"""Create fact_checker
Generated by Phi-Octopus Eden
2025-11-06 03:19:16.297989
"""

```python
from typing import List
from functools import lru_cache

class FactChecker:
    """
    A class that checks if a given statement is factually correct based on predefined facts.
    """

    def __init__(self):
        self.facts = {
            "earth_is_round": True,
            "water_boils_at_100_celsius": True,
            "pythagorean_theorem_exists": True
        }

    @lru_cache(maxsize=32)
    def check_fact(self, statement: str) -> bool:
        """
        Checks if the given statement is a fact according to predefined facts.
        
        :param statement: The statement to be checked as a string.
        :return: True if the statement is true, False otherwise.
        """
        return self.facts.get(statement.lower(), False)

def example_usage():
    """
    Demonstrates how to use the FactChecker class.
    """
    fact_checker = FactChecker()
    
    # Check predefined facts
    print(f"Earth is round: {fact_checker.check_fact('earth_is_round')}")  # Expected output: True
    print(f"Water boils at 100 celsius: {fact_checker.check_fact('water_boils_at_100_celsius')}")  # Expected output: True
    
    # Check an undefined fact
    print(f"Is gravity a myth?: {fact_checker.check_fact('gravity_is_myth')}")  # Expected output: False

if __name__ == "__main__":
    example_usage()
```

This Python code defines a `FactChecker` class that checks if statements are true based on predefined facts. It includes an `example_usage` function to demonstrate its usage and validate the implementation with some hardcoded checks.