
$\ll\mathcal P:=\max\{x^{\delta+10\eps}H^3,H^4\}/s_2$, times the supremum. The condition
$(\cdot,m)\le T$ in $\Sigma_4$ does not depend on the pair. So, by the triangle inequality, we may
instead sum bounds that depend on the pair:
$\Upsilon_T\le\sum_{(y,\widetilde y)}|\Sigma_4(\cdot,(y,\widetilde y))|$. The terms $\Upsilon_S(q)$ are not
changed. By the proof of Lemma~8, each $\Sigma_4(\cdot,(y,\widetilde y))$ is zero or equals $\Sigma_5(z)$ for a $z\in Z_5$ with $y=w_1\lambda$,
$\widetilde y=w_1\lt$ and $\Lambda=Y/w_1$. In particular
\[
\frac{\lambda^*}{|\lambda|}=\frac{(y,\widetilde y)}{|y|}.
\]
By \eqref{eq:S5}, the target $\Sigma_3\ll q_0^2(q_0,\ell)^2N^2x^{-27\eps}$ of Lemma~7 (hence of
Lemma~8) follows from
\begin{equation}\label{eq:target}
s_2\,T(z)\sum_{(y,\widetilde y)}F_\lambda\ll q_0^2N^2x^{-31\eps}.
\end{equation}
Condition~(3.29) of \cite{Sta23} is the same inequality with $\sum F_\lambda$ replaced by
$\mathcal P\,F$, where $F$ is the expression with $\Delta^*$ and $N$. By Step~2, the contribution of
the first and last terms of $F_\lambda$ is at most twice Stadlmann's. It is therefore covered by
(3.30) and (3.32) of \cite{Sta23}, which give the conditions
$\gamma\le1/2-4\varpi-2\delta-50\eps$ and $\gamma\ge1/4+12\varpi+4\delta+100\eps$. It remains to
replace (3.31), which concerns the two middle terms and needed $\gamma\ge32\varpi+10\delta+400\eps$.
By \eqref{eq:mid},
\[
\sum_{(y,\widetilde y)}\Bigl(\frac{\Delta_1}{\Delta^*_\lambda}N_\lambda+\Delta_1\Bigr)
\le2x^{5\eps}\Delta_1\,\#\{(y,\widetilde y)\}+N\sum_{(y,\widetilde y)}\frac{(y,\widetilde y)}{|y|}.
\]
For fixed $y\neq0$, the integers $\widetilde y\sim Y$ lie in an interval of length $|Y|$, and
$|y|\le2|Y|$. Hence
\[
\sum_{\widetilde y\sim Y}(y,\widetilde y)\le\sum_{e\mid y}e\Bigl(\frac{|Y|}{e}+1\Bigr)
\le\tau(y)|Y|+\sigma(y)\le3\tau(y)|Y| .
\]
Since $|y|\ge|Y|$, it follows that
$\sum_{(y,\widetilde y)}(y,\widetilde y)/|y|\ll x^{\eps}\,\mathcal Y$, where $\mathcal Y$ is the number of
admissible $y$. By the proof of Lemma~7, $\mathcal Y\ll\min\{H^2/s_2+H,\ |Y|/s_2+1\}$ and
$|Y|\ll x^{\delta+5\eps}H^2/(v_1,v_2)$. We also use
\[
T(z)\le\max\{s_2^{-1},H^{-1}\}\frac{x^{\delta+100\eps}H^2N}{(v_1,v_2)\Delta_1},\qquad
\Delta_1\gg\frac{N}{x^{\delta+55\eps}H^2},\qquad
H\ll\frac{x^{4\varpi+\delta+7\eps}}{q_0}.
\]

\emph{First contribution.} We have $\max\{1,s_2/H\}\mathcal P\le\max\{x^{\delta+10\eps}H^3,H^4\}\le
x^{\delta+10\eps}H^4$. Hence
\[
s_2T(z)\cdot x^{5\eps}\Delta_1\,\mathcal P\le x^{2\delta+115\eps}H^6N .
\]
This is $\ll q_0^2N^2x^{-31\eps}$ provided $x^{2\delta+146\eps}H^6\ll q_0^2N$, which follows from
$\gamma\ge24\varpi+8\delta+190\eps$.

