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Yes! Define $$ k = \left[ \begin{array}{} 1 & 1 & 1 \\ 1 & 9 & 1 \\ 1 & 1 & 1 \end{array} \right] $$ and $$ f(u) = \begin{cases} 1 & \text{if } u \in \{3, 11, 12\} \\ 0 & \text{else} \end{cases} $$ This corresponds to the previous parameterization via $(0,3)\to 3$, $(1,2)\to 11$, $(1,3)\to 12$. Now the system can be written simply as $u_{t+1} = f \circ (k * u_t)$.

This works because the value of 9 in the center cell outweighs the sum of its neighbors, so $k*u_t(x)\geq9$ guarantees $u_t(x)=1$, and likewise $k*u_t(x)<9$ guarantees $u_t(x)=0$, with $k*u_t(x)\bmod 9$ being the neighbor count.

Yes! Define $$ k = \left[ \begin{array}{} 1 & 1 & 1 \\ 1 & 9 & 1 \\ 1 & 1 & 1 \end{array} \right] $$ and $$ f(u) = \begin{cases} 1 & \text{if } u \in \{3, 11, 12\} \\ 0 & \text{else} \end{cases} $$ This corresponds to the previous parameterization via $(0,3)\to 3$, $(1,2)\to 11$, $(1,3)\to 12$. This works because $k*u_t(x)\geq9$ guarantees $u_t(x)=1$, and likewise $k*u_t(x)<9$ guarantees $u_t(x)=0$, with $k*u_t(x)\bmod 9$ being the neighbor count.

It can - basically, redefine the kernel $k$ as follows: $$ k = \left[ \begin{array}{} 1 & 1 & 1 \\ 1 & 9 & 1 \\ 1 & 1 & 1 \end{array} \right] $$ Then redefine $f$ as $$ f(u) = \begin{cases} 1 & \text{if } u \in \{3, 11, 12\} \\ 0 & \text{else} \end{cases} $$ This corresponds to the previous parameterization via $(0,3)\to 3$, $(1,2)\to 11$, $(1,3)\to 12$. Now the system can be written simply as $u_{t+1} = f \circ (k * u_t)$.

This works because the value of 9 in the center cell outweighs the sum of its neighbors, so $k*u_t(x)\geq9$ guarantees $u_t(x)=1$, and likewise $k*u_t(x)<9$ guarantees $u_t(x)=0$, with $k*u_t(x)\bmod 9$ being the neighbor count.

Some comments: I began thinking about this after trying to implement Game of Life on this cool web app called neuralpatterns.io which only supports scalar reaction-diffusion systems. It suggests a family of similar systems with sharply mean-centric dispersion kernels and bimodal activation functions which display Life-like dynamics. Also, the scalar form is that of the somewhat obscure integro-difference equation.

Yes! Define $$ k = \left[ \begin{array}{} 1 & 1 & 1 \\ 1 & 9 & 1 \\ 1 & 1 & 1 \end{array} \right] $$ and $$ f(u) = \begin{cases} 1 & \text{if } u \in \{3, 11, 12\} \\ 0 & \text{else} \end{cases} $$ This corresponds to the previous parameterization via $(0,3)\to 3$, $(1,2)\to 11$, $(1,3)\to 12$. Now the system can be written simply as $u_{t+1} = f \circ (k * u_t)$.

This works because the value of 9 in the center cell outweighs the sum of its neighbors, so $k*u_t(x)\geq9$ guarantees $u_t(x)=1$, and likewise $k*u_t(x)<9$ guarantees $u_t(x)=0$, with $k*u_t(x)\bmod 9$ being the neighbor count.

It can - basically, redefine the kernel $k$ as follows: $$ k = \left[ \begin{array}{} 1 & 1 & 1 \\ 1 & 9 & 1 \\ 1 & 1 & 1 \end{array} \right] $$ Then redefine $f$ as $$ f(u) = \begin{cases} 1 & \text{if } u \in \{3, 11, 12\} \\ 0 & \text{else} \end{cases} $$ This corresponds to the previous parameterization via $(0,3)\to 3$, $(1,2)\to 11$, $(1,3)\to 12$.

  Now the system can be written simply as $u_{t+1} = f \circ (k * u_t)$, which is the form of the somewhat obscure integro-difference equation. How cool! It also suggests a family of similar systems with sharply mean-centric dispersion kernels and bimodal activation functions which display Life-like dynamics.

It can - basically, redefine the kernel $k$ as follows: $$ k = \left[ \begin{array}{} 1 & 1 & 1 \\ 1 & 9 & 1 \\ 1 & 1 & 1 \end{array} \right] $$ Then redefine $f$ as $$ f(u) = \begin{cases} 1 & \text{if } u \in \{3, 11, 12\} \\ 0 & \text{else} \end{cases} $$ This corresponds to the previous parameterization via $(0,3)\to 3$, $(1,2)\to 11$, $(1,3)\to 12$. Now the system can be written simply as $u_{t+1} = f \circ (k * u_t)$.

This works because the value of 9 in the center cell outweighs the sum of its neighbors, so $k*u_t(x)\geq9$ guarantees $u_t(x)=1$, and likewise $k*u_t(x)<9$ guarantees $u_t(x)=0$, with $k*u_t(x)\bmod 9$ being the neighbor count.

Some comments: I began thinking about this after trying to implement Game of Life on this cool web app called neuralpatterns.io which only supports scalar reaction-diffusion systems. It suggests a family of similar systems with sharply mean-centric dispersion kernels and bimodal activation functions which display Life-like dynamics. Also, the scalar form is that of the somewhat obscure integro-difference equation.