Finiteness, Prescribed Gaps, and Extremal Growth in Edge Multiset Dimension: manuscript, verification code, and exact small-order census

Manuscript and complete verification package for the paper "Finiteness, Prescribed Gaps, and Extremal Growth in Edge Multiset Dimension". The paper resolves Problems 22–24 and contributes to Problem 21 of the survey by Farhan, Klavžar, Kuziak and Yero (arXiv:2607.10311). Main results: (1) the one-subdivision of every complete graph Kn, n ≥ 3, has infinite edge multiset dimension (Problem 24); (2) every integer is realized as mdim(G) − medim(G) with both parameters finite (Problem 22), with the minimum orders for the first gaps determined by exact census; (3) the extremal order function satisfies 5⌊(n−1)/7⌋ ≤ M(n) ≤ n for all n ≥ 29, with the exact value M(10) = 9 (Problem 23); (4) a parity-free comparison theorem mdim(G) ≤ medim(G) for bipartite graphs of diameter at most three with finite edge multiset dimension; (5) a counterexample to a published radius-two nonexistence lemma [Ikhlaq, Ismail, Siddiqui, Nadeem, Symmetry 15 (2023), Art. 762, Lemma 3] and a complete repair for trees of edge-distance radius at most two. Contents: the manuscript (paper.tex, paper.pdf), nine standard-library Python verification programs, two C++17 census sources (dual_census.cpp, scan_edge_full.cpp), the complete connected graph6 catalogs of orders 1–9, the order-ten extremal scan output and reproducibility metadata, and a SHA-256 manifest. See README.txt and CENSUS_README.txt for usage. The 117 MB raw order-ten catalog is not bundled; its generating command, record count, byte count, and SHA-256 digest are documented for exact reproduction.