Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$.

Let $\Gamma$ be the graph of the inverse of geometric Frobenius $F:=\phi\otimes\overline{\mathbf{F}}_p$, with $\phi$ the $p$-th power map on $\mathbf{P}^1_{\mathbf{F}_p}$.

The intersection $\Gamma\cap \Delta$, with $\Delta$ the diagonal, is $X(\mathbf{F}_p)$, although intuitively each of the elements of $X(\mathbf{F}_p)$ should be counted with multiplicity $1/p$.

Is this intuition correct and is there a way to make it precise?

For example, if one could talk about cycles on $X$, perhaps one could argue that $$(1\times F)^*(\Gamma\cap\Delta)=\Delta\cap\Gamma_F$$ and since $(1\times F)$ multiplies cycles by $p$ (hopefully), then $\Gamma\cap\Delta = (1/p)(\Gamma_F\cap\Delta)$, with $\Gamma_F$ the graph of $F$. Now $\Gamma_F\cap\Delta$ is supported on $X(\mathbf{F}_p)$ with its elements counted with multiplicity one.

I would be interested in a reference, if any.