A structural obstruction to (1,1,3,3,4)-packing colorings

This paper gives a negative answer to Problem 1 posed by Mortada, El Zein, and Al Hajjar concerning (1,1,3,3,4)-packing colorability of connected claw-free subcubic graphs. For every odd integer k ≥ 3, we construct a connected cubic, claw-free, and diamond-free graph on 12k vertices that does not admit a (1,1,3,3,4)-packing coloring. The construction is obtained by applying triangle expansion to a cyclic ring of k diamonds. The noncolorability result is established through a structural mathematical argument rather than exhaustive search. Any hypothetical packing coloring would induce a proper three-coloring of the underlying cubic graph. The distance-four requirement imposes an additional private-neighbor condition, which is incompatible with the parity structure of an odd diamond ring. The smallest member of the constructed family has 36 vertices. This fixed counterexample is additionally verified in Lean using actual graph distances and universally quantified coloring functions. The formalization covers this instance, while the infinite-family theorem is proved mathematically in the manuscript. Included materials (version 0.3): clawfree-packing-v0.3.pdf — Complete research manuscript. clawfree-packing-v0.3-latex-source.zip — LaTeX source and bibliography. clawfree-packing-v0.3-reproducibility.zip — Lean formalization, graph certificates, verification programs, reproducibility instructions, and verification records. Licensing: The manuscript and LaTeX source are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). Repository-owned verification software is covered by Apache License 2.0 where applicable. Other included materials retain their respective original licenses, as documented in the reproducibility archive. Research status: This is a publicly distributed research preprint, not an externally peer-reviewed publication. AI assistance in mathematical research, formal verification, software development, and manuscript preparation is disclosed in the manuscript.