Large-scale transformers achieve impressive results on program synthesis benchmarks, yet their true generalization capabilities remain obscured by data contamination and opaque training corpora. To rigorously assess whether models are truly generalizing or merely retrieving memorized templates, we introduce a strictly controlled program synthesis environment based on a domain-specific arithmetic grammar. By systematically enumerating and evaluating millions of unique programs, we construct interpretable syntactic and semantic metric spaces. This allows us to precisely map data distributions and sample train and test splits that isolate specific distributional shifts. Our experiments demonstrate that optimizing density generalization -- through diverse sampling over both semantic and syntactic spaces -- induces robust out-of-distribution generalization. Conversely, evaluating support generalization reveals that transformers severely struggle with extrapolation, experiencing a performance drop of over 30% when forced to generate syntactically novel programs. While steadily scaling up compute improves generalization, the gains follow a strictly log-linear relationship. We conclude that robust generalization requires maximizing training diversity across multiple manifolds, and our findings indicate the necessity for novel search-based approaches to break through current log-linear scaling bottlenecks.