What potential-dispersion minimization constrains in small-body interiors

Paper V of the shape-based surface gravity series. This record contains both the manuscript and the data-and-code archive that produced every number in it. Richardson & Bowling (2014) argued that on a small body whose surface has relaxed under downslope regolith flow the bulk density can be estimated by minimizing the dispersion of the surface geopotential, and Kanamaru & Sasaki (2019) extended the argument to a two-region interior with the total mass held fixed. This paper asks what that extension actually constrains. Five bodies are examined — Itokawa, Eros, Bennu, 67P/Churyumov-Gerasimenko and Arrokoth — together with eight uniform synthetic contact binaries and a homogeneous ellipsoid whose surface gravity has a closed form. Of the five, only Itokawa yields a dispersion improvement above the mesh-resolution sensitivity envelope: 16.9% at the detected neck, with a contrast of 1.47 and a centre-of-mass offset of 16.5 +/- 2.9 m at the plane used by Kanamaru & Sasaki, consistent with the 21 +/- 12 m derived from YORP spin-up by Lowry et al. The principal result concerns identifiability. The mass constraint makes the excess mass identically M_tot - rho_rest V_tot, so the quantity the method determines in the strong-signal regime is the background density, not the density of the region: over the eleven admissible planes of an Itokawa sweep the region density varies by +/-13.2% and the background density by +/-0.43%,a ratio of 32 of which 11.8 is algebraic and 7.5 reflects genuine compensation. Dispersion minimization does not constrain its own dividing plane, which must come from geomorphology and a material bound. On Arrokoth, which has no measured mass, the sense of the contrast reverses within the published range of bulk density. Seven failure modes are catalogued, all of which produce plausible-looking numbers, and no numerical diagnostic tested here detects amisplaced dividing plane. The archive includes the region solver, the manifold-preserving quadric decimator, the synthetic generator, the analytic ellipsoid test, the containment test for multi-component shape models, the figure scripts, and a verification script that checks the values quoted in the paper against the archived run records. Shape models are not redistributed; their sources arelisted in the deposit. Released under CC BY 4.0. Development is tracked at https://github.com/praponkan/shape-based-surface-gravity