Classical chance-constrained programs are solved by safe approximations based on the empirical CVaR, which uses a uniform measure over scenarios and systematically underweights tail events under heavy-tailed distributions. We introduce \emph{q-CCP}, a non-extensive safe approximation grounded in the Riemannian geometry of the Tsallis statistical manifold: the rank-based q-CVaR escort weights are the $g^{(q)}$-geodesic projection onto the tail simplex face, and the q-CCP feasible set is a Tsallis-divergence ball (Proposition~12). This geometric foundation yields three results. First, q-CCP is a provable strict tightening of CVaR-CCP for all $q > 1$ (Theorem~7). Second, the empirical violation ratio satisfies $ρ(q) = [1-(1-\varepsilon)^{q+1}]/\varepsilon$, independent of the tail index $ν$ (Proposition~10). Third, the feasible-region volume cost is monotone increasing in $q$ and $ν$ (Proposition~11), providing a data-adaptive safety knob. The formulation inherits convexity and coherence from the q-CVaR functional and admits an iterative LP reformulation converging in 2--3 iterations. Experiments on 15 Ibovespa equities confirm the theory (violation ratio $0.241$, $q^* = 1.50$); an M5 inventory newsvendor experiment generalises the method to supply chain ($q^* = 1.88$, cost premium $1.155\times$, zero OOS stockout violations).
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