To solve the given optimization problem, we need to define the variables and constraints in a way that can be translated into Gurobi code. The objective function is defined as minimizing the value of `4.94 * vitamin_B12 + 9.67 * iron + 2.41 * zinc`. 

Given the complexity of the problem with multiple constraints, we'll identify the key components:
- Variables: `vitamin_B12`, `iron`, `zinc`
- Objective Function: Minimize `4.94 * vitamin_B12 + 9.67 * iron + 2.41 * zinc`
- Constraints:
  - Cognitive performance index constraints
  - Immune support index constraints
  - Muscle growth index constraints
  - Specific linear inequality constraints involving the variables

We will use Gurobi's Python interface to model and solve this problem.

```python
from gurobipy import *

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

# Define variables
vitamin_B12 = m.addVar(vtype=GRB.INTEGER, name="vitamin_B12")  # Whole number of milligrams
iron = m.addVar(vtype=GRB.CONTINUOUS, name="iron")  # Fractional number allowed
zinc = m.addVar(vtype=GRB.INTEGER, name="zinc")  # Non-fractional (whole) number

# Objective function: Minimize the total value
m.setObjective(4.94 * vitamin_B12 + 9.67 * iron + 2.41 * zinc, GRB.MINIMIZE)

# Constraints:
# Cognitive performance index constraints
m.addConstr(5.39 * vitamin_B12 + 7.24 * zinc >= 9, name="cognitive_min_vitamin_zinc")
m.addConstr(5.39 * vitamin_B12 + 3.4 * iron >= 13, name="cognitive_min_vitamin_iron")
m.addConstr(5.39 * vitamin_B12 + 3.4 * iron + 7.24 * zinc >= 13, name="cognitive_min_all")
m.addConstr(5.39 * vitamin_B12 + 3.4 * iron <= 34, name="cognitive_max_vitamin_iron")
m.addConstr(3.4 * iron + 7.24 * zinc <= 65, name="cognitive_max_iron_zinc")

# Immune support index constraints
m.addConstr(8.08 * vitamin_B12 + 5.23 * iron >= 29, name="immune_min_vitamin_iron")
m.addConstr(8.08 * vitamin_B12 + 5.23 * iron + 8.59 * zinc >= 29, name="immune_min_all")
m.addConstr(8.08 * vitamin_B12 + 5.23 * iron <= 194, name="immune_max_vitamin_iron")
m.addConstr(8.08 * vitamin_B12 + 5.23 * iron + 8.59 * zinc <= 194, name="immune_max_all")

# Muscle growth index constraints
m.addConstr(10.29 * vitamin_B12 + 13.7 * zinc >= 20, name="muscle_min_vitamin_zinc")
m.addConstr(10.29 * vitamin_B12 + 5.27 * iron >= 49, name="muscle_min_vitamin_iron")
m.addConstr(10.29 * vitamin_B12 + 5.27 * iron + 13.7 * zinc >= 49, name="muscle_min_all")
m.addConstr(10.29 * vitamin_B12 + 5.27 * iron + 13.7 * zinc <= 116, name="muscle_max_all")

# Specific linear inequality constraints
m.addConstr(6 * vitamin_B12 - 5 * iron >= 0, name="specific_constraint1")
m.addConstr(-3 * vitamin_B12 + 3 * zinc >= 0, name="specific_constraint2")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Vitamin B12: {vitamin_B12.x}")
    print(f"Iron: {iron.x}")
    print(f"Zinc: {zinc.x}")
else:
    print("No optimal solution found.")

```
This code directly translates the problem description into Gurobi's modeling language, defining variables, objective functions, and constraints as specified. It then solves the model and prints out the optimal values for `vitamin_B12`, `iron`, and `zinc` if an optimal solution is found.