To solve this optimization problem using Gurobi, we first need to understand and translate the given natural language description into a mathematical model. The objective is to minimize the function: $1 \times \text{milligrams of vitamin B1} + 6 \times \text{milligrams of potassium}$, subject to several constraints involving cognitive performance indices and immune support indices for both milligrams of vitamin B1 and potassium.

Let's denote:
- $x_0$ as the quantity of milligrams of vitamin B1,
- $x_1$ as the quantity of milligrams of potassium.

Given information:
- Cognitive performance index per milligram of vitamin B1: 9
- Immune support index per milligram of vitamin B1: 2
- Cognitive performance index per milligram of potassium: 2
- Immune support index per milligram of potassium: 14

Constraints:
1. Total cognitive performance index $\geq 9$
2. Total immune support index $\geq 16$
3. $-10x_0 + 4x_1 \geq 0$
4. Total cognitive performance index $\leq 19$
5. Total immune support index $\leq 31$

The mathematical formulation of the problem can be written as:
- Minimize: $x_0 + 6x_1$
- Subject to:
  - $9x_0 + 2x_1 \geq 9$
  - $9x_0 + 2x_1 \leq 19$
  - $2x_0 + 14x_1 \geq 16$
  - $2x_0 + 14x_1 \leq 31$
  - $-10x_0 + 4x_1 \geq 0$
- $x_0$ can be non-integer (continuous),
- $x_1$ must be integer.

Here is the Gurobi code that captures this problem:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x0 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="milligrams_of_vitamin_B1")
x1 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.INTEGER, name="milligrams_of_potassium")

# Define the objective function
m.setObjective(x0 + 6*x1, GRB.MINIMIZE)

# Add constraints
m.addConstr(9*x0 + 2*x1 >= 9, "cognitive_performance_min")
m.addConstr(9*x0 + 2*x1 <= 19, "cognitive_performance_max")
m.addConstr(2*x0 + 14*x1 >= 16, "immune_support_min")
m.addConstr(2*x0 + 14*x1 <= 31, "immune_support_max")
m.addConstr(-10*x0 + 4*x1 >= 0, "additional_constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of Vitamin B1: {x0.x}")
    print(f"Milligrams of Potassium: {x1.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found")
```