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The result is actually false, for $m=6$. (One can bring it down to $m=3$ with a bit of effort.)

Let $n$ be arbitrarily large, and $\phi_n(\vec q)$ be a Jankov–De Jongh frame formula of $\def\p#1{\langle#1\rangle}\def\C{\mathcal C}\C_n$, so that for any frame $F$, $$F\nvDash\phi_n\iff\text{$\C_n$ is a p-morphic image of a generated subframe of $F$}.$$ Let $V$ be the valuation in $\C_n$ to the six variables $p_{(i,j)}$, $(i,j)\in\C_n$, $i\le1$, such that $$V(p_x)=\{y\in\C_n:y\nleq x\}.$$ We define a formula $\alpha_x(\vec p)$ for each $x\in\C_n$ by induction on $i$: $$\alpha_{(i,j)}=\begin{cases} p_{(i,j)}&i\le1,\\ \alpha_{(i-1,j)}\to\bigvee_{j'\ne j}\alpha_{(i-1,j')}&2\le i\le n-2,\\ \bigvee_{j'}\alpha_{(i-1,j')}&i=n-1. \end{cases}$$ Then we see by induction on $i$ that for all $x,y\in\C_n$, $$\p{\C_n,V},y\models\alpha_x\iff y\nleq x.$$ It follows that if $U$ is an arbitrary upper subset of $\C_n$, then $$y\in U\iff\p{\C_n,V},y\models\beta_U,$$ where $$\beta_U=\bigwedge_{x\notin U}\alpha_x.$$

Since $\C_n$ is a p-morphic image of itself, there is a valuation $W$ to the variables $\vec q$ such that $\p{\C_n,W}\nvDash\phi_n$. Let $\sigma$ be the substitution $$\sigma(q_i)=\beta_{W(q_i)}.$$ Then $$\p{\C_n,W},x\models\chi\iff\p{\C_n,V},x\models\sigma(\chi)$$ for all $x\in\C_n$ and all formulas $\chi(\vec q)$.

Thus, $\psi_n=\sigma(\phi_n)$ is a formula in the six variables $\vec p$, and by construction, $\C_n\nvDash\psi_n$ under $V$. However, for all $s\le n-2$, we have $\C_n(s)\models\phi_n$, and a fortiori $\C_n(s)\models\psi_n$, as $\C_n$ is not a p-morphic image of a generated subframe of $\C_n(s)$ (or of any other frame of size strictly smaller than $|\C_n|$, for that matter).

To reduce the required number of variables to three, consider in place of $V$ a valuation that makes one variable true in $\{(0,0),(0,1),(1,0),(1,1)\}$, one in $\{(0,0),(0,1),(1,0),(1,2)\}$, and one in $\{(0,0)\}$. Show that you can still define suitable formulas $\alpha_x$.

The result is actually false, for $m=6$. (One can bring it down to $m=2$ with a bit of effort.)

Let $n$ be arbitrarily large, and $\phi_n(\vec q)$ be a Jankov–De Jongh frame formula of $\def\p#1{\langle#1\rangle}\def\C{\mathcal C}\C_n$, so that for any frame $F$, $$F\nvDash\phi_n\iff\text{$\C_n$ is a p-morphic image of a generated subframe of $F$}.$$ Let $V$ be the valuation in $\C_n$ to the six variables $p_{(i,j)}$, $(i,j)\in\C_n$, $i\le1$, such that $$V(p_x)=\{y\in\C_n:y\nleq x\}.$$ We define a formula $\alpha_x(\vec p)$ for each $x\in\C_n$ by induction on $i$: $$\alpha_{(i,j)}=\begin{cases} p_{(i,j)}&i\le1,\\ \alpha_{(i-1,j)}\to\bigvee_{j'\ne j}\alpha_{(i-1,j')}&2\le i\le n-2,\\ \bigvee_{j'}\alpha_{(i-1,j')}&i=n-1. \end{cases}$$ Then we see by induction on $i$ that for all $x,y\in\C_n$, $$\p{\C_n,V},y\models\alpha_x\iff y\nleq x.$$ It follows that if $U$ is an arbitrary upper subset of $\C_n$, then $$y\in U\iff\p{\C_n,V},y\models\beta_U,$$ where $$\beta_U=\bigwedge_{x\notin U}\alpha_x.$$

Since $\C_n$ is a p-morphic image of itself, there is a valuation $W$ to the variables $\vec q$ such that $\p{\C_n,W}\nvDash\phi_n$. Let $\sigma$ be the substitution $$\sigma(q_i)=\beta_{W(q_i)}.$$ Then $$\p{\C_n,W},x\models\chi\iff\p{\C_n,V},x\models\sigma(\chi)$$ for all $x\in\C_n$ and all formulas $\chi(\vec q)$.

