\section{The dimension estimate and the choice of parameters}
\label{sec:conclusion}

We can now assemble the argument. Section~\ref{sec:arithmetic} made the
forms integral, Section~\ref{sec:estimates} bounded their coefficients
and values, and Section~\ref{sec:rank} supplied the rank statement.
We first apply the criterion with all auxiliary parameters fixed, then
choose them as the number of beta values increases.

For each admissible $n$, form the integer matrix with rows the coefficients
of $\Lambda_{n,p,k}$ in
\[
 1,\beta(2),\ldots,\beta(2\lfloor(a+h)/2\rfloor).
\]
It satisfies \eqref{eq:weak-rank}.
Indeed, a rational vector annihilating all these rows lifts to one for
the rows $Q_{k,i}^{[p]}(-1)$ by putting zero in every odd coordinate and
multiplying coordinate $i/2$ by $-2^{i+1}$ for even $i\geq2$.
Its constant coordinate is unchanged.
The row multipliers $\Dt_{d,k-1}/(k-1)!$ are nonzero, so
Proposition~\ref{prop:rank} forces that constant coordinate to be zero.

If $\log\alpha<0$, Proposition~\ref{prop:criterion} gives
\[
 \dim_\Q\Span_\Q\{1,\beta(2),\ldots,
                      \beta(2\lfloor(a+h)/2\rfloor)\}
       \geq1-\frac{\log\alpha}{\log B}.
\]
Removing $1$ loses at most one dimension. Thus
\begin{equation}\label{eq:parameter-bound}
 D(2\lfloor(a+h)/2\rfloor)\geq-\frac{\log\alpha}{\log B}.
\end{equation}

We choose the parameters in the order of the constraints they must meet.
First fix $r\geq2$, $\kappa$, $\omega$ and a proportion $0<h/a\leq1$
such that
\[
 2r<\kappa<\omega,\qquad \frac{h}{a}(\kappa-2r)>1.
\]
The first condition gives a nonempty range of derivative rows with
summable tails; the second ensures the rank condition.
Next choose $\Omega$ so that its coefficient cost $\Omega^2/a$ has
the same order as the denominator cost $\log a$.
Thus $\Omega$ has order $\sqrt{a\log a}$, and its decay contribution
$\Omega\log r$ dominates both costs. These choices leave $\log B$
of order $\log a$ and $-\log\alpha$ of order $\sqrt{a\log a}$.

For the numerical constants, use the choice of \cite[Section~4.6]{F26}.
Let $a$ tend to infinity through positive multiples of $25$, and set
\begin{equation}\label{eq:parameters}
 h=\frac{9a}{25},\quad
 r=\frac{39}{10},\quad
 \kappa=\frac{529}{50},\quad
 \omega=12,\quad
 \Omega=\left\lfloor r\sqrt{a\log a}\right\rfloor.
\end{equation}
For sufficiently large $a$ these satisfy
$a>\Omega\geq\omega>\kappa>2r$ and $0\leq h\leq a$.
The rank condition has a small strict margin, visible in its slope:
\[
 \frac{h}{a}(\kappa-2r)
   =\frac9{25}\left(\frac{529}{50}-2\cdot\frac{39}{10}\right)
   =\frac{1251}{1250}>1.
\]
Take $n$ through sufficiently large multiples of $50$.
For each fixed $a$, all needed integer conditions and the weak rank
statement then hold.

As $a\to\infty$, the $\Omega^2$ term supplies the leading part of
$\log\chi$, and $\kappa\Ht_{a+h}\sim\kappa\log a$ supplies the
denominator contribution to $\log B$. The term $-\Omega\log r$
dominates $\log\alpha$. Equations \eqref{eq:coeff-small}, \eqref{eq:B}
and \eqref{eq:alpha} therefore give
\begin{align}
 \log\chi&\sim\frac{r^2\log r}{2}\log a,\notag\\
 \log B&\sim\left(\frac{r^2\log r}{2}+\kappa\right)\log a,
                                                   \label{eq:rates-asymptotic}\\
 \log\alpha&\sim-r\log r\sqrt{a\log a}.\notag
\end{align}
In particular, $\log\alpha<0$ for sufficiently large $a$.
With $s=a+h=34a/25$, \eqref{eq:parameter-bound} implies
\begin{equation}\label{eq:cstar}
 \liminf_{\substack{s\to\infty\\34\mid s}}
   \frac{D(s)}{\sqrt{s/\log s}}
 \geq
 c_*:=\frac{r\log r}
   {\left(r^2\log r/2+\kappa\right)\sqrt{34/25}}
 =0.2174562348\ldots .
\end{equation}
The strict inequality $c_*>0.21$ requires little numerical precision:
$1.36<\log(3.9)$ and $\sqrt{1.36}<1.167$ already suffice,
since $x/(r^2x/2+\kappa)$ increases with $x>0$.

Finally $D(s)$ is nondecreasing, and successive multiples of $34$ have
bounded gaps. For an arbitrary large even $s$, apply
\eqref{eq:cstar} to $34\lfloor s/34\rfloor$.
The quotient of the corresponding factors $\sqrt{s/\log s}$ tends to $1$.
The strict slack between $c_*$ and $0.21$ proves
Theorem~\ref{thm:main}.

\begin{remark}
The limiting assertion \eqref{eq:cstar} extends to all even $s$ in the
same way. We retain the simpler constant $0.21$ in the theorem.
No explicit threshold for \emph{sufficiently large} is asserted;
in particular, the multiplicity constant has not been tracked.
\end{remark}

\section*{Acknowledgements and preparation}

The proof uses Fischler's rational-function construction, refined
linear-independence criterion and multiplicity method. The numerical
parameters and the organization of the rank argument are taken from his
proof, with the adaptations identified in the text.
Mistiaen's explanation of the conductor restriction suggested the
evaluation-point question addressed here.

This research note was prepared with OpenAI Codex, using Astra-6-Max for
the exposition revisions. An independent proof audit by a second model,
reported on 3 November 2026, found no substantive mathematical gap in the
19-page revision. The audit checked the denominator and rank arguments,
the application of Fischler's multiplicity estimate and the final bound;
the supplied and additional computational checks were reported to pass.
The audit used the cited results and was not formal verification.