## Step 1: Define the symbolic representation of the problem

Let's define the symbolic variables:
- $x_1$ represents the number of cans of Iota paint.
- $x_2$ represents the number of cans of Lambda paint.

The objective is to minimize the cost, which can be represented as $8x_1 + 12x_2$.

The constraints based on the minimum requirements are:
- Dye: $3x_1 + 5x_2 \geq 10$
- Thinner: $5x_1 + 8x_2 \geq 12$
- Oil: $6x_1 + 3x_2 \geq 15$
- Non-negativity: $x_1 \geq 0, x_2 \geq 0$

## Step 2: Convert the problem into a Gurobi-compatible format

The symbolic representation can be directly translated into Gurobi code.

## Step 3: Write down the symbolic representation in the required format

```json
{
    'sym_variables': [('x1', 'cans of Iota paint'), ('x2', 'cans of Lambda paint')],
    'objective_function': '8*x1 + 12*x2',
    'constraints': [
        '3*x1 + 5*x2 >= 10',
        '5*x1 + 8*x2 >= 12',
        '6*x1 + 3*x2 >= 15',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Step 4: Implement the optimization problem using Gurobi

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="Iota", lb=0, vtype=gurobi.GRB.CONTINUOUS)
    x2 = model.addVar(name="Lambda", lb=0, vtype=gurobi.GRB.CONTINUOUS)

    # Define the objective function
    model.setObjective(8 * x1 + 12 * x2, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(3 * x1 + 5 * x2 >= 10, name="Dye")
    model.addConstr(5 * x1 + 8 * x2 >= 12, name="Thinner")
    model.addConstr(6 * x1 + 3 * x2 >= 15, name="Oil")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: x1 = {x1.varValue}, x2 = {x2.varValue}")
        print(f"Minimum cost: {model.objVal}")
    else:
        print("The problem is infeasible")

solve_paint_mixture_problem()
```