Unconventional Approaches to Lepton Collider Analysis: Clifford Groups, p-adics, and B-series.

# Example of a simple Clifford element in matrix form
# (This would represent a specific spin rotation in our system)
import numpy as np

def simple_clifford_matrix(angle):
    """Create a basic Clifford group element representing rotation"""
    return np.array([
        [np.cos(angle/2), -np.sin(angle/2)],
        [np.sin(angle/2), np.cos(angle/2)]
    ])

B(g, h) = sum_T (h^|T|/σ(T)) * a_T * φ_T(g)

# Example usage of our dependency manager
dep_manager = DependencyManager()
if dep_manager.check_requirements([('sympy', 'for p-adic calculations')]):
    # We can use p-adic functionality
    Qp = dep_manager.get('sympy', 'Qp')
    p_field = Qp(3, prec=10)  # Create 3-adic field with precision 10

def phi_T(tree, g):
    """Calculate phi_T(g) for a given tree structure"""
    prod = 1.0
    for c in tree.children_count:
        prod *= g(c)
    return prod

def bseries_evaluation(g, h, chi, trees, coords):
    """Evaluate a B-series expansion"""
    result = 0.0
    for tree in trees:
        w = tree.weight
        sig = tree.sigma
        a_t = tree.coeff(chi)
        phi = phi_T(tree, g)
        term = (h ** w) / sig * a_t * phi
        result += term
    return result

# SageMath pseudocode example:
Qp = Qp(p, prec=precision)
converted_count = 0
for lambda_val in eigenvalues:
    try:
        ev_padic = Qp(lambda_val)
        if ev_padic.valuation() < 1:
            return False
        converted_count += 1
    except Exception as e:
        return False
return converted_count > 0

def sample_toric_probability(self, n_samples=1000):
    """
    Estimate the probability that a randomly generated beam transport or spin rotation
    matrix (Clifford group element) satisfies the p-adic toric stability condition.
    This is relevant for assessing the statistical likelihood of numerically stable
    transformations in lepton collider simulations.
    """
    if n_samples <= 0:
        return 0.0

    valid_count = 0
    for i in range(n_samples):
        try:
            # Generate a random Clifford group element (e.g., beam transport or spin rotation)
            random_element = self.generate_random_element()
            matrix = self.get_matrix_representation(random_element)
            # Check the p-adic toric condition (all eigenvalues satisfy v_p(λ) ≥ 1)
            if self.check_toric_condition(matrix):
                valid_count += 1
        except Exception:
            # Ignore numerically unstable samples (e.g., singular matrices or eigenvalue computation failure)
            continue

    return valid_count / n_samples

def feynman_to_bseries_tree(diagram_topology):
    """Convert a Feynman diagram topology to a B-series tree"""
    # Simplified example - real implementation would be more complex
    vertices = count_vertices(diagram_topology)
    propagators = count_propagators(diagram_topology)
    weight = propagators  # In this simplified model

    # Map electron-positron interactions to tree structures
    children_counts = []
    for vertex in diagram_topology.vertices:
        children_counts.append(len(vertex.outgoing) - 1)

    # Define coefficient function based on diagram type
    def coeff(chi):
        # For e+e- -> Higgs processes, for example
        return chi ** (vertices - 1) * (1 - chi) ** propagators

    return BSeriesTree(
        tree_id=f"feynman_{hash(diagram_topology)}",
        weight=weight,
        sigma=calculate_symmetry_factor(diagram_topology),
        children_count=children_counts,
        coeff=coeff
    )

class LeptonSpinAnalyzer:
    """Analyze spin precession in magnetic fields using Clifford groups"""

    def __init__(self, n_particles=1000, field_strength=1.0):
        self.analyzer = CliffordAnalyzer(n_qubits=1)  # 1 qubit for a single spin-1/2
        self.n_particles = n_particles
        self.field_strength = field_strength

    def simulate_spin_precession(self, time_steps=100):
        """Simulate spin precession in a uniform magnetic field"""
        # Initialize spin-1/2 states for all leptons in the beam
        spins = [self.analyzer.generate_random_element() for _ in range(self.n_particles)]

        results = []
        for t in range(time_steps):
            # Compute precession angle due to external magnetic field (Larmor precession)
            angle = self.field_strength * t * 0.01  # Simplified Larmor frequency model
            precession = simple_clifford_matrix(angle)

            # Apply spin rotation to each lepton (matrix multiplication in Clifford algebra)
            spins = [precession @ spin for spin in spins]

            # Measure longitudinal polarization (z-projection) of the beam
            polarization = self.measure_polarization(spins)
            results.append(polarization)

        return results

    def measure_polarization(self, spins):
        """Calculate net polarization of the beam"""
        # Simplified implementation
        return sum(self.get_z_projection(spin) for spin in spins) / self.n_particles

def padic_regularize_amplitude(amplitude_terms, p=3, precision=10):
    """Apply p-adic regularization to potentially divergent terms"""
    qp = Qp(p, prec=precision)

    regularized_terms = []
    for term in amplitude_terms:
        # Convert to p-adic
        term_padic = qp(term)

        # Check valuation
        val = term_padic.valuation()

        if val < 0:
            # Divergent term detected (negative p-adic valuation)
            # Apply regularization to suppress infrared/ultraviolet divergence
            regularized = 1.0 / (p ** abs(val))
        else:
            # Term is numerically stable, retain original value
            regularized = float(term)

        regularized_terms.append(regularized)

    return regularized_terms

def borel_resum_bseries(g, trees, coords, borel_parameter=0.5):
    """Apply Borel resummation to a B-series expansion"""
    # Transform original series into Borel-transformed series
    borel_trees = []
    for tree in trees:
        # Create modified tree with weight-adjusted coefficient
        def borel_coeff(chi):
            return tree.coeff(chi) / math.factorial(tree.weight)

        borel_tree = BSeriesTree(
            tree_id=f"borel_{tree.tree_id}",
            weight=tree.weight,
            sigma=tree.sigma,
            children_count=tree.children_count,
            coeff=borel_coeff
        )
        borel_trees.append(borel_tree)

    # Evaluate Borel transform at t=borel_parameter
    borel_sum = bseries_evaluation(g, borel_parameter, 1.0, borel_trees, coords)

    # Multiply by integration factor
    result = borel_sum * math.exp(-borel_parameter)

    return result

def fit_to_experimental_data(model, experimental_data, parameter_ranges):
    """Find best-fit parameters for our model against experimental data"""
    best_params = None
    best_fit_score = float('inf')

    # Simple grid search for illustration
    for params in grid_search(parameter_ranges):
        predictions = model(params)
        fit_score = mean_squared_error(experimental_data, predictions)

        if fit_score < best_fit_score:
            best_fit_score = fit_score
            best_params = params

    return best_params, best_fit_score

def create_padic_visualization(values, p=3, max_power=10):
    """Create visualization of p-adic valuations"""
    import matplotlib.pyplot as plt

    # Calculate valuations
    valuations = [padic_valuation(v, p) for v in values if v != 0]

    # Create histogram
    plt.figure(figsize=(10, 6))
    plt.hist(valuations, bins=range(min(min(valuations), -max_power), max_power+1))
    plt.xlabel(f"p-adic valuation (p={p})")
    plt.ylabel("Frequency")
    plt.title("Distribution of p-adic valuations")
    plt.grid(True, alpha=0.3)
    plt.axvline(x=0, color='red', linestyle='--', alpha=0.5)

    return plt