To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the information provided.

### Symbolic Representation:

Let's denote:
- \(x_1\) as the number of round watches made by team A.
- \(x_2\) as the number of square watches made by team B.

The objective is to maximize profit. Given that the profit per round watch is $1000 and the profit per square watch is $1250, the objective function can be represented as:
\[ \text{Maximize: } 1000x_1 + 1250x_2 \]

The constraints are based on the production capacity of each team and the quality check limit:
1. Team A can make at most 5 round watches a day: \( x_1 \leq 5 \)
2. Team B can make at most 6 square watches a day: \( x_2 \leq 6 \)
3. The senior watchmaker can check at most 8 watches in total per day: \( x_1 + x_2 \leq 8 \)

Additionally, the number of watches cannot be negative:
4. \( x_1 \geq 0 \) and \( x_2 \geq 0 \)

### Symbolic Representation in JSON Format:

```json
{
    'sym_variables': [('x1', 'number of round watches'), ('x2', 'number of square watches')],
    'objective_function': '1000*x1 + 1250*x2',
    'constraints': ['x1 <= 5', 'x2 <= 6', 'x1 + x2 <= 8', 'x1 >= 0', 'x2 >= 0']
}
```

### Gurobi Code in Python:

To solve this linear programming problem, we'll use the Gurobi Python interface. First, ensure you have Gurobi installed. If not, you can install it via `pip install gurobipy`.

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="round_watches")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="square_watches")

# Set the objective function
m.setObjective(1000*x1 + 1250*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 <= 5, "team_a_capacity")
m.addConstr(x2 <= 6, "team_b_capacity")
m.addConstr(x1 + x2 <= 8, "quality_check_limit")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Round watches: {x1.x}")
    print(f"Square watches: {x2.x}")
    print(f"Maximum profit: ${m.objVal:.2f}")
else:
    print("No optimal solution found")
```