\emph{Second contribution.} We claim that $\max\{1,s_2/H\}\,\mathcal Y\ll x^{\delta+5\eps}H^2$. If
$s_2\le H$, then $\mathcal Y\le2H^2$. If $s_2>H$, we use $s_2\le2|Y|$: indeed $s_2\mid w_2\mid y$ for
every admissible $y\ne0$, and the sum is empty otherwise. Then
$(s_2/H)(|Y|/s_2+1)\le3|Y|/H\ll x^{\delta+5\eps}H$. Therefore
\[
s_2T(z)\cdot x^{\eps}N\mathcal Y\ll x^{2\delta+106\eps}\frac{H^4N^2}{\Delta_1}\ll x^{3\delta+161\eps}H^6N ,
\]
which is $\ll q_0^2N^2x^{-31\eps}$ provided $x^{3\delta+192\eps}H^6\ll q_0^2N$. This follows from
$\gamma\ge24\varpi+9\delta+240\eps$.

\subsection*{Step 4: conclusion (Lemma~17 and Proposition~1)}
Lemma~17 therefore holds with $\max\{1/4+12\varpi+4\delta+100\eps,\ 24\varpi+9\delta+400\eps\}\le\gamma\le
1/2-4\varpi-2\delta-50\eps$. The final step of the proof of Proposition~1 is unchanged. It covers
$\gamma$ near $1/2$ by \cite[Thm.~2.8(ii),(iv)]{PM8a} when $68\varpi+14\delta<1$ and
$72\varpi+24\delta+200\eps<1$. Hence the range $\gamma\in[1/2-\sigma,1/2]$ is covered as soon as
$1/4+12\varpi+4\delta<1/2-\sigma$, $24\varpi+9\delta<1/2-\sigma$ and $72\varpi+24\delta<1$, with $\eps$
small enough. These are the conditions of Theorem~\ref{thm:main}. \qed

\section{Consequences}

We use the general Harman-sieve construction of Stadlmann \cite[Def.~9 and Prop.~2]{Sta26}, with
parameters $\xi_1,\xi_2,\xi_3$. It is the construction of \cite[\S4]{Sta23}, with the numerical
values $0.33856$, $0.40481$, $0.59519$, $0.19$, $0.405$, $0.595$ replaced by $\xi_1$, $\xi_2$,
$1-\xi_2$, $1-2\xi_3$, $\xi_3$, $1-\xi_3$. Its proof requires the five inequalities
\begin{equation}\label{eq:five}
2\xi_1+3\xi_2<2,\qquad\xi_2\le\xi_3,\qquad\xi_1+9\xi_2<4,\qquad2\xi_1+\xi_2>1,\qquad17\xi_2<7 .
\end{equation}
Its input is equidistribution, on the family of moduli under consideration, for three types of
convolutions:
\begin{itemize}
\item[(I)] smooth factors of length $\ge x^{\xi_1}$;
\item[(II)] bilinear forms with both factors Siegel--Walfisz and one length in
$[x^{\xi_2},x^{1-\xi_2}]$;
\item[(III)] three smooth factors of lengths in $[x^{1-2\xi_3},x^{\xi_3}]$ with pairwise products
$\ge x^{1-\xi_3}$.
\end{itemize}
In \cite{Sta26} it is stated for the moduli $Q^*$ of that paper; its proof, like that of
\cite[\S4]{Sta23} where the moduli are $x^\delta$-smooth, uses these inputs on the family of moduli
concerned. In our application the coefficients of type (I) with lengths in $[x^{\xi_2},x^{1-\xi_2}]$
are treated by (II), which uses that they satisfy the Siegel--Walfisz property \cite[Lemma~A.8]{OAI}. We include the corrections of
OpenAI \cite[footnotes~2--4 and Lemmas~A.8--A.9]{OAI}, written there for fixed numerical values.
The numerical inequalities in their arguments reduce to \eqref{eq:five}, $6(1-2\xi_2)>1$,
$\xi_2>2/5$ and a level of distribution at most $x^{0.53}$; for instance their two grouping gaps are
$1-\xi_1-3\xi_2/2$ and $4-\xi_1-9\xi_2$. The
minorant $\rho$ has mass $1-c_1(\xi_2)$, where $c_1(\xi_2)=I_1+I_2$ is the explicit sum of two
four-dimensional integrals in \cite[Prop.~2]{Sta26}. Every constraint in these integrals becomes
stricter when $\xi_2$ decreases, so $c_1$ is nondecreasing in $\xi_2$.

