## Step 1: Define the symbolic representation of the problem

Let's define the symbolic variables:
- $x_1$ = number of rings
- $x_2$ = number of necklaces

The objective function is to maximize profit: $50x_1 + 75x_2$

The constraints are:
- Heating machine: $x_1 + 3x_2 \leq 15$
- Polishing machine: $2x_1 + 4x_2 \leq 12$
- Non-negativity: $x_1 \geq 0, x_2 \geq 0$

## Step 2: Convert the problem into a Gurobi code

We will use the Gurobi library in Python to solve this linear programming problem.

## Step 3: Write the Gurobi code

```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(lb=0, name="x1")  # number of rings
x2 = model.addVar(lb=0, name="x2")  # number of necklaces

# Define the objective function
model.setObjective(50 * x1 + 75 * x2, gurobi.GRB.MAXIMIZE)

# Define the constraints
model.addConstr(x1 + 3 * x2 <= 15, name="heating_machine")
model.addConstr(2 * x1 + 4 * x2 <= 12, name="polishing_machine")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution:")
    print(f"Number of rings: {x1.varValue}")
    print(f"Number of necklaces: {x2.varValue}")
    print(f"Max profit: {model.objVal}")
else:
    print("No optimal solution found")
```

## Step 4: Provide the symbolic representation in JSON format

```json
{
    'sym_variables': [('x1', 'rings'), ('x2', 'necklaces')],
    'objective_function': '50*x1 + 75*x2',
    'constraints': [
        'x1 + 3*x2 <= 15',
        '2*x1 + 4*x2 <= 12',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

The final answer is: 

```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(lb=0, name="x1")  # number of rings
x2 = model.addVar(lb=0, name="x2")  # number of necklaces

# Define the objective function
model.setObjective(50 * x1 + 75 * x2, gurobi.GRB.MAXIMIZE)

# Define the constraints
model.addConstr(x1 + 3 * x2 <= 15, name="heating_machine")
model.addConstr(2 * x1 + 4 * x2 <= 12, name="polishing_machine")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution:")
    print(f"Number of rings: {x1.varValue}")
    print(f"Number of necklaces: {x2.varValue}")
    print(f"Max profit: {model.objVal}")
else:
    print("No optimal solution found")
```