Algebraic non-contractibility criterion for weighted barycentric spaces

Let $\Sigma$ be a compact surface with singular points $p_1,\ldots,p_m$ and ordered weights $-1<\alpha_1\leq\cdots\leq\alpha_m<0$. For $\rho>0$, let $\Sigma_{\rho,\underline{\alpha}}$ be the corresponding space of formal barycenters. Algebraic version. The space $\Sigma_{\rho,\underline{\alpha}}$ is not contractible if and only if there exist $n\in\mathbb{N}$ and a set $\iota\subseteq\{2,3,\ldots,m\}$ with $\operatorname{card}(\iota)\geq1$ such that

$$\rho>4\pi\sum_{i\in\iota}(1+\alpha_i)\quad\wedge\quad\rho<4\pi\sum_{i\in\{1\}\cup\iota}(1+\alpha_i).$$

This is presented as an algebraic reformulation of the preceding topological conjecture and is intended to classify the parameter regimes in which the formal-barycenter space is non-contractible. The statement includes an existential $n$ that does not occur in the displayed inequalities, so its role should be checked against the source’s definitions.