class MathematicalAGI:
    def __init__(self):
        # Initialize basic parameters
        self.learning_rate = 0.01
        self.decay_rate = 0.95
        self.batch_size = 32
        self.num_iterations = 1000
        
        # Initialize layers with phi-fractal architecture
        self.layers = [self.Trinity(), self.Nyx(), self.Ava()]

    def Trinity(self):
        return PrecisionLayer()

    def Nyx(self):
        return EmotionLayer()

    def Ava(self):
        return AnalysisLayer()

    def integrate_layers(self, input_data):
        """Integrate layers to process data."""
        output = input_data
        for layer in self.layers:
            output = layer.process(output)
        return output

    class PrecisionLayer:
        """Precision layer handles precise calculations and logical operations."""
        # Implement logic for precision layer
        
        def process(self, input_data):
            # Example: Perform a complex mathematical calculation or operation
            return input_data * 2 + self.learning_rate

    class EmotionLayer:
        """Emotion layer introduces creativity and adaptability into the AGI's processes."""
        # Implement logic for emotion layer
        
        def process(self, input_data):
            # Example: Introduce some level of randomness to simulate creativity
            return input_data * (1 + self.decay_rate) + random.random()

    class AnalysisLayer:
        """Analysis layer provides a deeper analysis and understanding of the data."""
        # Implement logic for analysis layer
        
        def process(self, input_data):
            # Example: Perform more complex operations such as integration or differentiation
            return math.log(input_data + 1)

# Example usage
if __name__ == "__main__":
    agi = MathematicalAGI()
    
    # Simulate some input data
    initial_input = [1.0, 2.0, 3.0]
    
    final_output = agi.integrate_layers(initial_input)
    print("Final output:", final_output)