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class WaveletKnotAnalyzer:
    """
    Combines wavelet geometric optimization with knot distance matrices
    to extract robust topological invariants
    """

    def __init__(self, knot_curve, wavelet_basis='db4'):
        self.knot = knot_curve
        self.wavelet_basis = wavelet_basis
        self.distance_matrix = None
        self.wavelet_coefficients = None

    def compute_wavelet_distance_features(self, n_samples=200):
        # Sample knot uniformly
        points = self.sample_knot(n_samples)

        # Compute distance matrix
        D = self.compute_distance_matrix(points)

        # Apply 2D wavelet transform to distance matrix
        coeffs = pywt.wavedec2(D, self.wavelet_basis, level=5)

        # Extract multi-scale features
        features = self.extract_multiscale_features(coeffs)

        # Apply autoadaptive permutation to find optimal basis
        optimized_features = self.autoadaptive_optimization(features)

        return optimized_features

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def wavelet_persistent_homology(distance_matrix, wavelet='sym4'):
    """
    Compute persistent homology at multiple wavelet scales
    """
    # Wavelet decomposition
    coeffs = pywt.wavedec2(distance_matrix, wavelet, level=4)

    persistence_features = []

    # For each scale
    for level in range(len(coeffs)):
        # Reconstruct distance matrix at this scale
        D_scale = wavelet_reconstruct_level(coeffs, level)

        # Compute persistence
        dgm = ripser(D_scale, maxdim=1)['dgms']

        # Extract scale-specific features
        features = {
            'max_persistence': np.max(dgm[1][:, 1] - dgm[1][:, 0]) if len(dgm[1]) > 0 else 0,
            'total_persistence': np.sum(dgm[1][:, 1] - dgm[1][:, 0]) if len(dgm[1]) > 0 else 0,
            'betti_1': len(dgm[1]),
            'persistence_entropy': compute_persistence_entropy(dgm[1])
        }

        persistence_features.append(features)

    return persistence_features

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class AutoadaptiveKnotBasis:
    """
    Discovers optimal wavelet bases for different knot families
    through geometric optimization
    """

    def __init__(self, knot_family):
        self.knot_family = knot_family
        self.basis_wavelets = self.initialize_wavelet_dictionary()
        self.optimal_permutation = None

    def optimize_basis(self, training_knots):
        """
        Find optimal wavelet basis for distinguishing knots in family
        """
        # Initialize with standard wavelet families
        candidate_bases = ['db4', 'sym4', 'coif2', 'bior3.5']

        best_score = -np.inf
        best_basis = None

        for basis in candidate_bases:
            # Compute features for all training knots
            features = []
            for knot in training_knots:
                D = compute_distance_matrix(knot)
                coeffs = pywt.wavedec2(D, basis)
                features.append(self.extract_invariant_features(coeffs))

            # Measure discriminative power
            score = self.compute_fisher_score(features, knot_labels)

            if score > best_score:
                best_score = score
                best_basis = basis

        # Apply autoadaptive permutation refinement
        self.refine_with_permutation(best_basis, training_knots)

        return best_basis

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def optimize_knot_embedding(knot, n_points=200):
    """
    Find optimal point distribution on knot for invariant extraction
    """
    # Initial uniform sampling
    points = uniform_sample(knot, n_points)

    # Define optimization objective
    def objective(point_params):
        # Reconstruct points from parameters
        points = params_to_points(point_params, knot)

        # Compute distance matrix
        D = distance_matrix(points)

        # Objective combines multiple criteria
        spectral_reg = np.sum(np.abs(np.linalg.eigvals(D)[:10]))  # Spectral sparsity
        crossing_clarity = compute_crossing_score(D, knot)         # Clear crossings
        isotopy_stability = compute_stability_score(D)             # Stable under perturbation

        return spectral_reg + 0.5 * crossing_clarity + 0.3 * isotopy_stability

    # Optimize point positions along knot
    result = minimize(objective, points.flatten(), method='L-BFGS-B')

    return result.x.reshape(-1, 3)

