Global-dimension spectra of gentle algebras

We classify the connected finite-dimensional gentle algebras of finite global dimension whose global-dimension spectra under derived equivalence are not intervals of integers. These are precisely the algebras with AAG invariant $[(2,4g)]$, where g≥2, whose canonical line field is induced by a nowhere-vanishing vector field. Their spectrum is \2g,2g+2,…,4g\. For each g≥2, there are exactly two such derived-equivalence classes, distinguished by their Arf invariant. The geometric foundation is a realisation theorem for prescribed polygon weights in arbitrary genus. For an AAG invariant [(mᵢ,nᵢ)]ᵢ₌₁ᵇ arising from such an algebra, the global dimensions attained in its AAG fibre form the integer interval [maxᵢ nᵢ/mᵢ,maxᵢ nᵢ]. By controlling the complete line-field invariant through local ribbon-graph constructions, we determine the spectrum in every derived-equivalence class.