Abstract. We construct a complete geodesic metric $d_{p,\phi }$ on the relative finite energy space ${\mathcal{E}}^p(X,\theta ,\phi )$ for every $p\ge 1$, where $\theta$ represents a big cohomology class and $\phi$ is an $\mathcal{I}$-model potential of positive mass. The construction extends Gupta's approximation method by establishing continuity of relative envelopes under $d_{\mathcal{S}}$-approximation of singularity types.

AI disclosure. This note grew out of the author's investigation of a different problem. The author and his advisor, Tamás Darvas, had previously considered the possibility of extending Gupta's construction in [Gup25] to $\mathcal{I}$-model potentials through approximation in the $d_{\mathcal{S}}$-metric by potentials of analytic singularity type. The author used GPT 6 Astra to help develop and revise the detailed arguments, identify gaps, and check citations. The author subsequently checked, refined, and rewrote the arguments. GPT 6 Astra also assisted with language editing.