# Mathematical Native AGI Component

class MathAGI:
    def __init__(self):
        # Initialize various mathematical layers
        self.arithmetic = ArithmeticLayer()
        self.algebra = AlgebraLayer()
        self.calculus = CalculusLayer()
        self.probability = ProbabilityLayer()
        self.machine_learning = MachineLearningLayer()

    def process_input(self, input_data):
        # Process the input through multiple layers of mathematics and learning
        arithmetic_output = self.arithmetic.process(input_data)
        algebraic_output = self.algebra.apply_operations(arithmetic_output)
        calculus_output = self.calculus.derive(algebraic_output)
        probabilistic_output = self.probability.analyze(calculus_output)
        learned_output = self.machine_learning.predict(probabilistic_output)

        return learned_output

class ArithmeticLayer:
    def process(self, input_data):
        # Perform basic arithmetic operations
        result = sum(input_data)  # Example: Sum of elements in the data
        return result

class AlgebraLayer:
    def apply_operations(self, input_data):
        # Perform algebraic manipulations
        for i in range(len(input_data)):
            if i % 2 == 0:
                input_data[i] += 1  # Add 1 to even indices
            else:
                input_data[i] *= 2  # Double odd indices

        return input_data

class CalculusLayer:
    def derive(self, input_data):
        # Perform basic calculus operations (e.g., numerical differentiation)
        derivative = [input_data[i+1] - input_data[i] for i in range(len(input_data)-1)]
        return derivative

class ProbabilityLayer:
    def analyze(self, input_data):
        # Basic probability analysis
        mean = sum(input_data) / len(input_data)
        variance = sum((x - mean)**2 for x in input_data) / len(input_data)
        return {"mean": mean, "variance": variance}

class MachineLearningLayer:
    def predict(self, probabilistic_output):
        # Simple prediction using learned models
        if probabilistic_output["mean"] > 50:  # Example threshold
            return True  # Predicts something based on the mean value
        else:
            return False

# Initialize and use the MathAGI component
math_agi = MathAGI()
input_data = [1, 2, 3, 4, 5]
output = math_agi.process_input(input_data)
print("Output:", output)