To solve the given 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 algebraically.

Let's define:
- \(x_1\) as the number of cleaners,
- \(x_2\) as the number of receptionists.

The objective is to minimize the weekly wage bill, which can be represented as:
\[500x_1 + 350x_2\]

The constraints given are:
1. The total number of workers must be at least 100: \(x_1 + x_2 \geq 100\)
2. There must be at least 20 receptionists: \(x_2 \geq 20\)
3. The number of receptionists should be at least a third of the number of cleaners: \(x_2 \geq \frac{1}{3}x_1\)
4. The weekly wage bill should be below $30,000: \(500x_1 + 350x_2 \leq 30000\)

Now, let's represent this problem in the requested JSON format:

```json
{
    'sym_variables': [('x1', 'number of cleaners'), ('x2', 'number of receptionists')],
    'objective_function': '500*x1 + 350*x2',
    'constraints': ['x1 + x2 >= 100', 'x2 >= 20', 'x2 >= (1/3)*x1', '500*x1 + 350*x2 <= 30000']
}
```

Given this symbolic representation, we can now formulate the problem in Gurobi code to find the minimum wage bill:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(name='cleaners', vtype=GRB.INTEGER, lb=0)
x2 = m.addVar(name='receptionists', vtype=GRB.INTEGER, lb=0)

# Set the objective function
m.setObjective(500*x1 + 350*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(x1 + x2 >= 100, name='total_workers')
m.addConstr(x2 >= 20, name='min_receptionists')
m.addConstr(x2 >= (1/3)*x1, name='receptionist_to_cleaner_ratio')
m.addConstr(500*x1 + 350*x2 <= 30000, name='max_wage_bill')

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Minimum wage bill: {m.objVal}")
    print(f"Number of cleaners: {x1.x}")
    print(f"Number of receptionists: {x2.x}")
else:
    print("Model is infeasible")
```