To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints using these variables.

Let's define:
- $x_1$ as the number of new cooks,
- $x_2$ as the number of senior cooks.

The objective is to minimize the weekly wage bill, which can be represented as $500x_1 + 1000x_2$ since new cooks earn $500 a week and senior cooks earn $1000 a week.

The constraints given are:
1. The total wage bill must be kept below $50,000: $500x_1 + 1000x_2 \leq 50000$.
2. There must be at least 30 total cooks: $x_1 + x_2 \geq 30$.
3. At least 5 of the cooks must be senior: $x_2 \geq 5$.
4. The number of senior cooks should be at least a third the number of new cooks: $x_2 \geq \frac{1}{3}x_1$.

In symbolic notation, our problem can be represented as:
```json
{
  'sym_variables': [('x1', 'number of new cooks'), ('x2', 'number of senior cooks')],
  'objective_function': '500*x1 + 1000*x2',
  'constraints': [
    '500*x1 + 1000*x2 <= 50000',
    'x1 + x2 >= 30',
    'x2 >= 5',
    'x2 >= (1/3)*x1'
  ]
}
```

To solve this problem using Gurobi in Python, we'll use the following code:

```python
from gurobipy import *

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

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

# Set the objective function
m.setObjective(500*x1 + 1000*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(500*x1 + 1000*x2 <= 50000, "total_wage_bill")
m.addConstr(x1 + x2 >= 30, "total_cooks")
m.addConstr(x2 >= 5, "senior_cooks_minimum")
m.addConstr(x2 >= (1/3)*x1, "senior_to_new_ratio")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of new cooks: {x1.x}")
    print(f"Number of senior cooks: {x2.x}")
    print(f"Total wage bill: ${500*x1.x + 1000*x2.x}")
else:
    print("No optimal solution found")
```