To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. Let's denote:

- $x_1$ as the number of rings produced,
- $x_2$ as the number of necklaces produced.

The objective function aims to maximize profit. Given that each ring brings in $50 and each necklace brings in $75, the objective function can be represented algebraically as:
\[ 50x_1 + 75x_2 \]

Now, let's formulate the constraints based on the availability of machines:

1. The heating machine is available for at most 15 hours. Since it takes 1 hour to make a ring and 3 hours to make a necklace, this constraint can be represented as:
\[ x_1 + 3x_2 \leq 15 \]

2. The polishing machine is available for at most 12 hours. Given that it takes 2 hours to polish a ring and 4 hours to polish a necklace, the constraint for the polishing machine is:
\[ 2x_1 + 4x_2 \leq 12 \]

Additionally, we should consider non-negativity constraints since production cannot be negative:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Thus, our symbolic representation in JSON format is:

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

To solve this linear programming problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

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

# Add variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="number_of_rings_produced", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="number_of_necklaces_produced", lb=0)

# Set the objective function
m.setObjective(50*x1 + 75*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + 3*x2 <= 15, "heating_machine_constraint")
m.addConstr(2*x1 + 4*x2 <= 12, "polishing_machine_constraint")

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of rings produced: {x1.x}")
    print(f"Number of necklaces produced: {x2.x}")
    print(f"Maximum profit: ${50*x1.x + 75*x2.x:.2f}")
else:
    print("No optimal solution found.")

```