To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. Let's define the variables as follows:

- $x_1$: Number of international employees
- $x_2$: Number of local employees

The objective function aims to minimize the total weekly wage bill. Given that international employees earn $500 per week and local employees earn $1200 per week, the objective function can be represented algebraically as:

Minimize: $500x_1 + 1200x_2$

The constraints based on the problem description are:

1. The total number of employees should be at least 50: $x_1 + x_2 \geq 50$
2. At least 12 employees should be local: $x_2 \geq 12$
3. The number of local employees should be at least a third of the number of international employees: $x_2 \geq \frac{1}{3}x_1$
4. The total weekly wage bill should not exceed $40,000: $500x_1 + 1200x_2 \leq 40000$

Thus, the symbolic representation of the problem is:

```json
{
    'sym_variables': [('x1', 'Number of international employees'), ('x2', 'Number of local employees')],
    'objective_function': 'Minimize: 500*x1 + 1200*x2',
    'constraints': [
        'x1 + x2 >= 50',
        'x2 >= 12',
        'x2 >= (1/3)*x1',
        '500*x1 + 1200*x2 <= 40000'
    ]
}
```

Now, let's implement this problem using Gurobi in Python:

```python
from gurobipy import *

# Create a new model
m = Model("Wage_Bill_Optimization")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="international_employees")
x2 = m.addVar(vtype=GRB.INTEGER, name="local_employees")

# Set the objective function
m.setObjective(500*x1 + 1200*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(x1 + x2 >= 50, "total_employees")
m.addConstr(x2 >= 12, "minimum_local_employees")
m.addConstr(x2 >= (1/3)*x1, "local_to_international_ratio")
m.addConstr(500*x1 + 1200*x2 <= 40000, "wage_bill_limit")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"International Employees: {x1.x}")
    print(f"Local Employees: {x2.x}")
    print(f"Total Wage Bill: ${500*x1.x + 1200*x2.x}")
else:
    print("No optimal solution found")
```