Thus, $\psi_n=\sigma(\phi_n)$ is a formula in the six variables $\vec p$, and by construction, $\C_n\nvDash\psi_n$ under $V$. However, for all $s\le n-2$, we have $\C_n(s)\models\phi_n$, and a fortiori $\C_n(s)\models\psi_n$, as $\C_n$ is not a p-morphic image of a generated subframe of $\C_n(s)$ (or of any other frame of size strictly smaller than $|\C_n|$, for that matter).

To reduce $m$ from $6$ to $3$, replace $V$ with a valuation of $\{p_0,p_1,p_2\}$ such that $V(p_0)=\{(0,0),(0,1),(1,0),(1,1)\}$, $V(p_1)=\{(0,0),(0,1),(1,0),(1,2)\}$, and $V(p_2)=\{(0,0)\}$. Show that you can still define suitable formulas $\alpha_x$.

To reduce it further to $m=2$, drop $p_2$, and show that you can define every upset $U\subseteq\C_n$ that does not differentiate the points $(0,0)$, $(0,1)$, and $(1,0)$. This allows you to carry out the argument above with $\phi_n$ replaced with the frame formula of the frame $\C'_n$ obtained from $\C_n$ by identifying $(0,0)$, $(0,1)$, and $(1,0)$. Note that $\C'_n$ still has strictly larger cardinality than $\C(s)$.

Finally, it is easy to show that the result holds for $m=1$, as there are only a constant number of formulas in one variable (up to equivalence) not valid in any $\C_n$.

The result is actually false, for $m=6$. (One can bring it down to $m=3$ with a bit of effort.)

Let $n$ be arbitrarily large, and $\phi_n(\vec q)$ be a Jankov–De Jongh frame formula of $\def\p#1{\langle#1\rangle}\def\C{\mathcal C}\C_n$, so that for any frame $F$, $$F\nvDash\phi_n\iff\text{$\C_n$ is a p-morphic image of a generated subframe of $F$}.$$ Let $V$ be the valuation in $\C_n$ to the six variables $p_{(i,j)}$, $(i,j)\in\C_n$, $i\le1$, such that $$V(p_x)=\{y\in\C_n:y\nleq x\}.$$ We define a formula $\alpha_x(\vec p)$ for each $x\in\C_n$ by induction on $i$: $$\alpha_{(i,j)}=\begin{cases} p_{(i,j)}&i\le1,\\ \alpha_{(i-1,j)}\to\bigvee_{j'\ne j}\alpha_{(i-1,j')}&2\le i\le n-2,\\ \bigvee_{j'}\alpha_{(i-1,j')}&i=n-1. \end{cases}$$ Then we see by induction on $i$ that for all $x,y\in\C_n$, $$\p{\C_n,V},y\models\alpha_x\iff y\nleq x.$$ It follows that if $U$ is an arbitrary upper subset of $\C_n$, then $$y\in U\iff\p{\C_n,V},y\models\beta_U,$$ where $$\beta_U=\bigwedge_{x\notin U}\alpha_x.$$

Since $\C_n$ is a p-morphic image of itself, there is a valuation $W$ to the variables $\vec q$ such that $\p{\C_n,W}\nvDash\phi_n$. Let $\sigma$ be the substitution $$\sigma(q_i)=\beta_{\{x:\p{C_n,W},x\models q_i\}}.$$ Then $$\p{\C_n,W},x\models\chi\iff\p{\C_n,V},x\models\sigma(\chi)$$ for all $x\in\C_n$ and all formulas $\chi(\vec q)$.

Thus, $\psi_n=\sigma(\phi_n)$ is a formula in the six variables $\vec p$, and by construction, $\C_n\nvDash\psi_n$ under $V$. However, for all $s\le n-2$, we have $\C_n(s)\models\phi_n$, and a fortiori $\C_n(s)\models\psi_n$, as $\C_n$ is not a p-morphic image of a generated subframe of $\C_n(s)$ (or of any other frame of size strictly smaller than $|\C_n|$, for that matter).

