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Quantitatively mapping enzyme sequence-catalysis landscapes remains a critical challenge in understanding enzyme function, evolution, and design. Here, we expand an emerging microfluidic platform to measure catalytic constants- and -for hundreds of diverse naturally occurring sequences and mutants of the model enzyme Adenylate Kinase (ADK). This enables us to dissect the sequence-catalysis landscape's topology, navigability, and mechanistic underpinnings, revealing distinct catalytic peaks organized by structural motifs. These results challenge long-standing hypotheses in enzyme adaptation, demonstrating that thermophilic enzymes are not slower than their mesophilic counterparts. Combining the rich representations of protein sequences provided by deep-learning models with our custom high-throughput kinetic data yields semi-supervised models that significantly outperform existing models at predicting catalytic parameters of naturally occurring ADK sequences. Our work demonstrates a promising strategy for dissecting sequence-catalysis landscapes across enzymatic evolution and building family-specific models capable of accurately predicting catalytic constants, opening new avenues for enzyme engineering and functional prediction.