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class MultiScaleKnotSignature:
    def __init__(self, knot):
        self.knot = knot
        self.signatures = {}

    def compute_signature(self, scales=[50, 100, 200, 400]):
        """
        Compute knot signatures at multiple sampling densities
        """
        for n in scales:
            # Sample at this scale
            points = optimized_sample(self.knot, n)
            D = distance_matrix(points)

            # Wavelet transform
            coeffs = pywt.wavedec2(D, 'db4', level=int(np.log2(n))-2)

            # Extract scale-invariant features
            self.signatures[n] = {
                'energy_distribution': self.compute_energy_distribution(coeffs),
                'cross_scale_correlation': self.compute_cross_scale_correlation(coeffs),
                'topological_moments': self.compute_topological_moments(coeffs)
            }

        # Combine across scales
        return self.fuse_multiscale_signatures()

    def compute_energy_distribution(self, coeffs):
        """Energy distribution across wavelet scales"""
        energies = []
        for level_coeffs in coeffs:
            if isinstance(level_coeffs, tuple):
                energy = sum(np.sum(c**2) for c in level_coeffs)
            else:
                energy = np.sum(level_coeffs**2)
            energies.append(energy)
        return np.array(energies) / np.sum(energies)

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def discover_knot_families(knot_database, n_families=None):
    """
    Automatically discover knot families through wavelet feature clustering
    """
    # Compute wavelet features for all knots
    features = []
    for knot in knot_database:
        D = compute_distance_matrix(knot)

        # Multi-wavelet decomposition
        feature_vec = []
        for wavelet in ['db4', 'sym4', 'coif2']:
            coeffs = pywt.wavedec2(D, wavelet)
            feature_vec.extend(extract_statistical_features(coeffs))

        features.append(feature_vec)

    # Apply dimensionality reduction
    features_reduced = UMAP(n_components=10).fit_transform(features)

    # Hierarchical clustering to discover families
    if n_families is None:
        # Automatic selection using silhouette score
        n_families = select_optimal_clusters(features_reduced)

    clusters = AgglomerativeClustering(n_clusters=n_families).fit_predict(features_reduced)

    # Characterize each family
    families = {}
    for i in range(n_families):
        family_knots = [knot_database[j] for j in np.where(clusters == i)[0]]
        families[f'family_{i}'] = {
            'knots': family_knots,
            'characteristic_wavelet': find_characteristic_wavelet(family_knots),
            'invariant_signature': compute_family_signature(family_knots)
        }

    return families

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class QuantumKnotInvariant:
    """
    Compute knot invariants inspired by quantum wavelet states
    """

    def __init__(self, knot):
        self.knot = knot
        self.hilbert_dim = 256  # Dimension of quantum Hilbert space

    def compute_quantum_invariant(self):
        # Distance matrix as quantum observable
        D = compute_distance_matrix(self.knot, n=self.hilbert_dim)

        # Normalize to quantum density matrix
        rho = D / np.trace(D)
        rho = (rho + rho.T) / 2  # Ensure Hermitian

        # Eigendecomposition gives quantum states
        eigenvalues, eigenvectors = np.linalg.eigh(rho)

        # Wavelet transform of eigenvectors
        wavelet_states = []
        for vec in eigenvectors.T:
            coeffs = pywt.wavedec(vec, 'db4', level=4)
            wavelet_states.append(coeffs)

