## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'signal platoons' and 'light infantry companies', which we can denote as $x_1$ and $x_2$ respectively. The objective function to maximize is $4x_1 + 4x_2$. The constraints are as follows:

- $4x_1 + 12x_2 \geq 78$ (total fuel demand of at least 78 gallons/day)
- $5x_1 + 9x_2 \geq 41$ (total combined fun factor of at least 41)
- $13x_1 + 2x_2 \geq 33$ (total combined mobility rating of at least 33)
- $17x_1 + 8x_2 \geq 17$ (total combined deployment weight of at least 17 metric tons)
- $12x_1 + 14x_2 \geq 17$ (total logistics footprint of at least 17 logistics units)
- $-x_1 + 9x_2 \geq 0$
- $4x_1 + 12x_2 \leq 112$ (total fuel demand of at most 112 gallons/day)
- $5x_1 + 9x_2 \leq 126$ (total combined fun factor of at most 126)
- $13x_1 + 2x_2 \leq 62$ (total combined mobility rating of at most 62)
- $17x_1 + 8x_2 \leq 48$ (total combined deployment weight of at most 48 metric tons)
- $12x_1 + 14x_2 \leq 25$ (total logistics footprint of at most 25 logistics units)
- $x_1, x_2 \geq 0$ and are integers.

## Step 2: Convert the problem into a Gurobi code
We will use Gurobi to solve this optimization problem. The Gurobi code will define the variables, the objective function, and the constraints.

## Step 3: Write down the symbolic representation in JSON format
```json
{
    'sym_variables': [('x1', 'signal platoons'), ('x2', 'light infantry companies')],
    'objective_function': '4*x1 + 4*x2',
    'constraints': [
        '4*x1 + 12*x2 >= 78',
        '5*x1 + 9*x2 >= 41',
        '13*x1 + 2*x2 >= 33',
        '17*x1 + 8*x2 >= 17',
        '12*x1 + 14*x2 >= 17',
        '-1*x1 + 9*x2 >= 0',
        '4*x1 + 12*x2 <= 112',
        '5*x1 + 9*x2 <= 126',
        '13*x1 + 2*x2 <= 62',
        '17*x1 + 8*x2 <= 48',
        '12*x1 + 14*x2 <= 25'
    ]
}
```

## Step 4: Implement the optimization problem using Gurobi
```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="signal_platoons", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="light_infantry_companies", vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(4 * x1 + 4 * x2, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(4 * x1 + 12 * x2 >= 78, name="fuel_demand_constraint")
    model.addConstr(5 * x1 + 9 * x2 >= 41, name="fun_factor_constraint")
    model.addConstr(13 * x1 + 2 * x2 >= 33, name="mobility_rating_constraint")
    model.addConstr(17 * x1 + 8 * x2 >= 17, name="deployment_weight_constraint")
    model.addConstr(12 * x1 + 14 * x2 >= 17, name="logistics_footprint_constraint")
    model.addConstr(-x1 + 9 * x2 >= 0, name="light_infantry_fun_factor_constraint")
    model.addConstr(4 * x1 + 12 * x2 <= 112, name="fuel_demand_upper_bound_constraint")
    model.addConstr(5 * x1 + 9 * x2 <= 126, name="fun_factor_upper_bound_constraint")
    model.addConstr(13 * x1 + 2 * x2 <= 62, name="mobility_rating_upper_bound_constraint")
    model.addConstr(17 * x1 + 8 * x2 <= 48, name="deployment_weight_upper_bound_constraint")
    model.addConstr(12 * x1 + 14 * x2 <= 25, name="logistics_footprint_upper_bound_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Signal platoons: {x1.varValue}")
        print(f"Light infantry companies: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```