To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables for the quantities of large and small phones produced, formulating the objective function that represents the profit, and listing the constraints based on the available manufacturing and testing time.

Let's denote:
- \(x_1\) as the number of large size phones,
- \(x_2\) as the number of small size phones.

The objective function is to maximize the total profit, which can be represented as:
\[400x_1 + 300x_2\]

Given the constraints:
1. Manufacturing belt time: \(20x_1 + 15x_2 \leq 1500\)
2. Testing time: \(12x_1 + 10x_2 \leq 1000\)
3. Non-negativity constraint (since we cannot produce a negative number of phones): \(x_1, x_2 \geq 0\)

Thus, the symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'number of large size phones'), ('x2', 'number of small size phones')],
    'objective_function': '400*x1 + 300*x2',
    'constraints': ['20*x1 + 15*x2 <= 1500', '12*x1 + 10*x2 <= 1000', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we'll use the following code:
```python
from gurobipy import *

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

# Add variables
x1 = m.addVar(lb=0, vtype=GRB.INTEGER, name="large_phones")
x2 = m.addVar(lb=0, vtype=GRB.INTEGER, name="small_phones")

# Set the objective function
m.setObjective(400*x1 + 300*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(20*x1 + 15*x2 <= 1500, "manufacturing_belt")
m.addConstr(12*x1 + 10*x2 <= 1000, "testing_time")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print(f"Optimal production: {x1.varName} = {int(x1.x)}, {x2.varName} = {int(x2.x)}")
    print(f"Maximum profit: ${400*int(x1.x) + 300*int(x2.x)}")
else:
    print("No optimal solution found.")
```