To reduce the required number of variables to three, consider in place of $V$ a valuation that makes one variable true in $\{(0,0),(0,1),(1,0),(1,1)\}$, one in $\{(0,0),(0,1),(1,0),(1,2)\}$, and one in $\{(0,0)\}$. Show that you can still define suitable formulas $\alpha_x$.

The result is actually false, for $m=6$. (One can bring it down to $m=3$ with a bit of effort.)

Let $n$ be arbitrarily large, and $\phi_n(\vec q)$ be a Jankov–De Jongh frame formula of $\def\p#1{\langle#1\rangle}\def\C{\mathcal C}\C_n$, so that for any frame $F$, $$F\nvDash\phi_n\iff\text{$\C_n$ is a p-morphic image of a generated subframe of $F$}.$$ Let $V$ be the valuation in $\C_n$ to the six variables $p_{(i,j)}$, $(i,j)\in\C_n$, $i\le1$, such that $$V(p_x)=\{y\in\C_n:y\nleq x\}.$$ We define a formula $\alpha_x(\vec p)$ for each $x\in\C_n$ by induction on $i$: $$\alpha_{(i,j)}=\begin{cases} p_{(i,j)}&i\le1,\\ \alpha_{(i-1,j)}\to\bigvee_{j'\ne j}\alpha_{(i-1,j')}&2\le i\le n-2,\\ \bigvee_{j'}\alpha_{(i-1,j')}&i=n-1. \end{cases}$$ Then we see by induction on $i$ that for all $x,y\in\C_n$, $$\p{\C_n,V},y\models\alpha_x\iff y\nleq x.$$ It follows that if $U$ is an arbitrary upper subset of $\C_n$, then $$y\in U\iff\p{\C_n,V},y\models\beta_U,$$ where $$\beta_U=\bigwedge_{x\notin U}\alpha_x.$$

Since $\C_n$ is a p-morphic image of itself, there is a valuation $W$ to the variables $\vec q$ such that $\p{\C_n,W}\nvDash\phi_n$. Let $\sigma$ be the substitution $$\sigma(q_i)=\beta_{W(q_i)}.$$ Then $$\p{\C_n,W},x\models\chi\iff\p{\C_n,V},x\models\sigma(\chi)$$ for all $x\in\C_n$ and all formulas $\chi(\vec q)$.

Thus, $\psi_n=\sigma(\phi_n)$ is a formula in the six variables $\vec p$, and by construction, $\C_n\nvDash\psi_n$ under $V$. However, for all $s\le n-2$, we have $\C_n(s)\models\phi_n$, and a fortiori $\C_n(s)\models\psi_n$, as $\C_n$ is not a p-morphic image of a generated subframe of $\C_n(s)$ (or of any other frame of size strictly smaller than $|\C_n|$, for that matter).

To reduce the required number of variables to three, consider in place of $V$ a valuation that makes one variable true in $\{(0,0),(0,1),(1,0),(1,1)\}$, one in $\{(0,0),(0,1),(1,0),(1,2)\}$, and one in $\{(0,0)\}$. Show that you can still define suitable formulas $\alpha_x$.

The result is actually false, for $m=6$.

Let $n$ be arbitrarily large, and $\phi_n(\vec q)$ be a Jankov–De Jongh frame formula of $\def\p#1{\langle#1\rangle}\def\C{\mathcal C}\C_n$, so that for any frame $F$, $$F\nvDash\phi_n\iff\text{$\C_n$ is a p-morphic image of a generated subframe of $F$}.$$ Let $V$ be the valuation in $\C_n$ to the six variables $p_{(i,j)}$, $(i,j)\in\C_n$, $i\le1$, such that $$V(p_x)=\{y\in\C_n:y\nleq x\}.$$ We define a formula $\alpha_x(\vec p)$ for each $x\in\C_n$ by induction on $i$: $$\alpha_{(i,j)}=\begin{cases} p_{(i,j)}&i\le1,\\ \alpha_{(i-1,j)}\to\bigvee_{j'\ne j}\alpha_{(i-1,j')}&2\le i\le n-2,\\ \bigvee_{j'}\alpha_{(i-1,j')}&i=n-1. \end{cases}$$ Then we see by induction on $i$ that for all $x,y\in\C_n$, $$\p{\C_n,V},y\models\alpha_x\iff y\nleq x.$$ It follows that if $U$ is an arbitrary upper subset of $\C_n$, then $$y\in U\iff\p{\C_n,V},y\models\beta_U,$$ where $$\beta_U=\bigwedge_{x\notin U}\alpha_x.$$

