The Multiset Dimension of Square King Grids

We prove that the multiset dimension of the square king grid equals 4 for every n ≥ 5, answering the open question of Hakanen and Yero (Fundamenta Informaticae 191 (2024), 315–330), recorded as Problem 1 in the survey of Farhan, Klavžar, Kuziak and Yero (arXiv:2607.10311). Together with the known small cases this determines the multiset dimension of every square king grid: infinite for n = 2, 3; six for n = 4; four for n ≥ 5. The proof is elementary and constructive. After a 45-degree change of coordinates the Chebyshev metric becomes half the Manhattan metric on a parity sublattice of a diamond; four boundary inequalities reduce every potentially resolving three-landmark set to two geometric cases, and in each case an explicit pair of distinct vertices with the same distance multiset is exhibited. The manuscript also reports why the method does not extend verbatim to rectangular king grids, together with a computational survey of rectangles showing a non-monotone row, and leaves the rectangular case open. This record contains the manuscript (LaTeX source and PDF), four independent verification programs (exhaustive search over all three-landmark sets for n = 5..12, machine-check of every collision identity in the normal-form table, a rectangular scan, and independent spot-checks), their outputs, and SHA-256 checksums. The proof is independent of computation; all programs are sanity checks, not extrapolation.