Transition for boxes/lines over page break

Question

Situation: Currently, I implemented a theorem environment that has a gray line on the left hand side (the implementation should not be too relevant for now). For an example see below. I am quite happy with the result, but would like to optimise the page break behaviour.

Hope: I would like a "fading transition" of the line if it is broken (as a visual indicator that the environment continues). So the line might get more transparent or might transition into a dashed line.

In general I would be interested into such "transitions" for boxes and frames. I don't have any preferences on the package that is used for generating the lines/boxes/frames and would be happy about any solution that works nicely with theorem environments.

Answer 1

Like this?

\documentclass{article}
\usepackage[most]{tcolorbox}
\usepackage{mathtools}
\usepackage{lipsum}

\newtcbtheorem[auto counter]{PeterTheorem}{Theorem}%
{theorem style=plain, enhanced, breakable, 
empty,
coltitle=black, 
fonttitle=\upshape\scshape,
boxrule=0.0pt,
overlay unbroken={\fill[gray](frame.north west)rectangle ([xshift=1.5mm]frame.south west);},
overlay first={\fill[top color=gray, bottom color=white](frame.north west)rectangle ([xshift=1.5mm]frame.south west);},
overlay last={\fill[top color=white, bottom color=gray](frame.north west)rectangle ([xshift=1.5mm]frame.south west);},
overlay middle={\fill[top color=white, bottom color=white, middle color=gray](frame.north west)rectangle ([xshift=1.5mm]frame.south west);},
}{theo}


\begin{document}

\lipsum[1-4]

\begin{PeterTheorem}{}{theo1-3}%
Consider the n.n.f. Insing model on $Z^d,\ d\geq 1$. There exists acritical inverse temperature $\beta_c=\beta_c(d) \in[0,\infty]$, such that the Ising undergoes a \emph{sharp} ferromagnetic phase transistion at $\beta_c$:
\begin{equation*}
(\sigma_0)^+_{\beta;\Lambda_n}
\begin{cases}\leq e^{-cn} & $for$\ \beta<\beta_c\ $with$\ c=c(\beta)>0\\
\geq \sqrt{1-(\beta_c/\beta)^2} & $for$\ \beta\geq \beta_c \end{cases}
\end{equation*}
uniformly in $n\geq0$, where $\Lambda_n:=Z^d\cap[-n,n]^d$ denotes the box of size $n$ around the origin. Moreover, for\ $d\geq2$, the phase transition is \emph{non-trivial}, meaning that $0<\beta_c<\infty$.
\end{PeterTheorem}

\end{document}