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Graph neural networks have become central to computational materials science, yet their application to periodic crystals remains limited by the absence of robust positional encoding. Without explicit representation of absolute atomic placement, symmetry-equivalent sites with identical local environments collapse into indistinguishable states, undermining predictive accuracy. This limitation becomes critical in large supercells, where extended defects, domain structures, and long-range ordering dominate material behavior. Existing encoding strategies either violate symmetry constraints or fail to scale, leaving a persistent gap between theoretical requirements and practical implementation. This work develops a theory of symmetry-aware positional encoding tailored to large supercell simulations. Six fundamental requirements are identified, spanning invariance, completeness, scalability, differentiability, and interpretability, and are shown to arise directly from the structure of periodic crystals. Analysis of current approaches reveals that none simultaneously satisfies these conditions at scale. A hierarchical framework is therefore introduced, distributing positional information across global, symmetry-aware, and local levels. This structure reconciles representational completeness with computational efficiency while preserving physical consistency. The resulting perspective establishes positional encoding as a central design principle for graph neural networks and provides a foundation for modeling properties that depend on absolute atomic arrangement in complex crystalline systems.

Multi-property graph neural networks (GNNs) have emerged as the dominant paradigm in computational materials science, enabling a single equivariant backbone to simultaneously predict energies, forces, stresses, elastic tensors, band gaps, and other key properties from atomic structures. Proponents emphasize gains in data efficiency, regularization, and inductive bias transfer across tasks. However, this work reveals a fundamental and previously underappreciated limitation: equivariant error cancellation within shared representations. By enforcing rotational, translational, and permutation symmetries, equivariant GNN backbones map atomic structures into a constrained feature space where different physical properties transform under distinct irreducible representations. We demonstrate theoretically and conceptually that gradients arising from weakly correlated or anti-correlated properties frequently conflict within these shared layers. Such conflicts drive error cancellation—reductions in the aggregate multi-task loss that mask stagnant or deteriorating accuracy on individual targets—particularly when properties map to incompatible tensorial characters (scalars, vectors, rank-2 and higher tensors). We identify three regimes of representation sharing: beneficial (strongly aligned gradients), neutral (orthogonal gradients), and harmful (opposing gradients leading to representational compromise). Proof sketches show that equivariance tightens the feasible representation space and bounds gradient alignment more severely than in invariant architectures, systematically increasing the risk of harmful cancellation. Detection diagnostics and architectural mitigation strategies are proposed to identify and alleviate these effects. This analysis reframes the design of multi-property materials GNNs, demonstrating that joint training is not universally beneficial and can actively degrade specialization. It provides a principled foundation for next-generation architectures that preserve data efficiency while safeguarding per-property accuracy, advancing reliable data-driven materials discovery and design.