## Step 1: Define the symbolic representation of the problem
The problem involves variables: ['milligrams of calcium', 'milligrams of vitamin B6', 'milligrams of vitamin A', 'milligrams of vitamin B5']. Let's denote them as $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $1.27x_1x_2 + 9.65x_3^2 + 9.62x_2 + 4.91x_3 + 6.7x_4$.

## 3: List the semi-algebraic constraints
Constraints:
1. $8.07x_1 \geq 0$ (Implicit, as $x_1$ is non-negative)
2. $16.22x_1 \geq 0$ (Implicit, as $x_1$ is non-negative)
3. $14.06x_2 \geq 0$ (Implicit, as $x_2$ is non-negative)
4. $17.19x_2 \geq 0$ (Implicit, as $x_2$ is non-negative)
5. $8.43x_3 \geq 0$ (Implicit, as $x_3$ is non-negative)
6. $18.17x_3 \geq 0$ (Implicit, as $x_3$ is non-negative)
7. $14.41x_4 \geq 0$ (Implicit, as $x_4$ is non-negative)
8. $14.31x_4 \geq 0$ (Implicit, as $x_4$ is non-negative)
9. $8.07x_1 + 14.06x_2 \geq 27$
10. $14.06^2x_2^2 + 8.43^2x_3^2 \geq 35$
11. $8.07^2x_1^2 + 14.41^2x_4^2 \geq 24$
12. $14.06^2x_2^2 + 14.41^2x_4^2 \geq 26$
13. $14.06x_2 + 8.43x_3 + 14.41x_4 \geq 32$
14. $8.07^2x_1^2 + 14.06^2x_2^2 + 14.41^2x_4^2 \geq 32$
15. $8.07^2x_1^2 + 14.06^2x_2^2 + 8.43^2x_3^2 \geq 32$
16. $14.06x_2 + 8.43x_3 + 14.41x_4 \geq 20$
17. $8.07x_1 + 14.06x_2 + 14.41x_4 \geq 20$
18. $8.07^2x_1^2 + 14.06^2x_2^2 + 8.43^2x_3^2 \geq 20$
19. $14.06x_2 + 8.43x_3 + 14.41x_4 \geq 20$
20. $8.07x_1 + 14.06x_2 + 14.41x_4 \geq 20$
21. $8.07x_1 + 14.06x_2 + 8.43x_3 + 14.41x_4 \geq 20$
22. $18.17x_3 + 14.31x_4 \geq 37$
23. $17.19x_2 + 18.17x_3 \geq 35$
24. $16.22x_1 + 18.17x_3 \geq 13$
25. $16.22x_1 + 17.19x_2 \geq 38$
26. $16.22^2x_1^2 + 14.41^2x_4^2 \geq 18$
27. $17.19^2x_2^2 + 14.41^2x_4^2 \geq 35$
28. $16.22x_1 + 17.19x_2 + 18.17x_3 + 14.31x_4 \geq 35$
29. $-3x_2 + 7x_3 \geq 0$
30. $-10x_1 + 8x_3 \geq 0$
31. $8.43x_3 + 14.41x_4 \leq 119$
32. $16.22x_1 + 17.19x_2 \leq 47$
33. $16.22^2x_1^2 + 14.41^2x_4^2 \leq 67$
34. $17.19^2x_2^2 + 18.17^2x_3^2 + 14.41^2x_4^2 \leq 97$
35. $16.22x_1 + 17.19x_2 + 14.31x_4 \leq 156$
36. $16.22^2x_1^2 + 17.19^2x_2^2 + 18.17^2x_3^2 \leq 51$

## 4: Define the symbolic representation in JSON format
```json
{
    'sym_variables': [
        ('x1', 'milligrams of calcium'), 
        ('x2', 'milligrams of vitamin B6'), 
        ('x3', 'milligrams of vitamin A'), 
        ('x4', 'milligrams of vitamin B5')
    ], 
    'objective_function': '1.27*x1*x2 + 9.65*x3^2 + 9.62*x2 + 4.91*x3 + 6.7*x4', 
    'constraints': [
        '8.07*x1 + 14.06*x2 >= 27',
        '14.06^2*x2^2 + 8.43^2*x3^2 >= 35',
        '8.07^2*x1^2 + 14.41^2*x4^2 >= 24',
        '14.06^2*x2^2 + 14.41^2*x4^2 >= 26',
        '14.06*x2 + 8.43*x3 + 14.41*x4 >= 32',
        '8.07^2*x1^2 + 14.06^2*x2^2 + 14.41^2*x4^2 >= 32',
        '8.07^2*x1^2 + 14.06^2*x2^2 + 8.43^2*x3^2 >= 32',
        '14.06*x2 + 8.43*x3 + 14.41*x4 >= 20',
        '8.07*x1 + 14.06*x2 + 14.41*x4 >= 20',
