To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the problem statement.

Let's denote:
- \(x_1\) as the number of sapphire rings made,
- \(x_2\) as the number of ruby rings made.

The objective is to maximize profit. Given that the profit per sapphire ring is $500 and the profit per ruby ring is $400, the objective function can be represented algebraically as:
\[ \text{Maximize: } 500x_1 + 400x_2 \]

Now, let's consider the constraints based on the problem description:
1. The team making sapphire rings can make at most 10 sapphire rings per day: \( x_1 \leq 10 \)
2. The team making ruby rings can make at most 15 ruby rings per day: \( x_2 \leq 15 \)
3. The master jeweler can check at most 15 rings of either type per day: \( x_1 + x_2 \leq 15 \)

Additionally, we have non-negativity constraints since the number of rings cannot be negative:
- \( x_1 \geq 0 \)
- \( x_2 \geq 0 \)

The symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'number of sapphire rings'), ('x2', 'number of ruby rings')],
    'objective_function': '500*x1 + 400*x2',
    'constraints': ['x1 <= 10', 'x2 <= 15', 'x1 + x2 <= 15', 'x1 >= 0', 'x2 >= 0']
}
```

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

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="sapphire_rings")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="ruby_rings")

# Set the objective function
m.setObjective(500*x1 + 400*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 <= 10, "sapphire_limit")
m.addConstr(x2 <= 15, "ruby_limit")
m.addConstr(x1 + x2 <= 15, "jeweler_limit")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of sapphire rings: {x1.x}")
    print(f"Number of ruby rings: {x2.x}")
    print(f"Maximum profit: ${500*x1.x + 400*x2.x:.2f}")
else:
    print("No optimal solution found.")
```