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

- $x_1$ as the number of jars of pasta sauce produced.
- $x_2$ as the number of jars of barbecue sauce produced.

The objective function aims to maximize profit. Given that the profit per jar of pasta sauce is $3 and the profit per jar of barbecue sauce is $5, the objective function can be represented as:

$$\text{Maximize: } 3x_1 + 5x_2$$

The constraints are based on the availability of the filling and jarring machines. Each jar of pasta sauce takes 1 minute on the filling machine and 3 minutes on the jarring machine, while each jar of barbecue sauce takes 3 minutes on the filling machine and 4 minutes on the jarring machine. The filling machine is available for 12,500 minutes, and the jarring machine is available for 20,000 minutes. Therefore, the constraints can be represented as:

- Filling machine constraint: $x_1 + 3x_2 \leq 12500$
- Jarring machine constraint: $3x_1 + 4x_2 \leq 20000$

Additionally, since we cannot produce a negative number of jars, we have non-negativity constraints:

- $x_1 \geq 0$
- $x_2 \geq 0$

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'number of jars of pasta sauce'), ('x2', 'number of jars of barbecue sauce')],
    'objective_function': '3*x1 + 5*x2',
    'constraints': ['x1 + 3*x2 <= 12500', '3*x1 + 4*x2 <= 20000', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this linear programming problem, we can use the Gurobi Python interface. Here is how you could implement it:

```python
from gurobipy import *

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

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

# Set the objective function
m.setObjective(3*x1 + 5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + 3*x2 <= 12500, "filling_machine")
m.addConstr(3*x1 + 4*x2 <= 20000, "jarring_machine")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Produce {x1.x} jars of pasta sauce")
    print(f"Produce {x2.x} jars of barbecue sauce")
    print(f"Maximum profit: ${m.objVal}")
else:
    print("No optimal solution found")
```