To solve this optimization problem, we first need to convert the natural language description into a symbolic representation. Let's define the variables as follows:

- $x_1$: The number of furniture items.
- $x_2$: The number of carpet items.

The objective function is to maximize profit. Given that the profit per furniture sold is $40 and the profit per carpet sold is $30, the objective function can be represented algebraically as:

$40x_1 + 30x_2$

Now, let's list out the constraints based on the problem description:

1. Space constraint: Each furniture takes 12 square feet of space, and each carpet takes 7 square feet of space. The company has 1200 square feet of space available.
   - $12x_1 + 7x_2 \leq 1200$

2. Budget constraint: Buying a furniture costs the store $300, and buying a carpet costs the store $80. The company has a budget of $30,000.
   - $300x_1 + 80x_2 \leq 30000$

3. Furniture proportion constraint: At least 20% of items in stock have to be furniture.
   - $\frac{x_1}{x_1 + x_2} \geq 0.20$
   - To linearize this, we can multiply both sides by $(x_1 + x_2)$ (assuming $x_1 + x_2 > 0$) to get:
     $x_1 \geq 0.20(x_1 + x_2)$
     Simplifying gives:
     $0.80x_1 \geq 0.20x_2$
     Or more usefully for our purposes:
     $4x_1 \geq x_2$

However, to ensure that we're comparing the proportions correctly and maintaining linearity in our constraints, we can rearrange this constraint to better fit into a linear programming framework by ensuring it's in a standard form. Since we've established our variables and objective function, let's correct and simplify the representation for clarity:

```json
{
  'sym_variables': [('x1', 'number of furniture items'), ('x2', 'number of carpet items')],
  'objective_function': '40*x1 + 30*x2',
  'constraints': [
    '12*x1 + 7*x2 <= 1200', 
    '300*x1 + 80*x2 <= 30000', 
    'x1 >= 0.2*(x1 + x2)'
  ]
}
```

Given the correction needed for linear programming and to adhere strictly to the format requested without additional explanations here, let's proceed directly to the Gurobi code implementation in Python:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="furniture_items")
x2 = m.addVar(vtype=GRB.INTEGER, name="carpet_items")

# Set the objective function
m.setObjective(40*x1 + 30*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(12*x1 + 7*x2 <= 1200, "space_constraint")
m.addConstr(300*x1 + 80*x2 <= 30000, "budget_constraint")
m.addConstr(x1 >= 0.2*(x1 + x2), "furniture_proportion_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of furniture items: {x1.x}")
    print(f"Number of carpet items: {x2.x}")
else:
    print("No optimal solution found")
```