# Example validation pattern
import mpmath as mp
mp.mp.dps = 50  # Required precision

def validate_mathematical_claim(theory_function, empirical_data):
    """Validate mathematical theory against empirical data"""
    # Theoretical prediction
    prediction = theory_function(empirical_data)
    
    # Statistical validation
    from scipy import stats
    t_stat, p_value = stats.ttest_1samp(prediction, expected_value)
    
    # Significance requirement
    assert p_value < 1e-6, f"Insufficient significance: p={p_value}"
    
    # Precision requirement
    precision_error = abs(prediction - expected_value)
    assert precision_error < 1e-16, f"Precision error: {precision_error}"
    
    return True

def framework_algorithm(parameters):
    """Template for framework algorithm implementation"""
    # Input validation
    if not validate_inputs(parameters):
        raise ValueError("Invalid input parameters")
    
    # High-precision calculation
    import mpmath as mp
    mp.mp.dps = 50
    
    try:
        # Algorithm implementation
        result = high_precision_calculation(parameters)
        
        # Result validation
        if not validate_result(result):
            raise ValueError("Invalid result computed")
            
        return result
        
    except Exception as e:
        # Robust error handling
        logger.error(f"Algorithm error: {e}")
        raise

class FrameworkExtension:
    """Template for framework extension classes"""
    
    def __init__(self, config):
        """Initialize extension with configuration"""
        self.validate_config(config)
        self.setup_precision()
    
    def validate_config(self, config):
        """Validate configuration parameters"""
        required_params = ['precision', 'validation_threshold']
        for param in required_params:
            if param not in config:
                raise ValueError(f"Missing required parameter: {param}")
    
    def setup_precision(self):
        """Set up high-precision arithmetic"""
        import mpmath as mp
        mp.mp.dps = 50

import time
import memory_profiler

def benchmark_optimization(func, test_data, iterations=1000):
    """Benchmark function performance"""
    # Memory usage
    mem_before = memory_profiler.memory_usage()[0]
    
    # Timing
    start_time = time.time()
    for _ in range(iterations):
        result = func(test_data)
    end_time = time.time()
    
    mem_after = memory_profiler.memory_usage()[0]
    
    return {
        'avg_time': (end_time - start_time) / iterations,
        'memory_usage': mem_after - mem_before,
        'result_sample': result
    }

import unittest
import mpmath as mp

class TestFrameworkFunction(unittest.TestCase):
    
    def setUp(self):
        """Set up test environment"""
        mp.mp.dps = 50
        self.test_data = generate_test_data()
        self.expected_results = load_expected_results()
    
    def test_mathematical_correctness(self):
        """Test mathematical correctness"""
        result = framework_function(self.test_data)
        expected = self.expected_results['mathematical']
        
        # High-precision comparison
        precision_error = abs(result - expected)
        self.assertLess(precision_error, mp.mpf('1e-16'))
    
    def test_statistical_significance(self):
        """Test statistical significance"""
        results = [framework_function(data) for data in self.test_data]
        
        # Statistical validation
        from scipy import stats
        t_stat, p_value = stats.ttest_1samp(results, expected_mean)
        self.assertLess(p_value, 1e-6)
    
    def test_edge_cases(self):
        """Test edge cases and error handling"""
        with self.assertRaises(ValueError):
            framework_function(invalid_input)
        
        # Boundary conditions
        boundary_result = framework_function(boundary_input)
        self.assertTrue(validate_boundary_result(boundary_result))

def framework_function(parameter_a, parameter_b, precision=50):
    """
    Calculate framework-specific transformation.
    
    This function implements the core Z Framework transformation
    using high-precision arithmetic to ensure numerical stability.
    
    Mathematical Foundation:
    The transformation follows the universal form:
    
    .. math::
        Z = A \cdot \left(\frac{B}{c}\right)
    
    Where:
    - A: Frame-dependent measured quantity
    - B: Rate or transformation parameter  
    - c: Universal invariant (speed of light)
    
    Parameters
    ----------
    parameter_a : float or mpmath.mpf
        Frame-dependent quantity A in the universal form
    parameter_b : float or mpmath.mpf
        Rate parameter B in the universal form
    precision : int, optional
        Decimal precision for calculations (default: 50)
    
    Returns
    -------
    mpmath.mpf
        Transformed value Z with specified precision
    
    Raises
    ------
    ValueError
        If parameters violate physical constraints
    TypeError
        If parameters are not numeric types
    
    Examples
    --------
    >>> import mpmath as mp
    >>> mp.mp.dps = 50
    >>> result = framework_function(1.0, 1e6)
    >>> print(f"Z = {result}")
    Z = 3.33564095198e-03
    
    See Also
    --------
    universal_form : Basic universal form calculation
    validate_parameters : Parameter validation utility
    
    References
    ----------
    .. [1] Z Framework Documentation, Mathematical Model Section
    .. [2] Core Principles, Universal Invariance Axiom
    """
    # Implementation with documentation standards
    pass

# Always set precision early
import mpmath as mp
mp.mp.dps = 50

# Use mpmath for all critical calculations
def critical_calculation(x):
    return mp.exp(mp.log(x) * mp.pi)

# Validate precision maintenance
def validate_precision(result, expected, tolerance=mp.mpf('1e-16')):
    error = abs(result - expected)
    assert error < tolerance, f"Precision loss: {error}"

def validate_statistical_claim(data, expected, alpha=1e-6):
    """Validate statistical claim with required significance"""
    from scipy import stats
    
    # Choose appropriate statistical test
    if len(data) > 30:
        stat, p_value = stats.ttest_1samp(data, expected)
    else:
        stat, p_value = stats.wilcoxon(data - expected)
    
    # Verify significance
    assert p_value < alpha, f"Insufficient significance: p={p_value}"
    
    # Report effect size
    effect_size = (np.mean(data) - expected) / np.std(data)
    return p_value, effect_size

# Cache expensive calculations
from functools import lru_cache

@lru_cache(maxsize=10000)
def expensive_calculation(n):
    """Cached calculation for performance"""
    return complex_mathematical_operation(n)

# Use vectorization when possible
import numpy as np

def vectorized_operation(data):
    """Vectorized operation for performance"""
    return np.vectorize(framework_function)(data)

def robust_framework_function(parameters):
    """Robust framework function with comprehensive error handling"""
    try:
        # Validate inputs
        validated_params = validate_and_convert(parameters)
        
        # Perform calculation
        result = core_calculation(validated_params)
        
        # Validate output
        if not is_valid_result(result):
            raise ValueError("Invalid result computed")
            
        return result
        
    except (ValueError, TypeError) as e:
        logger.error(f"Parameter error: {e}")
        raise
    except Exception as e:
        logger.error(f"Unexpected error: {e}")
        raise RuntimeError(f"Calculation failed: {e}")

class FrameworkConfig:
    """Framework configuration management"""
    
    DEFAULT_PRECISION = 50
    DEFAULT_VALIDATION_THRESHOLD = 1e-6
    
    def __init__(self, **kwargs):
        self.precision = kwargs.get('precision', self.DEFAULT_PRECISION)
        self.threshold = kwargs.get('threshold', self.DEFAULT_VALIDATION_THRESHOLD)
        self.validate_config()
    
    def validate_config(self):
        """Validate configuration parameters"""
        if self.precision < 16:
            raise ValueError("Minimum precision is 16 decimal places")
        if self.threshold >= 1e-3:
            raise ValueError("Validation threshold too high")