When 3D Gaussian Splatting Recovers Real Surfaces

When does 3D Gaussian Splatting (3DGS) recover the truescene surface rather than just overfitting view-dependent appearance? Weanswer this by developing a mathematical framework—based on a first-hitrendering abstraction—that cleanly isolates geometry from appearance.We prove that geometric misalignment forcefully converts spatial texturesinto high-frequency angular signals via parallax. This establishes a strictidentifiability window: if angular capacity is bounded, surface-consistentsolutions are mathematically preferred; if unrestricted, the same imagescan be perfectly explained by an incorrect, opaque “billboard” geometry.Experiments on synthetic stress tests confirm this prediction, showingbillboard failures emerge precisely at high angular capacities. Conversely,in the real-world datasets we evaluate under standard capture proto-cols, reconstructions remain surface-consistent even at high SH degrees,which is consistent with the prediction that rich spatial texture can pushbillboard solutions outside the tested angular-capacity range.