ζ(s) = Tr(ρ^s) where ρ is a quantum density matrix

def quantum_zeta_construction():
    """
    Construct quantum system whose partition function is ζ(s)
    
    Key insight: Prime factorization → Quantum circuit decomposition
    """
    # Quantum register representing prime factorizations
    n_qubits = log2(max_prime_considered)
    
    # Hamiltonian encoding prime structure
    H = construct_prime_hamiltonian()
    
    # Density matrix with eigenvalues 1/n^s
    rho = expm(-beta * H)  # β relates to s
    
    # Verify: Tr(rho^s) = ζ(s)
    return verify_zeta_trace(rho, s_values)

ζ(s) = ⟨0|0⟩_s  (vacuum amplitude in s-dependent theory)

K(prime_sequence_n) ∼ ζ(s) behavior at height n

def algorithmic_rh_approach():
    """
    Use algorithmic information theory to attack RH
    """
    def kolmogorov_complexity_estimate(sequence):
        # Estimate K(sequence) using compression
        compressed = best_compression(sequence)
        return len(compressed)
    
    def prime_sequence_complexity(n):
        primes = sieve_of_eratosthenes(n)
        binary_sequence = primality_indicator(n)
        return kolmogorov_complexity_estimate(binary_sequence)
    
    def zeta_complexity_relation(s):
        # Theoretical connection between K(primes) and ζ(s)
        return complex_relationship(s)
    
    # Verify complexity patterns predict zero locations
    return verify_complexity_predictions()

import dionysus as d
import numpy as np

def topological_zero_analysis(zeros):
    """
    Apply persistent homology to Riemann zeros
    """
    # Convert zeros to point cloud
    points = [(z.real, z.imag) for z in zeros]
    
    # Build Rips complex
    f = d.fill_rips(points, k=2, r=max_radius)
    
    # Compute persistence
    m = d.homology_persistence(f)
    
    # Analyze persistence diagrams
    dgms = d.init_diagrams(m, f)
    
    return analyze_topological_features(dgms)

def topological_rh_proof():
    """
    Attempt proof using topological constraints
    """
    # Conjecture: Critical line zeros have specific topological signature
    critical_signature = compute_critical_line_topology()
    
    # Prove: Only critical line can support this signature
    return prove_topological_uniqueness(critical_signature)

import torch
import torch.nn as nn

class ZetaPatternNet(nn.Module):
    """
    Deep neural network for discovering patterns in ζ(s) zeros
    """
    def __init__(self, input_dim=2, hidden_dim=512):
        super().__init__()
        self.encoder = nn.Sequential(
            nn.Linear(input_dim, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, hidden_dim//2),
            nn.ReLU(),
            nn.Linear(hidden_dim//2, 64)
        )
        
        self.classifier = nn.Linear(64, 2)  # On critical line or not
        
    def forward(self, zeros):
        features = self.encoder(zeros)
        predictions = self.classifier(features)
        return predictions

def train_pattern_recognition():
    """
    Train neural network to recognize zero patterns
    """
    model = ZetaPatternNet()
    
    # Training data: known zeros + random complex numbers
    training_data = generate_training_data()
    
    # Train to distinguish actual zeros from random points
    train_model(model, training_data)
    
    # Use trained model to predict unknown zero properties
    return discover_patterns(model)

class ProofSearchAgent:
    """
    RL agent for searching mathematical proof space
    """
    def __init__(self):
        self.q_network = build_proof_q_network()
        self.proof_environment = MathematicalProofEnv()
    
    def search_proof_space(self, theorem="riemann_hypothesis"):
        state = self.proof_environment.reset(theorem)
        
        while not self.proof_environment.done:
            # Choose next proof step
            action = self.epsilon_greedy_action(state)
            
            # Apply logical step
            next_state, reward, done = self.proof_environment.step(action)
            
            # Update Q-network
            self.update_q_values(state, action, reward, next_state)
            
            state = next_state
        
        return self.proof_environment.get_proof()

H = (V, E) where:
V = {primes, composite numbers, arithmetic functions}
E = {multiplicative relationships, Dirichlet convolutions, ...}

def hypergraph_rh_approach():
    """
    Model number theory as hypergraph and study invariants
    """
    # Vertices: arithmetic objects
    vertices = generate_arithmetic_objects()
    
    # Hyperedges: mathematical relationships
    hyperedges = [
        multiplicative_relation,
        additive_relation,
        dirichlet_convolution,
        mobius_inversion
    ]
    
