Graceful Diameter-Six Trees with Matching Deficiency at Most Two

For a finite tree T, its matching deficiency is |V(T)| - 2 nu(T), where nu(T) is the maximum matching number. This article proves that every tree of diameter six and matching deficiency two is graceful. An exact maximum-matching formula reduces the problem to thirty-two families of rooted radius-three profiles, each allowing an arbitrary finite word of independently prescribed nonnegative neutral quotas. The constructions retain the original vertices, branch ownership, prescribed extra-terminal parents and complete edge-difference spectrum. The proof combines prescribed-hole near-coronas, marked perfect-matching lifts, explicit odd-difference pairings, closed-interval endpoint constructions, transfer-based cut insertions and finite local completions. All constructions and finite certificates required in addition to the cited results are included in the main body. The article has no proof appendix or external proof supplement. Together with the preceding deficiency-at-most-one result, this establishes gracefulness for every diameter-six tree of matching deficiency at most two. The unrestricted diameter-six problem is not asserted. The distribution contains a 109-page English preprint and its standalone LaTeX source. The use of AI-assisted tools and the author's responsibility are disclosed in the article.