Outer Multiset Dimension of Three-Dimensional Grids: A Universal Lower Bound and an Exact Two-Parameter Classification

Ryutaro Yonezu, Independent Researcher. Research No.59. Preprint v1.0, 8 November 2026. For every a, b, c ≥ 2, the outer multiset dimension of P_a □ P_b □ P_c is at least 4. For 2 ≤ m ≤ n, the value for P_2 □ P_m □ P_n is 4 when n ≥ 6 or (m, n) is (2, 5) or (4, 5), and 5 otherwise. The computer-assisted proof uses analytic lemmas, finite exhaustive checks, and exact-rational 22-branch certificates. No.58 established the m = 3, n ≥ 6 subfamily. Its B(t) landmark family and even-N interior collision witnesses overlap the earlier F3 (x-type) construction in No.59. The shared construction is credited to No.58. This preprint is a candidate partial response to Farhan et al., Problem 10. No.58 GitHub Release · No.58 Zenodo version DOI. Independent AI audit PASS within the author-reported Rev2 differential scope: mathematical invariance, the 23-file source manifest, frozen v0.5/v0.6 reproduction, and the SymPy certificate for 22 branches (264 coordinate bounds, 154 distinctness and exteriority conditions, and 26 absolute-value sign checks). Windows PowerShell 5.1 reproduction PASS is author-reported and applies to the frozen v0.5 finite computation. REFERENCES PARTIAL. NOVELTY INCONCLUSIVE. Not human peer-reviewed. All rights reserved. PDF SHA256: d8ccbf1288fe0b8672a8eade015d0fa0953c893d1df56a66b71bd591eb1e41c3Reproducibility ZIP SHA256: f721843c2ba560c7ba1d7dcb23eea24b383fc3eb40ce2e2a3d6f9b26a5cf0bd2 GitHub repository · GitHub Release v1.0 · SHA256 manifest. The ZIP includes matching LaTeX, verifier sources, certificates, and execution instructions.