import math

def total_byte_entropy_stats(python_code: str) -> dict:
    """
    Calculates the total byte entropy of a given Python code string.

    The function encodes the input Python code string into UTF-8 bytes, counts the occurrences
    of each byte, and computes the entropy based on their frequency distribution. The entropy
    is a measure of the randomness or unpredictability of the byte distribution.

    Args:
        python_code (str): A string representing Python code.

    Returns:
        dict: A dictionary containing a single key, 'total_byte_entropy', which maps to
            the computed entropy value as a float.
    """
    # TODO: Implement this function
    pass