\begin{corollary}\label{cor:minorant}
Let $0<\varpi<7/544$ and let $\delta>0$ be small enough. Take $\xi_2=\xi_3$ with
\[
\max\Bigl\{\tfrac14+12\varpi+4\delta,\ \tfrac25\Bigr\}<\xi_2<\min\Bigl\{\tfrac49-\tfrac{28}{9}\varpi-\tfrac29\delta,\
\tfrac{55}{136}\Bigr\},
\]
and $\xi_1$ with
$\max\{\tfrac14+7\varpi+2\delta,\tfrac{1-\xi_2}2\}<\xi_1<\min\{1-\tfrac32\xi_2,\ 4-9\xi_2\}$.
These intervals are nonempty. Then the minorant $\rho$ of \cite[Prop.~2]{Sta26} with these
parameters has the following properties:
\begin{itemize}
\item $\rho\le\mathbf 1_{\mathbb P}$, and every prime factor of an $n$ with $\rho(n)\ne0$ is at least
$x^{1-2\xi_2}$;
\item $\rho$ has exponent of distribution $1/2+2\varpi$ to $x^\delta$-smooth squarefree moduli;
\item $\rho$ has mass $1-c_1(\xi_2)$ with $c_1(\xi_2)\le c_1(55/136)$.
\end{itemize}
\end{corollary}

\begin{proof}
\emph{The intervals are nonempty.} We have $1/4+12\varpi+4\delta<4/9-28\varpi/9-2\delta/9$ if and
only if $544\varpi+152\delta<7$; the two bounds meet at $\varpi=7/544$, $\xi_2=55/136$. Also
$1/4+12\varpi+4\delta<55/136$ for $\varpi<7/544$ and $\delta$ small, and $2/5<\min\{4/9-28\varpi/9-2\delta/9,
55/136\}$ since $\varpi<1/70$. So the interval for $\xi_2$ is nonempty. Since $2/5<\xi_2<55/136$, we
have $(1-\xi_2)/2<3/10$, $1-3\xi_2/2>0.393$ and $4-9\xi_2>49/136>0.3602$, while
$1/4+7\varpi+2\delta<0.3401$; so the interval for $\xi_1$ is nonempty. The program
\texttt{check\_parameters.py} checks all the inequalities used below in exact rational
arithmetic, at sample values of $(\varpi,\delta)$ (including values close to $7/544$ and small
values of $\varpi$) and at points close to the ends of the intervals for $\xi_1$ and $\xi_2$. In
general they follow from the displayed choices of $\xi_1,\xi_2$ by the elementary verifications
below.

\emph{Type II.} We use Theorem~\ref{thm:main} with $\sigma=1/2-\xi_2+\tau$ for a small $\tau>0$
(this also absorbs the fixed slack $\epsilon$ of \cite[Def.~9]{Sta26}, which we take small depending
on $\varpi$); both factors are Siegel--Walfisz, so they may be interchanged. Its conditions read
$\xi_2-\tau>1/4+12\varpi+4\delta$, $\xi_2-\tau>24\varpi+9\delta$ and $72\varpi+24\delta<1$, all satisfied
for $\tau$ small. With \eqref{eq:S} instead, one would also need $\xi_2>32\varpi+10\delta$; together
with the Type III bound $\xi_2<4/9-28\varpi/9$ this forces $\varpi<1/79$ (as $\delta\to0$), and
Stadlmann takes $\varpi=0.01265<1/79$.

\emph{Type III.} We use \cite[Thm.~2.8(v)]{PM8a} with $\sigma_3$ strictly between
$1/18+28\varpi/9+2\delta/9$ and $1/2-\xi_3$, the slack $\epsilon$ in the ranges of (III) being smaller
than these gaps, exactly as in \cite[Lemma~18(III)]{Sta23} and
\cite[proof of Prop.~2.22]{OAI}.

\emph{Type I.} Smooth factors of length between $x^{\xi_1}$ and $x^{1/2}$ are covered by
\cite[Lemma~2.20]{OAI}, which needs $\xi_1>1/4+7\varpi+2\delta$ and holds for singly densely divisible
moduli, hence for $x^\delta$-smooth squarefree moduli. The coefficients in the decomposition satisfy
the Siegel--Walfisz property \cite[Lemma~A.8]{OAI}. So lengths in $[x^{\xi_2},x^{1-\xi_2}]$ are
covered by Type II, and lengths above $x^{1-\xi_2}>x^{1/2+2\varpi}$ by \cite[(2.18)]{OAI}. This is
the argument of \cite[proof of Prop.~2.22]{OAI}, with $(a_*,b_*,c_*)=(\xi_2,1-\xi_2,\xi_1)$.

