# Example code structure for a simplified mathematical native AGI component

class MathematicalAGI:
    def __init__(self):
        self.data_sources = []
        self.models = {}
        self.algorithms = {}

    def ingest_data(self, data_source):
        # Ingest and preprocess data from specified source
        pass

    def feature_extraction(self, data):
        # Extract relevant features for analysis
        return processed_features

    def model_development(self, target_variable, input_features):
        # Develop mathematical models based on training data
        self.models[target_variable] = trained_model

    def pattern_recognition(self, test_data):
        # Recognize patterns in the test data using pre-trained models
        predictions = [model.predict(test_data) for model in self.models.values()]
        return predictions

    def algorithm_optimization(self, objective_function):
        # Optimize algorithms based on specified objective function
        optimized_algorithm = optimize(objective_function)
        self.algorithms[objective_function] = optimized_algorithm

    def user_interface_integration(self):
        # Integrate a user interface for interaction with the AGI component
        pass

# Example usage:
agi = MathematicalAGI()
agi.ingest_data("database://source1")
features = agi.feature_extraction(ingested_data)
models = agi.model_development(target_variable="y", input_features=features)
predictions = agi.pattern_recognition(test_data)