Finite-temperature topological diagnostics in quasiperiodic systems are subtle because quasiperiodicity removes the Bloch momentum torus on which mixed-state band indices are usually formulated. We study a quasiperiodic Qi-Wu-Zhang Chern insulator on finite Fibonacci approximants and use twisted boundary phases as an operational two-torus. A gap scan over L=21, 34, and 55 shows that the fixed-boundary spectral gap can remain finite while extended low-gap regions appear elsewhere on the twist torus. Projected-Uhlmann diagnostics reveal a sharper finite-size result: the stability of the projected winding is not determined by the scalar minimum twist gap alone. Along representative phason trajectories, the Uhlmann stability scale collapses as a twist-space bottleneck develops. However, selected phasons with comparable small minimum gaps retain n_U=1 up to the highest sampled temperature T=0.03, while a nearby contrast point collapses again. Direct phase-curve diagnostics show that collapse is associated with the loss or singularization of a 2π projected-Uhlmann phase winding, whereas stable counterexamples preserve the winding despite small spectral gaps. Within the finite approximants studied here, these results identify phason-resolved boundary-phase geometry and projected-holonomy sensitivity, rather than the fixed-boundary gap or scalar minimum gap alone, as the mechanism controlling this finite-size projected diagnostic.