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Automated detection of interictal epileptiform discharges in scalp electroencephalography (EEG) is clinically important, but recent high-performing deep-learning models often trade interpretability for accuracy. We introduce EEG-SpikeAgent, a closed-loop program-synthesis framework that uses a large language model (LLM) agentic system to generate signal-processing features for spike detection in scalp EEG. The system iteratively proposes one deterministic EEG feature module at a time, executes the resulting code on EEG to generate tabular features, evaluates performance via a tabular classifier, summarizes run-level metrics, and feeds structured diagnostics back to the model for refinement. Across iterations, EEG-SpikeAgent proposes and refines candidate signal features and decision rules informed by model performance. We evaluated EEG-SpikeAgent on VEPISET, a public 29-channel dataset of 4-second epochs containing 2,516 discharge-containing and 22,933 non-discharge epochs. Across five-fold cross-validation with a gradient-boosted tree classifier, agent-generated features achieved an area under the receiver operating characteristic curve of 0.935, balanced accuracy of 0.699, F1 score of 0.557, sensitivity of 0.401, and specificity of 0.996 at the default operating point. At an operating point with sensitivity 0.80, mean precision was 0.470 and mean specificity was 0.900. Artifact-aware feature generation improved balanced accuracy and F1 score over spike-only feature search. These results indicate that LLM-based program synthesis can automate EEG feature engineering in auditable and inspectable code-driven manner for clinical and methodological review.