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Artificial intelligence (AI) has become increasingly effective at predicting material properties from microstructure-informed representations, enabling rapid screening and accelerated decision-making. Yet, the “explanations” attached to these predictive systems frequently fail to support the kind of understanding required in microstructure–property science—namely, transferable mechanisms, intervention-relevant guidance, and defensible generalization under realistic shifts in processing, measurement, and operating regimes. This conceptual paper argues that explanation failure in materials AI is often structural rather than incidental: many popular explanation toolkits are optimized for interpreting model behavior rather than for producing scientifically legitimate accounts of why a microstructure yields a property outcome. We define the microstructure–property explanation gap as the persistent mismatch between what explainability tools can formally justify and what materials reasoning demands for action. To anatomize this gap, we identify four recurring causes: representational non-identifiability, confounding by processing history, multi-scale emergence, and instability under distribution shift. Building on this anatomy, we propose a novel theoretical framework—the Explanation Integrity Triad (EIT)—which evaluates any AI explanation along three axes: Representational Integrity, Causal Integrity, and Operational Integrity. The EIT provides a domain-specific vocabulary to prevent mechanistic overclaims and align explanation practices with scientific accountability in applied materials informatics.