## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'milligrams of calcium' and 'milligrams of vitamin E'. Let's denote 'milligrams of calcium' as $x_1$ and 'milligrams of vitamin E' as $x_2$. The objective function to minimize is $5.76x_1 + 4.41x_2$. The constraints are:
1. $20x_1 + 13x_2 \geq 48$
2. $-9x_1 + 10x_2 \geq 0$
3. $20x_1 + 13x_2 \leq 231$
4. $20x_1 + 13x_2 \geq 48$ is repeated, but it's the same as constraint 1.

## 2: Correct and consolidate the constraints
Upon review, we see that constraints 1 and 4 are identical, so we keep only one of them. The constraints are:
1. $20x_1 + 13x_2 \geq 48$
2. $-9x_1 + 10x_2 \geq 0$
3. $20x_1 + 13x_2 \leq 231$

## 3: Symbolic representation
The symbolic representation of the problem is:
- Variables: $x_1$ for 'milligrams of calcium', $x_2$ for 'milligrams of vitamin E'
- Objective function: $5.76x_1 + 4.41x_2$
- Constraints:
  1. $20x_1 + 13x_2 \geq 48$
  2. $-9x_1 + 10x_2 \geq 0$
  3. $20x_1 + 13x_2 \leq 231$

## 4: Gurobi code implementation
To implement this in Gurobi, we use the following Python code:

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="milligrams_of_calcium", lb=0)  # Assuming non-negative
    x2 = model.addVar(name="milligrams_of_vitamin_E", lb=0)  # Assuming non-negative

    # Objective function
    model.setObjective(5.76 * x1 + 4.41 * x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(20 * x1 + 13 * x2 >= 48, name="immune_support_index_min")
    model.addConstr(-9 * x1 + 10 * x2 >= 0, name="vitamin_E_calcium_balance")
    model.addConstr(20 * x1 + 13 * x2 <= 231, name="immune_support_index_max")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of calcium: {x1.varValue}")
        print(f"Milligrams of vitamin E: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```

## 5: JSON representation of the problem
The symbolic representation in JSON format is:

```json
{
  "sym_variables": [
    ["x1", "milligrams of calcium"],
    ["x2", "milligrams of vitamin E"]
  ],
  "objective_function": "5.76x1 + 4.41x2",
  "constraints": [
    "20x1 + 13x2 >= 48",
    "-9x1 + 10x2 >= 0",
    "20x1 + 13x2 <= 231"
  ]
}
```