Your notation is quite strange to me, and very non-standard, at least coming from differential geometry.
You go though some length to define the interior of the $n$-simplex $$\mathrm{int}(\Delta^n) := \bigl\{ x \in \mathbb{R}^{n+1} \mid \sum_{i=0}^n \xi = 1 ~~\text{and} ~~ x_i > 0 ~~\text{for all}~~ i\bigr\}.$$ This is an $n$-dimensional submanifold of $\mathbb{R}^{n+1}$ and it is flat with respect to the induced metric.
However, apparently you want to define a non-standard metric on this manifold. This is where things get incomprehensible for me. First of all, what is $\xi$? You write it is "the" global coordinate system. Obviously there are various coordinate symstems on $\mathrm{int}(\Delta^n)$ (e.g. those given by one of the projection maps along one of the axes), but I do not see how there would be a canonical one.
Secondly, what is $p_\xi$? You write that $\mathcal{P} = \{p_\xi\}$, but I do not see how to interpret this as a meaningful set. Then again, $p_\xi$ seems to be a scalar, because you take its $\log$ later on.
Thirdly, the formula for Christoffel symbols generally looks different from the one you write. Probably it is right in this context, but as I said, I have no idea what $p_\xi$ should be.
And even then, you define $l_t = \log \dot{\gamma}(t)$, which does not make sense, as $\gamma$ is a path in a manifold, where you cannot take the log of. Even if you identify $\dot{\gamma}(t)$ with the corresponding coordinates in you coordinate system, that is still a vector in $\mathbb{R}^n$, where I do not know how to take the $\log$ of.
So, quite frankly, the reason that you didn't get a response on math.StackExchange might be that you do not define a single thing you write.