To solve this problem, we first need to convert the natural language description into a symbolic representation. Let's define the variables as follows:

- $x_1$: The number of cans of soup.
- $x_2$: The number of sandwiches.

The objective function is to minimize the total cost, which can be represented algebraically as $1x_1 + 3x_2$, since each can of soup costs $1 and each sandwich costs $3.

The constraints are:
1. Calories: $200x_1 + 250x_2 \geq 2000$ (to ensure at least 2000 calories).
2. Protein: $5x_1 + 10x_2 \geq 100$ (to ensure at least 100 grams of protein).
3. Carbs: $4x_1 + 15x_2 \geq 100$ (to ensure at least 100 grams of carbs).

Non-negativity constraints are also implied since the number of cans of soup and sandwiches cannot be negative:
- $x_1 \geq 0$
- $x_2 \geq 0$

Thus, the symbolic representation in JSON format is:

```json
{
    'sym_variables': [('x1', 'number of cans of soup'), ('x2', 'number of sandwiches')], 
    'objective_function': '1*x1 + 3*x2', 
    'constraints': ['200*x1 + 250*x2 >= 2000', '5*x1 + 10*x2 >= 100', '4*x1 + 15*x2 >= 100', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's write the Gurobi code in Python to solve this linear programming problem:

```python
from gurobipy import *

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

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

# Set the objective function
m.setObjective(1*x1 + 3*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(200*x1 + 250*x2 >= 2000, "calories")
m.addConstr(5*x1 + 10*x2 >= 100, "protein")
m.addConstr(4*x1 + 15*x2 >= 100, "carbs")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of cans of soup: {x1.x}")
    print(f"Number of sandwiches: {x2.x}")
    print(f"Total cost: {m.objVal}")
else:
    print("No optimal solution found")
```