        '8.07^2*x1^2 + 14.06^2*x2^2 + 8.43^2*x3^2 >= 20',
        '14.06*x2 + 8.43*x3 + 14.41*x4 >= 20',
        '8.07*x1 + 14.06*x2 + 14.41*x4 >= 20',
        '8.07*x1 + 14.06*x2 + 8.43*x3 + 14.41*x4 >= 20',
        '18.17*x3 + 14.31*x4 >= 37',
        '17.19*x2 + 18.17*x3 >= 35',
        '16.22*x1 + 18.17*x3 >= 13',
        '16.22*x1 + 17.19*x2 >= 38',
        '16.22^2*x1^2 + 14.41^2*x4^2 >= 18',
        '17.19^2*x2^2 + 14.41^2*x4^2 >= 35',
        '16.22*x1 + 17.19*x2 + 18.17*x3 + 14.31*x4 >= 35',
        '-3*x2 + 7*x3 >= 0',
        '-10*x1 + 8*x3 >= 0',
        '8.43*x3 + 14.41*x4 <= 119',
        '16.22*x1 + 17.19*x2 <= 47',
        '16.22^2*x1^2 + 14.41^2*x4^2 <= 67',
        '17.19^2*x2^2 + 18.17^2*x3^2 + 14.41^2*x4^2 <= 97',
        '16.22*x1 + 17.19*x2 + 14.31*x4 <= 156',
        '16.22^2*x1^2 + 17.19^2*x2^2 + 18.17^2*x3^2 <= 51',
        'x1 % 1 == 0',
        'x3 % 1 == 0',
        'x4 % 1 == 0'
    ]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="x1", vtype=gp.GRB.INTEGER)  # milligrams of calcium
x2 = m.addVar(name="x2")  # milligrams of vitamin B6
x3 = m.addVar(name="x3", vtype=gp.GRB.INTEGER)  # milligrams of vitamin A
x4 = m.addVar(name="x4", vtype=gp.GRB.INTEGER)  # milligrams of vitamin B5

# Objective function
m.setObjective(1.27*x1*x2 + 9.65*x3**2 + 9.62*x2 + 4.91*x3 + 6.7*x4, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(8.07*x1 + 14.06*x2 >= 27)
m.addConstr(14.06**2*x2**2 + 8.43**2*x3**2 >= 35)
m.addConstr(8.07**2*x1**2 + 14.41**2*x4**2 >= 24)
m.addConstr(14.06**2*x2**2 + 14.41**2*x4**2 >= 26)
m.addConstr(14.06*x2 + 8.43*x3 + 14.41*x4 >= 32)
m.addConstr(8.07**2*x1**2 + 14.06**2*x2**2 + 14.41**2*x4**2 >= 32)
m.addConstr(8.07**2*x1**2 + 14.06**2*x2**2 + 8.43**2*x3**2 >= 32)
m.addConstr(14.06*x2 + 8.43*x3 + 14.41*x4 >= 20)
m.addConstr(8.07*x1 + 14.06*x2 + 14.41*x4 >= 20)
m.addConstr(8.07**2*x1**2 + 14.06**2*x2**2 + 8.43**2*x3**2 >= 20)
m.addConstr(14.06*x2 + 8.43*x3 + 14.41*x4 >= 20)
m.addConstr(8.07*x1 + 14.06*x2 + 14.41*x4 >= 20)
m.addConstr(8.07*x1 + 14.06*x2 + 8.43*x3 + 14.41*x4 >= 20)
m.addConstr(18.17*x3 + 14.31*x4 >= 37)
m.addConstr(17.19*x2 + 18.17*x3 >= 35)
m.addConstr(16.22*x1 + 18.17*x3 >= 13)
m.addConstr(16.22*x1 + 17.19*x2 >= 38)
m.addConstr(16.22**2*x1**2 + 14.41**2*x4**2 >= 18)
m.addConstr(17.19**2*x2**2 + 14.41**2*x4**2 >= 35)
m.addConstr(16.22*x1 + 17.19*x2 + 18.17*x3 + 14.31*x4 >= 35)
m.addConstr(-3*x2 + 7*x3 >= 0)
m.addConstr(-10*x1 + 8*x3 >= 0)
m.addConstr(8.43*x3 + 14.41*x4 <= 119)
m.addConstr(16.22*x1 + 17.19*x2 <= 47)
m.addConstr(16.22**2*x1**2 + 14.41**2*x4**2 <= 67)
m.addConstr(17.19**2*x2**2 + 18.17**2*x3**2 + 14.41**2*x4**2 <= 97)
m.addConstr(16.22*x1 + 17.19*x2 + 14.31*x4 <= 156)
m.addConstr(16.22**2*x1**2 + 17.19**2*x2**2 + 18.17**2*x3**2 <= 51)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
    print("x4: ", x4.varValue)
else:
    print("The model is infeasible")
```