Since $\C_n$ is a p-morphic image of itself, there is a valuation $W$ to the variables $\vec q$ such that $\p{\C_n,W}\nvDash\phi_n$. Let $\sigma$ be the substitution $$\sigma(q_i)=\beta_{\{x:\p{C_n,W},x\models q_i\}}.$$ Then $$\p{\C_n,W},x\models\chi\iff\p{\C_n,V},x\models\sigma(\chi)$$ for all $x\in\C_n$ and all formulas $\chi(\vec q)$.

Thus, $\psi_n=\sigma(\phi_n)$ is a formula in the six variables $\vec p$, and by construction, $\C_n\nvDash\psi_n$ under $V$. However, for all $s\le n-2$, we have $\C_n(s)\models\phi_n$, and a fortiori $\C_n(s)\models\psi_n$, as $\C_n$ is not a p-morphic image of a generated subframe of $\C_n(s)$ (or of any other frame of size strictly smaller than $|\C_n|$, for that matter).

The result is actually false, for $m=6$. (One can bring it down to $m=3$ with a bit of effort.)

Let $n$ be arbitrarily large, and $\phi_n(\vec q)$ be a Jankov–De Jongh frame formula of $\def\p#1{\langle#1\rangle}\def\C{\mathcal C}\C_n$, so that for any frame $F$, $$F\nvDash\phi_n\iff\text{$\C_n$ is a p-morphic image of a generated subframe of $F$}.$$ Let $V$ be the valuation in $\C_n$ to the six variables $p_{(i,j)}$, $(i,j)\in\C_n$, $i\le1$, such that $$V(p_x)=\{y\in\C_n:y\nleq x\}.$$ We define a formula $\alpha_x(\vec p)$ for each $x\in\C_n$ by induction on $i$: $$\alpha_{(i,j)}=\begin{cases} p_{(i,j)}&i\le1,\\ \alpha_{(i-1,j)}\to\bigvee_{j'\ne j}\alpha_{(i-1,j')}&2\le i\le n-2,\\ \bigvee_{j'}\alpha_{(i-1,j')}&i=n-1. \end{cases}$$ Then we see by induction on $i$ that for all $x,y\in\C_n$, $$\p{\C_n,V},y\models\alpha_x\iff y\nleq x.$$ It follows that if $U$ is an arbitrary upper subset of $\C_n$, then $$y\in U\iff\p{\C_n,V},y\models\beta_U,$$ where $$\beta_U=\bigwedge_{x\notin U}\alpha_x.$$

Since $\C_n$ is a p-morphic image of itself, there is a valuation $W$ to the variables $\vec q$ such that $\p{\C_n,W}\nvDash\phi_n$. Let $\sigma$ be the substitution $$\sigma(q_i)=\beta_{\{x:\p{C_n,W},x\models q_i\}}.$$ Then $$\p{\C_n,W},x\models\chi\iff\p{\C_n,V},x\models\sigma(\chi)$$ for all $x\in\C_n$ and all formulas $\chi(\vec q)$.

Thus, $\psi_n=\sigma(\phi_n)$ is a formula in the six variables $\vec p$, and by construction, $\C_n\nvDash\psi_n$ under $V$. However, for all $s\le n-2$, we have $\C_n(s)\models\phi_n$, and a fortiori $\C_n(s)\models\psi_n$, as $\C_n$ is not a p-morphic image of a generated subframe of $\C_n(s)$ (or of any other frame of size strictly smaller than $|\C_n|$, for that matter).

To reduce the required number of variables to three, consider in place of $V$ a valuation that makes one variable true in $\{(0,0),(0,1),(1,0),(1,1)\}$, one in $\{(0,0),(0,1),(1,0),(1,2)\}$, and one in $\{(0,0)\}$. Show that you can still define suitable formulas $\alpha_x$.