  Here is the Python code that solves the time-harmonic equation using adaptive methods:
```
import numpy as np
import scipy.sparse as sp
import scipy.linalg as la
import fealpy

# Define functions to recover the curl of a given solution and to calculate the least squares matrix for each node of a mesh
def curl_curl_solution(solution, mesh):
    # Calculate the curl of the solution
    curl_solution = np.zeros((mesh.number_of_vertices, mesh.number_of_faces))
    for i in range(mesh.number_of_faces):
        for j in range(mesh.number_of_vertices_of_faces[i]):
            curl_solution[i, j] = np.cross(mesh.vertices[i, j], mesh.vertices[i, (j+1)%mesh.number_of_vertices_of_faces[i]])
    return curl_solution

def least_squares_matrix(mesh):
    # Calculate the least squares matrix for each node of the mesh
    least_squares_matrix = np.zeros((mesh.number_of_vertices, mesh.number_of_faces))
    for i in range(mesh.number_of_faces):
        for j in range(mesh.number_of_vertices_of_faces[i]):
            least_squares_matrix[i, j] = np.dot(mesh.vertices[i, j], mesh.vertices[i, j])
    return least_squares_matrix

# Parse command-line arguments to set the degree of the first kind Nedelec element, the initial mesh size, the maximum number of adaptive iterations, and the theta parameter for adaptive iteration
degree = int(sys.argv[1])
initial_mesh_size = int(sys.argv[2])
max_adaptive_iterations = int(sys.argv[3])
theta = float(sys.argv[4])

# Initialize the problem using the CosSinData function from the fealpy library
data = fealpy.CosSinData(degree, initial_mesh_size)

# Create a 2D box mesh using the MeshFactory class from the fealpy library and remove the fourth quadrant of the mesh
mesh = fealpy.MeshFactory.create_box_mesh(data.domain, data.boundary, degree)
mesh.remove_cells(mesh.cells[mesh.number_of_cells//4:])

# Iterate over the maximum number of adaptive iterations, during each iteration:
for i in range(max_adaptive_iterations):
    # Define the function space using the FirstKindNedelecFiniteElementSpace2d class from the fealpy library
    space = fealpy.FirstKindNedelecFiniteElementSpace2d(mesh, degree)

    # Apply Dirichlet boundary conditions using the DirichletBC class from the fealpy library
    bc = fealpy.DirichletBC(space, data.boundary)

    # Solve the system of equations using the scipy library's spsolve function
    solution = sp.spsolve(bc.A, bc.b)

    # Calculate the L2 error between the solution and the exact solution, the curl of the solution and the exact curl, and the curl of the solution and the recovered curl
    l2_error = np.linalg.norm(solution - data.exact_solution)
    curl_error = np.linalg.norm(curl_curl_solution(solution, mesh) - data.exact_curl)
    recovered_curl_error = np.linalg.norm(curl_curl_solution(solution, mesh) - curl_curl_solution(data.exact_solution, mesh))

    # If not the last iteration, mark the cells for refinement based on the recovery error and refine the mesh
    if i < max_adaptive_iterations - 1:
        refinement_criteria = np.abs(recovered_curl_error - curl_error)
        mesh.mark_cells(refinement_criteria, 1)
        mesh.refine()

# Plot the error rates using the showmultirate function from the fealpy library
fealpy.showmultirate(l2_error, curl_error, recovered_curl_error)
```
Note that this code assumes that the `fealpy` library is installed and that the `sys` module is available. Additionally, this code uses the `CosSinData` function from the `fealpy` library to generate the initial mesh and the `FirstKindNedelecFiniteElementSpace2d` class to define the function space.