        # Quantum invariants
        invariants = {
            'von_neumann_entropy': -np.sum(eigenvalues * np.log(eigenvalues + 1e-10)),
            'quantum_fidelity': self.compute_fidelity(wavelet_states),
            'entanglement_spectrum': self.compute_entanglement(wavelet_states),
            'topological_entanglement_entropy': self.compute_tee(eigenvalues)
        }

        return invariants

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class WaveletKnotHash:
    """
    Fast knot recognition using wavelet-based locality-sensitive hashing
    """

    def __init__(self, hash_size=128):
        self.hash_size = hash_size
        self.hash_functions = self.generate_hash_functions()

    def compute_hash(self, knot):
        # Quick distance matrix computation
        points = uniform_sample(knot, n=64)  # Small for speed
        D = distance_matrix(points)

        # Fast wavelet transform
        coeffs = pywt.dwt2(D, 'db2')  # Single level for speed

        # Extract hash
        hash_bits = []
        for h in self.hash_functions:
            bit = h(coeffs) > 0
            hash_bits.append(bit)

        return np.packbits(hash_bits)

    def build_index(self, knot_database):
        """Build LSH index for fast retrieval"""
        self.index = {}

        for knot in knot_database:
            hash_val = self.compute_hash(knot)
            hash_key = hash_val.tobytes()

            if hash_key not in self.index:
                self.index[hash_key] = []
            self.index[hash_key].append(knot)

    def query(self, query_knot, k=10):
        """Find k most similar knots"""
        query_hash = self.compute_hash(query_knot)

        # Hamming distance to all hash keys
        distances = []
        for hash_key, knots in self.index.items():
            stored_hash = np.frombuffer(hash_key, dtype=np.uint8)
            dist = np.sum(np.unpackbits(query_hash ^ stored_hash))
            distances.append((dist, knots))

        # Return k nearest
        distances.sort(key=lambda x: x[0])
        results = []
        for dist, knots in distances[:k]:
            results.extend(knots)
            if len(results) >= k:
                break

        return results[:k]

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def knot_emergence_simulation():
    """
    Simulate how discrete knot types emerge from continuous optimization
    """
    # Start with random closed curve
    initial_curve = generate_random_closed_curve()

    # Define energy functional combining:
    # - Geometric packing (from optimization project)
    # - Topological complexity (from knot project)
    def knot_energy(curve_params):
        curve = params_to_curve(curve_params)

        # Geometric term: self-distance
        geometric_energy = compute_ropelength(curve)

        # Topological term: crossing complexity
        D = distance_matrix(sample_curve(curve))
        topo_complexity = compute_crossing_number_lower_bound(D)

        # Quantum term: wavelet coherence
        wavelet_coherence = compute_wavelet_coherence(D)

        return geometric_energy - 0.1 * topo_complexity + 0.05 * wavelet_coherence

    # Optimize
    result = minimize(knot_energy, initial_curve.params, method='L-BFGS-B')

    # Classify resulting knot
    final_knot = params_to_curve(result.x)
    knot_type = classify_knot(final_knot)

    return knot_type

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class UniversalTopologicalWavelet:
    """
    Search for universal wavelet basis for topological invariants
    """

    def __init__(self):
        self.basis = None
        self.optimization_history = []

    def meta_optimize(self, topology_examples):
        """
        Find wavelet basis that optimally represents various topological spaces
        """
        # Include examples from:
        # - Knots and links
        # - Manifolds of different dimensions
        # - Simplicial complexes
        # - Point clouds

        # Parameterize wavelet family
        def parameterized_wavelet(params):
            # Generate wavelet from parameters
            return construct_wavelet(params)

        # Objective: maximize topological discrimination
        def objective(wavelet_params):
            wavelet = parameterized_wavelet(wavelet_params)

            total_score = 0
            for example in topology_examples:
                # Compute wavelet features
                features = compute_wavelet_features(example, wavelet)

                # Measure topological information content
                info_content = compute_topological_information(features)

                # Measure stability
                stability = compute_feature_stability(features, example)

                total_score += info_content * stability

            return -total_score  # Minimize negative score

        # Optimize wavelet parameters
        result = differential_evolution(objective, bounds=wavelet_param_bounds)

        self.basis = parameterized_wavelet(result.x)
        return self.basis