30.10 纯数论的统一框架

# 统一数论编程语言的核心语法

class UnifiedNumberTheoryLanguage:
    def __init__(self):
        self.state_space = NumberStateSpace()
        self.evolution_engine = EvolutionEngine()
        self.observation_system = ObservationSystem()

    def create_superposition(self, numbers, weights=None):
        """创建数字叠加态"""
        if weights is None:
            weights = [1/len(numbers)] * len(numbers)

        state = NumberState()
        for n, w in zip(numbers, weights):
            state.add_component(n, complex(w))

        return state.normalize()

    def evolve(self, state, hamiltonian, time):
        """演化数字状态"""
        return self.evolution_engine.evolve(state, hamiltonian, time)

    def observe(self, state, property_function):
        """观察数论性质"""
        return self.observation_system.measure(state, property_function)

    def correlate(self, state1, state2):
        """计算状态关联"""
        return compute_correlation(state1, state2)

# 使用示例
unt = UnifiedNumberTheoryLanguage()

# 创建素数叠加态
prime_state = unt.create_superposition([2, 3, 5, 7, 11, 13])

# 观察奇偶性
parity_result = unt.observe(prime_state, lambda n: n % 2)

# 演化状态
evolved_state = unt.evolve(prime_state, prime_hamiltonian, time=1.0)

实现 2：统一数论算法库

class UnifiedNumberTheoryAlgorithms:
    """统一数论算法库"""

    def __init__(self):
        self.superposition_engine = SuperpositionEngine()
        self.correlation_analyzer = CorrelationAnalyzer()
        self.evolution_solver = EvolutionSolver()

    def parallel_primality_test(self, numbers):
        """并行素性检测"""
        # 创建叠加态
        state = self.superposition_engine.create_uniform_superposition(numbers)

        # 应用素性算符
        prime_state = self.superposition_engine.apply_primality_operator(state)

        # 测量结果
        results = self.superposition_engine.measure_all(prime_state)

        return results

    def correlation_enhanced_search(self, target_property, search_space):
        """利用关联增强的搜索"""
        # 分析搜索空间的关联结构
        correlation_graph = self.correlation_analyzer.build_correlation_graph(search_space)

        # 基于关联优化搜索策略
        optimal_path = self.correlation_analyzer.find_optimal_search_path(
            correlation_graph, target_property
        )

        # 执行关联感知搜索
        results = self.execute_correlated_search(optimal_path, target_property)

        return results

    def adaptive_evolution_algorithm(self, initial_state, target_state, max_time):
        """自适应演化算法"""
        current_state = initial_state
        time = 0

        while time < max_time and not self.converged(current_state, target_state):
            # 计算最优哈密顿算符
            optimal_hamiltonian = self.compute_optimal_hamiltonian(
                current_state, target_state
            )

            # 演化一小步
            dt = self.adaptive_timestep(current_state, optimal_hamiltonian)
            current_state = self.evolution_solver.evolve(
                current_state, optimal_hamiltonian, dt
            )

            time += dt

        return current_state, time

统一框架的性能评估

评估 1：计算效率比较

def benchmark_unified_vs_traditional():
    """比较统一数论与传统数论的计算效率"""

    test_problems = [
        ('prime_generation', generate_primes_up_to),
        ('factorization', factorize_number),
        ('discrete_log', discrete_logarithm),
        ('number_field_sieve', advanced_factoring)
    ]

    results = {}

    for problem_name, problem_func in test_problems:
        traditional_times = []
        unified_times = []

        for size in [100, 500, 1000, 5000]:
            # 传统方法
            start = time.time()
            traditional_result = problem_func(size, method='traditional')
            traditional_time = time.time() - start

            # 统一方法
            start = time.time()
            unified_result = problem_func(size, method='unified_superposition')
            unified_time = time.time() - start

            traditional_times.append(traditional_time)
            unified_times.append(unified_time)

            # 验证结果一致性
            assert verify_results_equivalent(traditional_result, unified_result)

        # 分析加速比
        speedups = [t / u for t, u in zip(traditional_times, unified_times)]

        results[problem_name] = {
            'traditional_times': traditional_times,
            'unified_times': unified_times,
            'speedups': speedups,
            'average_speedup': numpy.mean(speedups)
        }

    return results

评估 2：理论覆盖度分析

def analyze_theoretical_coverage():
    """分析统一框架的理论覆盖度"""

    # 传统数论的核心概念
    traditional_concepts = [
        'prime_numbers', 'divisibility', 'congruences', 'diophantine_equations',
        'quadratic_residues', 'continued_fractions', 'elliptic_curves',
        'modular_forms', 'zeta_functions', 'l_functions'
    ]

    coverage_analysis = {}

    for concept in traditional_concepts:
        # 检查统一框架是否覆盖
        unified_representation = find_unified_representation(concept)

        if unified_representation:
            # 分析表示的完整性
            completeness = analyze_representation_completeness(
                concept, unified_representation
            )

            # 分析新的洞察
            new_insights = discover_new_insights(
                concept, unified_representation
            )

            coverage_analysis[concept] = {
                'covered': True,
                'representation': unified_representation,
                'completeness': completeness,
                'new_insights': new_insights
            }
        else:
            coverage_analysis[concept] = {
                'covered': False,
                'reason': 'No unified representation found'
            }

    return coverage_analysis