\section{Prime dilation and nonvanishing}\label{sec:dilation}

The forms now tend to zero after denominator clearing. We must
ensure that the final rational combination is nonzero infinitely
often. At prime indices, its constant coefficient will survive
a scaling modulo the prime while all beta coefficients vanish.
This is the principle of Lai and Sprang's auxiliary-prime
criterion~\cite[Lemma~2.3]{LS25}; we adapt the dilation calculation
of~\cite[Section~6]{Tcount} to the present odd grids.

Put
\[
 \tau_0=a-E>0,\qquad \tau=\tau_0+k>k.
\]
These exponents describe scaling at prime indices; the common
denominator exponent remains $a+k$.

\begin{proposition}\label{prop:dilation}
For every sufficiently large odd prime $p$,
\begin{align}
 p^\tau\rho_\nu(p)&\in p^\nu\Zp,\label{eq:rho-dilation}\\
 p^\tau q_b(p)&\equiv\ch(p)q_b(1)\pmod p
                    \qquad(b=1,3,\ldots,M).\label{eq:q-dilation}
\end{align}
Here $\Zp$ consists of rationals with denominator prime to $p$.
\end{proposition}

\subsection{The rational-function congruence}

The factors divisible by $p$ will reproduce the seed at $n=1$
after rescaling; the remaining factors contribute units modulo $p$.
For a single grid, $B=1$ and $\Theta=\{1\}$, so
\[
 P_p(pt)=\prod_{j=0}^{Lp-1}\bigl(p(t-2r)+2j+1\bigr).
\]
For an odd prime $p$, the selected indices are exactly
$j=(p-1)/2+\ell p$, $0\le\ell<L$. At these indices,
\[
 p(t-2r)+2j+1=p(t-2r+2\ell+1),
\]
so their product is $p^LP_1(t)$. With several grids, division
by $p$ can change the fraction class. The set $\Theta$ contains
all the classes needed to keep the same mechanism valid.

Now take $p>M$ and larger than the fixed denominator primes.
In a block of $P_p(pt)$ with class $v/d$, a factor is selected
by $p$ when $2dj+v\equiv0\pmod p$. After division by $p$,
its new class $v'/d$ satisfies
\[
 pv'\equiv v\pmod{2d}.
\]
Multiplication by $p$ permutes the odd reduced classes modulo
$2d$. Each block has $L$ selected factors, and together they
give $p^{DL}P_1(t)$. The unselected factors in each block cover
every nonzero residue $L$ times. Wilson's theorem therefore
gives their total unit as $(-1)^{DL}=-1$.

The normalization $\kappa_p$ has valuation $a-mDL$.
Using Wilson's theorem for $(p-1)!$ and Fermat's theorem for
the fixed factors, its unit is
\[
 \kappa_p/p^{a-mDL}\equiv(-1)^{a-mDL}\kappa_1=-\kappa_1\pmod p.
\]
Raising the numerator to odd $m$ cancels the two minus signs.
The selected denominator indices are $h=0,p$, contributing
$p^{2a}[t(t+2)]^a$; the remaining denominator unit is
$(-1)^a=1$. The net valuation is $-a$, so
\begin{equation}\label{eq:T-dilation}
 p^aR_p(pt)\equiv R_1(t),\qquad
 p^{a-E}T_p(pt)\equiv T_1(t)\pmod p.
\end{equation}
The second congruence uses $W_p(pt)=p^E W_1(t)$.

\subsection{Local coefficient extraction}

To extract coefficients, reduction must commute with the local
series expansion. At a selected pole put $t=-2y+z$, where
$y=0,1$. Removing the pole gives
\[
 z^a p^{a-E}T_p(p(-2y+z))\in\Zp[[z]].
\]
After the displayed powers are canceled, every remaining
denominator factor has a unit constant term. Its inverse has
integral series coefficients in $\Zp$, so reduction commutes
with Taylor extraction.

The order-$i$ term from the selected pole is
\[
 \frac{c_{i,py}(p)}{(p(-2y+z)+2py)^i}
                  =p^{-i}c_{i,py}(p)z^{-i}.
\]
After multiplication by $z^ap^{a-E}$, its coefficient of
$z^{a-i}$ is $p^{a-E-i}c_{i,py}(p)$.
Terms from other poles are analytic at $z=0$; after multiplication
by $z^a$ they start at degree $a$ and cannot contribute, since
$i\ge1$. Comparing coefficients in \eqref{eq:T-dilation} gives
\[
 p^{a-E-i}c_{i,py}(p)\equiv c_{i,y}(1)\pmod p.
\]
Now multiply both sides by the integer $\binom{i+k-1}{k}$.
With $\nu=i+k$ and $\tau=a-E+k$, we have
$a-E-i=\tau-\nu$. Thus
\begin{equation}\label{eq:selected-coefficients}
 p^{\tau-\nu}A_{\nu,py}(p)\equiv A_{\nu,y}(1)\pmod p.
\end{equation}

At a nonselected pole $-2h$, $1\le h\le p-1$, all other
denominator factors are units. The numerator contains $mDL$
constant factors divisible by $p$, so its degree-$j$
coefficient has valuation at least $mDL-j$.
Multiplication by $\kappa_p$ adds $a-mDL$. The remaining denominator
series has integral coefficients, so the order-$i$ coefficient
of $R_p$ has valuation at least $i$.
Multiplication by the integer polynomial $W_p$ preserves this
bound: its degree-$j$ Taylor coefficient multiplies an
order-$(i+j)$ coefficient. Differentiation gives
\begin{equation}\label{eq:other-coefficients}
 v_p(A_{\nu,h}(p))\ge\nu-k\qquad(1\le h\le p-1).
\end{equation}
The estimate for a numerator coefficient remains valid when
$j>mDL$, when its lower bound is negative.
Equations~\eqref{eq:selected-coefficients}--\eqref{eq:other-coefficients}
imply \eqref{eq:rho-dilation}.

\subsection{The surviving rational constant}

Multiply \eqref{eq:q} at $n=p$ by $p^\tau$ and take $p>2M$.
Every finite-sum denominator $j<2bp$ is less than $p^2$.
At nonselected poles, even the terms with $p\mid j$ have
valuation at least $\tau-k>0$.
At selected poles, \eqref{eq:selected-coefficients} gives valuation
at least $\nu>0$ whenever $p\nmid j$.

Only $h=p$ and $j=pj'$ survive, with $1\le j'<2b$ odd.
The pole $h=0$ has an empty finite sum. Since $p$ is odd,
$(-1)^p=-1$ and $\ch(pj')=\ch(p)\ch(j')$.
Using \eqref{eq:selected-coefficients}, the surviving sum is
$\ch(p)$ times the seed constant formula.
This proves \eqref{eq:q-dilation} and Proposition~\ref{prop:dilation}.