Once-subdivided complete graphs have infinite edge multiset dimension

Problem 24, section 7.4, printed page 27 of the versioned source asks whether the edge multiset dimension of K_n^1 is infinity for every integer n at least three. The definition on printed page 21 inserts one midpoint into every complete-graph edge. Section 6.1 on printed page 20 uses minimum actual shortest endpoint distances and arbitrary landmark subsets of the full vertex set, including originals and midpoints. Profiles are multisets with multiplicity; infinity means that no resolving subset exists.

D5/S3/Combinatorics/Graph/Walks/CompleteSubdivisionResolution.result gives the full affirmative conclusion: for every n at least three and every arbitrary landmark set, two distinct actual graph edges have equal distance multisets, and the extended-natural edge multiset dimension is top.

The source's diameter-two obstruction covers n=3 but leaves n at least four, where a disjoint midpoint has edge distance three. The full landmark domain requires a free diagonal in the selection matrix, rather than a simple-graph adjacency convention.

Exact shortest-distance histograms force distinct augmented selected degrees for any hypothetical resolving set. The symmetric binary selection matrix has unrestricted diagonal. Complement and row-rank permutation reduce it to a proved staircase. Explicit even and odd incidence pairs have identical four-bin histograms, contradicting resolution. The odd case includes n=3; arbitrary sets are finite because the full vertex type is finite. All bridges are proved in the eight native modules.

An n at least three and a subset of the full subdivided vertex set whose actual minimum-shortest-endpoint distance multisets are pairwise distinct on all graph edges.

The native endpoint and its graph, reachability, distance, histogram, selection, complement, collision and staircase dependencies carry Lean proofs. The source endpoint states both the collision on actual edges and the resulting infinite dimension.

theorem: the formal mathematical conclusion is affirmative for the entire printed Problem 24 domain. This dossier makes no repository acceptance or solved-count claim.

Source-open is verified in the versioned source. The bounded literature scope is the source, arXiv:2608.09983v1, section 7.4 of arXiv:2607.08128v1 and the released repository candidate search. Worldwide current openness and priority remain unverified.