To solve this problem, we'll use the Gurobi optimization library in Python. The model will have variables representing the amounts of each vitamin and fat, and constraints to ensure that all conditions are met.

```python
from gurobi import *

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

# Define the variables (allowing for non-integer values)
vitamin_b2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Vitamin_B2")
vitamin_d = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Vitamin_D")
vitamin_c = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Vitamin_C")
vitamin_b5 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Vitamin_B5")
vitamin_b9 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Vitamin_B9")
fat = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Fat")

# Constraints for energy stability index
m.addConstr(vitamin_c + vitamin_b5 - 6 * vitamin_b5 >= 98)
m.addConstr(4 * vitamin_c - 6 * vitamin_b5 >= 0)

# Constraints for cognitive performance index
m.addConstr(vitamin_d + vitamin_b5 <= 709)
m.addConstr(vitamin_b2 + vitamin_b5 <= 670)
m.addConstr(vitamin_b2 + vitamin_b9 <= 332)
m.addConstr(vitamin_b2 + vitamin_c <= 571)
m.addConstr(vitamin_b9 + fat <= 224)
m.addConstr(vitamin_b2 + vitamin_c + vitamin_b5 <= 312)
m.addConstr(vitamin_d + vitamin_c + vitamin_b5 <= 204)
m.addConstr(vitamin_d + vitamin_b9 + fat <= 301)
m.addConstr(vitamin_b2 + vitamin_d + vitamin_b9 <= 190)
m.addConstr(vitamin_b2 + vitamin_b5 + fat <= 277)
m.addConstr(vitamin_b2 + vitamin_d + vitamin_c <= 481)
m.addConstr(vitamin_d + vitamin_b5 + vitamin_b9 <= 181)
m.addConstr(vitamin_d + vitamin_c + vitamin_b9 <= 433)
m.addConstr(vitamin_b2 + vitamin_d + vitamin_b5 <= 403)
m.addConstr(vitamin_b2 + vitamin_c + fat <= 670)
m.addConstr(vitamin_b2 + vitamin_d + fat <= 291)

# Objective: Maximize the total amount of nutrients
m.setObjective(vitamin_b2 + vitamin_d + vitamin_c + vitamin_b5 + vitamin_b9 + fat, GRB.MAXIMIZE)

# Solve the model
m.optimize()

```python
```