Grokking Geometry: The Phase Transition from Positional to Semantic Learning in 3D

While 3D point cloud networks achieve high performance on complex benchmarks, it remains unclear whether they learn intrinsic geometric semantics or merely exploit spurious positional correlations. To address this, we introduce the Coordinate Trap, a synthetically controlled benchmark designed not to simulate ecological complexity, but to serve as a rigorous atomic unit test for geometric reasoning. By imposing a strong spatial bias (e.g., Sphere-Left, Cube-Right), we decouple absolute position from local topology. We observe that geometric generalization does not emerge gradually but follows a sharp phase transition governed by sample complexity ($N$). We identify a critical threshold ($N_c$) separating two distinct regimes: a Positional Phase, where models minimize loss via coordinate memorization, and a Semantic Phase, where a symmetry breaking event occurs, realigning the decision boundary from embedding axes to intrinsic shape properties. Crucially, we distinguish this data-driven transition from optimization-time grokking: below $N_c$, extended training fails to yield generalization, indicating that sufficient data density is a prerequisite for the optimization dynamics to escape the positional attractor. Furthermore, we quantify the excess sample complexity for generic architectures (e.g., Transformers) compared to those with geometric inductive biases, proposing $N_c$ as a novel metric for the geometric affinity of neural networks, providing a theoretical blueprint for scaling robust 3D foundation models.