    # Build hypergraph
    H = construct_hypergraph(vertices, hyperedges)
    
    # Compute spectral properties
    spectrum = hypergraph_spectrum(H)
    
    # Connect to ζ(s) zeros
    return relate_spectrum_to_zeros(spectrum)

class MathematicalGenome:
    """
    Genetic representation of mathematical expressions
    """
    def __init__(self, expression_tree):
        self.tree = expression_tree
        self.fitness = None
    
    def mutate(self):
        # Random mathematical operations
        mutations = [
            add_term,
            change_coefficient,
            modify_exponent,
            introduce_special_function
        ]
        random.choice(mutations)(self.tree)
    
    def crossover(self, other):
        # Combine mathematical ideas
        child_tree = combine_expressions(self.tree, other.tree)
        return MathematicalGenome(child_tree)
    
    def evaluate_fitness(self):
        # How well does this approximate ζ(s)?
        self.fitness = zeta_approximation_quality(self.tree)

def evolve_zeta_insights(generations=10000):
    """
    Evolve mathematical expressions to gain insights into ζ(s)
    """
    population = initialize_mathematical_population()
    
    for generation in range(generations):
        # Evaluate fitness
        for genome in population:
            genome.evaluate_fitness()
        
        # Selection
        survivors = select_fittest(population)
        
        # Reproduction
        offspring = []
        for parent1, parent2 in pairs(survivors):
            child = parent1.crossover(parent2)
            child.mutate()
            offspring.append(child)
        
        population = survivors + offspring
        
        # Check for breakthrough discoveries
        check_for_insights(population)
    
    return extract_best_insights(population)

def artificial_mathematical_intuition():
    """
    Simulate mathematical intuition for RH breakthrough
    """
    # Combine multiple AI approaches
    pattern_recognizer = DeepPatternNet()
    logic_engine = SymbolicReasoningEngine()
    creativity_module = CreativeLeapGenerator()
    
    # Input: Current state of RH knowledge
    current_knowledge = load_rh_knowledge_base()
    
    # Pattern recognition phase
    patterns = pattern_recognizer.discover_patterns(current_knowledge)
    
    # Logical reasoning phase
    logical_steps = logic_engine.derive_implications(patterns)
    
    # Creative leap phase
    insights = creativity_module.generate_novel_connections(
        patterns, logical_steps
    )
    
    # Validation phase
    validated_insights = validate_mathematical_insights(insights)
    
    return validated_insights

class MetaGOAPMathematicalSolver:
    """
    Meta-level GOAP system that coordinates multiple novel approaches
    """
    def __init__(self):
        self.approaches = [
            QuantumInformationApproach(),
            AlgorithmicInformationApproach(),
            TopologicalDataAnalysis(),
            MachineLearningDiscovery(),
            HypergraphApproach(),
            EvolutionaryMathematics(),
            ArtificialIntuition()
        ]
        
        self.meta_optimizer = SublinearOptimizer()
    
    def solve_riemann_hypothesis(self):
        """
        Coordinate all approaches using meta-GOAP
        """
        # Build approach interaction matrix
        interaction_matrix = self.build_approach_synergies()
        
        # Optimize approach combination
        optimal_strategy = self.meta_optimizer.optimize(
            interaction_matrix,
            objective="prove_riemann_hypothesis",
            constraints=["computational_feasibility", "mathematical_rigor"]
        )
        
        # Execute optimal strategy
        results = self.execute_coordinated_approaches(optimal_strategy)
        
        # Synthesize insights
        breakthrough = self.synthesize_breakthrough(results)
        
        return breakthrough
    
    def build_approach_synergies(self):
        """
        Model how different approaches complement each other
        """
        # Quantum + Topological: Quantum topology of zeros
        # ML + Algorithmic: Pattern discovery in complexity
        # Evolutionary + Intuition: Creative mathematical evolution
        # etc.
        
        synergy_matrix = create_synergy_matrix(self.approaches)
        return synergy_matrix

def evaluate_approach_success(approach_results):
    """
    Evaluate the success of novel approaches
    """
    metrics = {
        'computational_progress': measure_verification_advance(),
        'theoretical_insight': assess_mathematical_novelty(),
        'interdisciplinary_value': evaluate_cross_field_impact(),
        'proof_proximity': estimate_distance_to_proof(),
        'methodology_innovation': assess_goap_effectiveness()
    }
    
    return weighted_success_score(metrics)