Deep learning models for multi-modal breast cancer diagnosis achieve high predictive accuracy but remain clinically unacceptable without actionable, counterfactual explanations. Attribution-based methods (LIME, SHAP) are categorically inapplicable to this purpose, as they generate no alternative instances and thus cannot be evaluated on counterfactual quality metrics. This investigation provides empirical evidence that FCA-Guided Counterfactual (FCA-CF) framework that uses a Formal Concept Analysis (FCA) concept lattice as a hard structural constraint on counterfactual search, operating over a multi-modal TCGA-BRCA dataset. We benchmark against four genuine counterfactual methods: Wachter-style CF, DiCE, FACE, and NICE, evaluated on 60 benign-predicted TCGA-BRCA instances. The FCA-CF framework achieves Validity = 1.0000 (100% of counterfactuals successfully flip the prediction), Sparsity = 2.37 features changed (best among all valid methods), and Proximity = 0.900 (normalised L2-based, matching NICE as joint best). The classifier achieves Accuracy = 0.980, F1 = 0.976, ROC-AUC = 0.9947. Ablation analysis confirms that the FCA lattice constraint is the primary sparsity driver (removing it increases sparsity by +40%, p < 0.001, Cohen's d = 0.78), while Phase C greedy refinement accounts for the largest individual contribution (+113% sparsity increase when disabled, p < 0.001, d = 5.01). FCA-guided counterfactual generation achieves a clinically important Pareto-dominant outcome; it is simultaneously the sparsest and among the most proximate of all valid methods, with perfect validity. The emergent sparsity property arising from lattice topology rather than numerical penalty terms constitutes a structurally novel contribution to the counterfactual explanation literature.

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