\emph{Construction.} The inequalities \eqref{eq:five} hold. Indeed,
$\xi_1+9\xi_2<4$, $2\xi_1+3\xi_2<2$ and $2\xi_1+\xi_2>1$ are imposed, $\xi_2=\xi_3$, and
$17\xi_2<17\cdot55/136=6.875<7$. Moreover $6(1-2\xi_2)>1$, $\xi_2>2/5$, and the level is
$1/2+2\varpi<0.53$. The three properties then follow from the proof of \cite[Prop.~2]{Sta26} and of
\cite[\S4]{Sta23}, with the corrections of \cite[\S A.6]{OAI}, run on the family of $x^\delta$-smooth
squarefree moduli with the inputs above. We have checked that the numerical inequalities used in
these proofs reduce to those listed; we have not rewritten the whole construction. Since
$\xi_2<55/136$, the monotonicity of $c_1$ gives $c_1(\xi_2)\le c_1(55/136)$.
\end{proof}

The value $c_1(55/136)$ is computed by the program \texttt{deficit\_c1.py}, in two ways. Nested
Gauss--Legendre quadrature, with each range split where its limits change formula, gives
$6.604\cdot10^{-6}$. A Monte Carlo integration over the regions written directly from the
summation conditions gives $(6.628\pm0.020)\cdot10^{-6}$. Thus $c_1(55/136)<6.7\cdot10^{-6}$; this
is a numerical, not a certified, value. As a check, the same program gives
$c_1(0.40481)\approx9.33\cdot10^{-6}$, in agreement with Stadlmann's published bound
$c_1(0.40481)\le2\cdot10^{-5}$ \cite[Prop.~2]{Sta23}.

For comparison, the minorant of \cite[Prop.~2]{Sta23} and \cite[Prop.~2.22]{OAI} has
$\varpi=253/20000=0.01265$. Here $7/544=0.0128676\ldots$, and the exponent of distribution tends
to $1/2+7/272=0.525735\ldots$.

\begin{corollary}\label{cor:Hm}
For every $c>3.80423$, $\liminf_{n\to\infty}(p_{n+m}-p_n)\ll_c e^{cm}$. Using only Stadlmann's
bound $c_1(0.40481)\le2\cdot10^{-5}$ in place of the numerical value of $c_1(55/136)$, the same
holds for every $c>3.80428$.
\end{corollary}

\begin{proof}
By the lemma of Baker and Irving \cite[Lemma~1]{BI}, \cite[Lemma~2]{Sta23}, a minorant with the
properties of Corollary~\ref{cor:minorant} gives
$H_m\ll_\epsilon\exp\bigl(2(1+\epsilon)m/(\theta(1-c_1))\bigr)$ with $\theta=1/2+2\varpi$, that is,
the exponent $(1+\epsilon)/((1/4+\varpi)(1-c_1))$. We let $\varpi\to7/544$ and use
$c_1(\xi_2)\le c_1(55/136)$, which holds because $\xi_2<55/136$. Since $1/(1/4+7/544)=544/143=3.8041958\ldots$, the exponent can be taken
arbitrarily close to $544/(143(1-c_1(55/136)))$. With $c_1(55/136)<6.7\cdot10^{-6}$ this is
$<3.804222$. Since $\xi_2<55/136<0.40481$, monotonicity also gives $c_1(\xi_2)\le c_1(0.40481)\le
2\cdot10^{-5}$, and then the bound is $544/(143(1-2\cdot10^{-5}))<3.804272$.
\end{proof}

\subsection*{Files}
\texttt{anc/check\_parameters.py} checks the inequalities of Corollary~\ref{cor:minorant} and the
two constants of Corollary~\ref{cor:Hm} in exact arithmetic. \texttt{anc/deficit\_c1.py} computes
$c_1$. The outputs are in the corresponding \texttt{.log} files.

\subsection*{Acknowledgement}
Technical assistance: Claude (Anthropic).

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\url{https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf}.
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