To solve the given optimization problem, we first need to translate 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 define:
- $x_1$ as the number of diamond necklaces
- $x_2$ as the number of gold necklaces

The profit per diamond necklace is $1500, and per gold necklace is $500. Thus, the objective function to maximize profit can be written as:
\[ \text{Maximize: } 1500x_1 + 500x_2 \]

Given constraints are:
1. Designing team's time constraint: Each diamond necklace takes 3 hours to design, and each gold necklace takes 5 hours to design. The designing team is available for 30 hours.
\[ 3x_1 + 5x_2 \leq 30 \]
2. Crafting team's time constraint: Each diamond necklace takes 10 hours to craft, and each gold necklace takes 2 hours to craft. The crafting team is available for 45 hours.
\[ 10x_1 + 2x_2 \leq 45 \]
3. Non-negativity constraints: Since we cannot produce a negative number of necklaces,
\[ x_1 \geq 0, x_2 \geq 0 \]

### Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'diamond necklaces'), ('x2', 'gold necklaces')],
    'objective_function': '1500*x1 + 500*x2',
    'constraints': ['3*x1 + 5*x2 <= 30', '10*x1 + 2*x2 <= 45', 'x1 >= 0', 'x2 >= 0']
}
```

### Gurobi Code

To solve this linear programming problem using Gurobi in Python, we will use the following code:

```python
from gurobipy import *

# Create a model
m = Model("Jewelry_Shop_Profit")

# Define variables
x1 = m.addVar(name="diamond_necklaces", lb=0)
x2 = m.addVar(name="gold_necklaces", lb=0)

# Set the objective function
m.setObjective(1500*x1 + 500*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(3*x1 + 5*x2 <= 30, name="design_time_constraint")
m.addConstr(10*x1 + 2*x2 <= 45, name="craft_time_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print(f"Optimal number of diamond necklaces: {x1.x}")
    print(f"Optimal number of gold necklaces: {x2.x}")
    print(f"Maximum profit: ${1500*x1.x + 500*x2.x:.2f}")
else:
    print("No optimal solution found")
```