## Symbolic Representation

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

Let's define the variables:
- $x_1$ = Number of jars of pasta sauce
- $x_2$ = Number of jars of barbecue sauce

The objective function to maximize profit is:
\[ \text{Maximize:} \quad 3x_1 + 5x_2 \]

The constraints based on the machine availability are:
- Filling machine: $1x_1 + 3x_2 \leq 12500$
- Jarring machine: $3x_1 + 4x_2 \leq 20000$
- Non-negativity: $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

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

## Gurobi Code in Python

```python
import gurobipy as gp

# Create a new model
model = gp.Model("Sauce_Production")

# Define variables
x1 = model.addVar(name="pasta_sauce", lb=0, vtype=gp.GRB.CONTINUOUS)  # jars of pasta sauce
x2 = model.addVar(name="barbecue_sauce", lb=0, vtype=gp.GRB.CONTINUOUS)  # jars of barbecue sauce

# Objective function: Maximize profit
model.setObjective(3*x1 + 5*x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(x1 + 3*x2 <= 12500, name="filling_machine_constraint")
model.addConstr(3*x1 + 4*x2 <= 20000, name="jarring_machine_constraint")

# Solve the model
model.optimize()

# Print the results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal production levels: {x1.varName} = {x1.x}, {x2.varName} = {x2.x}")
    print(f"Maximum profit: ${3*x1.x + 5*x2.x:.2f}")
else:
    print("The model is infeasible or unbounded.")
```