## Step 1: Define the symbolic representation of the problem
The problem involves minimizing an objective function subject to several constraints. The variables are 'protein bars', 'black beans', and 'ham sandwiches', which we can denote as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $8x_0 + 6x_1 + 9x_2$.

## 3: Define the symbolic variables and their corresponding natural language objects
The symbolic variables and their corresponding objects are:
- $x_0$ : protein bars
- $x_1$ : black beans
- $x_2$ : ham sandwiches

## 4: List the constraints in symbolic notation
The constraints are:
- $5x_0 + 14x_1 + x_2 \geq 20$ (at least 20 grams of carbohydrates from all sources)
- $14x_1 + x_2 \geq 6$ (at least 6 grams of carbohydrates from black beans and ham sandwiches)
- $5x_0 + 14x_1 \geq 15$ (at least 15 grams of carbohydrates from protein bars and black beans)
- $14x_1 + 14x_2 \geq 24$ (total umami index from black beans and ham sandwiches at least 24)
- $x_0 + 14x_1 \geq 10$ (total umami index from protein bars and black beans at least 10)
- $x_0 + 14x_1 + 14x_2 \geq 10$ (total umami index from all sources at least 10)
- $5x_0 + 14x_1 \leq 45$ (up to 45 grams of carbohydrates from protein bars and black beans)
- $14x_1 + x_2 \leq 58$ (up to 58 grams of carbohydrates from black beans and ham sandwiches)
- $x_0 + 14x_1 + 14x_2 \leq 48$ (total umami index from all sources at most 48)

## 5: Construct the Gurobi model
We will use Gurobi to solve this linear programming problem.

## 6: Implement the Gurobi code
```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x0 = model.addVar(name="protein_bars", lb=0)  # protein bars
x1 = model.addVar(name="black_beans", lb=0)  # black beans
x2 = model.addVar(name="ham_sandwiches", lb=0)  # ham sandwiches

# Define the objective function
model.setObjective(8 * x0 + 6 * x1 + 9 * x2, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(5 * x0 + 14 * x1 + x2 >= 20)  # at least 20 grams of carbohydrates
model.addConstr(14 * x1 + x2 >= 6)  # at least 6 grams of carbohydrates from black beans and ham sandwiches
model.addConstr(5 * x0 + 14 * x1 >= 15)  # at least 15 grams of carbohydrates from protein bars and black beans
model.addConstr(14 * x1 + 14 * x2 >= 24)  # umami index from black beans and ham sandwiches
model.addConstr(x0 + 14 * x1 >= 10)  # umami index from protein bars and black beans
model.addConstr(x0 + 14 * x1 + 14 * x2 >= 10)  # umami index from all sources
model.addConstr(5 * x0 + 14 * x1 <= 45)  # up to 45 grams of carbohydrates from protein bars and black beans
model.addConstr(14 * x1 + x2 <= 58)  # up to 58 grams of carbohydrates from black beans and ham sandwiches
model.addConstr(x0 + 14 * x1 + 14 * x2 <= 48)  # umami index from all sources

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objVal)
    print("Protein bars: ", x0.varValue)
    print("Black beans: ", x1.varValue)
    print("Ham sandwiches: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 7: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'protein bars'), ('x1', 'black beans'), ('x2', 'ham sandwiches')],
    'objective_function': '8*x0 + 6*x1 + 9*x2',
    'constraints': [
        '5*x0 + 14*x1 + x2 >= 20',
        '14*x1 + x2 >= 6',
        '5*x0 + 14*x1 >= 15',
        '14*x1 + 14*x2 >= 24',
        'x0 + 14*x1 >= 10',
        'x0 + 14*x1 + 14*x2 >= 10',
        '5*x0 + 14*x1 <= 45',
        '14*x1 + x2 <= 58',
        'x0 + 14*x1 + 14*x2 <= 48'
    ]
}
```