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We prove that for any possibly-punctured surface with non-empty boundary$\mathbf{\Sigma}=(\Sigma, \mathbb{M}, \mathbb{P})$, and any taggedtriangulation $T$ of $\mathbf{\Sigma}$ in the sense ofFomin--Shapiro--Thurston, the coefficient-free bangle functions ofMusiker--Schiffler--Williams coincide with the coefficient-free genericCaldero--Chapoton functions arising from the Jacobian algebra of the quiverwith potential $(Q(T), W(T))$ associated to $T$ by Cerulli Irelli and thesecond author.  When the set of boundary marked points $\mathbb{M}$ has at least twoelements, Schr\"oer and the first two authors have shown, relying heavily onresults of Mills, Muller and Qin, that the generic coefficient-freeCaldero-Chapoton functions form a basis of the coefficient-free (upper) clusteralgebra $\mathcal{A}(\mathbf{\Sigma})=\mathcal{U}(\mathbf{\Sigma})$. So, theset of bangle functions proposed by Musiker--Schiffler--Williams over ten yearsago is indeed a basis.

Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary