To solve the given optimization problem, we first need to formulate it mathematically and then translate it into Gurobi code. The problem involves minimizing an objective function subject to several constraints.

### Mathematical Formulation:

**Variables:**
- \(x_0\): milligrams of vitamin B1
- \(x_1\): milligrams of potassium

**Objective Function:**
Minimize \(1x_0 + 6x_1\)

**Constraints:**
1. \(9x_0 + 2x_1 \geq 9\)
2. \(2x_0 + 14x_1 \geq 16\)
3. \(-10x_0 + 4x_1 \geq 0\)
4. \(9x_0 + 2x_1 \leq 19\)
5. \(2x_0 + 14x_1 \leq 31\)
6. \(x_0\) is continuous (can be non-integer)
7. \(x_1\) is integer (non-fractional)

### Gurobi Code:

```python
import gurobi

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

    # Define variables
    x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="milligrams_of_vitamin_B1")
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="milligrams_of_potassium", vtype=gurobi.GRB.INTEGER)

    # Objective function: Minimize 1*x0 + 6*x1
    model.setObjective(1*x0 + 6*x1, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(9*x0 + 2*x1 >= 9, name="cognitive_performance_index_min")
    model.addConstr(2*x0 + 14*x1 >= 16, name="immune_support_index_min")
    model.addConstr(-10*x0 + 4*x1 >= 0, name="linear_constraint")
    model.addConstr(9*x0 + 2*x1 <= 19, name="cognitive_performance_index_max")
    model.addConstr(2*x0 + 14*x1 <= 31, name="immune_support_index_max")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin B1: {x0.varValue}")
        print(f"Milligrams of potassium: {x1.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("No feasible solution exists.")
    else:
        print("Optimization failed.")

solve_optimization_problem()
```

This code defines the optimization problem as specified, using Gurobi to find an optimal solution. Note that the `vtype` parameter for `x1` (milligrams of potassium) is set to `gurobi.GRB.INTEGER` to ensure it is treated as an integer variable. The model is then solved, and the solution (if one exists) is printed out.