## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Mary', 'hours worked by George', and 'hours worked by Paul', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $1.6x_0^2 + 2.23x_0x_1 + 5.54x_0x_2 + 6.78x_1^2 + 7.89x_1x_2 + 6.63x_2^2 + 8.77x_0 + 1.15x_2$.

## 3: List the constraints in symbolic notation
The constraints are:
- $5x_0 \leq 113$ (productivity rating for Mary, but it's given as a fixed value, so $x_0$ is not bounded by this directly)
- $10x_0 \leq 78$ (paperwork competence rating for Mary)
- $8x_1 \leq 113$ (productivity rating for George)
- $13x_1 \leq 78$ (paperwork competence rating for George)
- $12x_2 \leq 113$ (productivity rating for Paul)
- $10x_2 \leq 78$ (paperwork competence rating for Paul)
- $5x_0 + 8x_1 \geq 20$ (total combined productivity rating from Mary and George)
- $5^2x_0^2 + 12^2x_2^2 \geq 27$ (total combined productivity rating from Mary and Paul squared)
- $8x_1 + 12x_2 \geq 15$ (total combined productivity rating from George and Paul)
- $5x_0 + 8x_1 + 12x_2 \geq 15$ (total combined productivity rating from all)
- $13x_1^2 + 10x_2^2 \geq 8$ (total combined paperwork competence rating from George and Paul squared)
- $10x_0 + 10x_2 \geq 26$ (total combined paperwork competence rating from Mary and Paul)
- $10x_0 + 13x_1 + 10x_2 \geq 13$ (total combined paperwork competence rating from all)
- $10x_0 + 13x_1 + 10x_2 \geq 13$ is the same as above, so we keep $10x_0 + 13x_1 + 10x_2 \geq 13$
- $-x_1^2 + 4x_2^2 \geq 0$
- $5x_0 + 12x_2 \leq 85$ (total combined productivity rating from Mary and Paul)
- $5x_0 + 8x_1 \leq 85$ (total combined productivity rating from Mary and George)
- $5x_0 + 8x_1 + 12x_2 \leq 79$ (total combined productivity rating from all)
- $10^2x_0^2 + 13^2x_1^2 \leq 47$ (total combined paperwork competence rating from Mary and George squared)

## 4: Correct and simplify constraints based on given attributes
Given attributes:
- $r0$: productivity rating with $x_0 = 5$, $x_1 = 8$, $x_2 = 12$
- $r1$: paperwork competence rating with $x_0 = 10$, $x_1 = 13$, $x_2 = 10$

So, the actual constraints from attributes are:
- $5x_0 = 5 \cdot x_0$ (Mary's productivity)
- $10x_0 = 10 \cdot x_0$ (Mary's paperwork)

## 5: Formulate the problem in Gurobi
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define variables
x0 = m.addVar(name="x0", lb=0, type=gurobi.GRB.INTEGER)  # hours worked by Mary
x1 = m.addVar(name="x1", lb=0, type=gurobi.GRB.INTEGER)  # hours worked by George
x2 = m.addVar(name="x2", lb=0)  # hours worked by Paul

# Objective function
m.setObjective(1.6*x0**2 + 2.23*x0*x1 + 5.54*x0*x2 + 6.78*x1**2 + 7.89*x1*x2 + 6.63*x2**2 + 8.77*x0 + 1.15*x2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(5 * x0 <= 113, name="Mary_productivity")
m.addConstr(10 * x0 <= 78, name="Mary_paperwork")
m.addConstr(8 * x1 <= 113, name="George_productivity")
m.addConstr(13 * x1 <= 78, name="George_paperwork")
m.addConstr(12 * x2 <= 113, name="Paul_productivity")
m.addConstr(10 * x2 <= 78, name="Paul_paperwork")

m.addConstr(5 * x0 + 8 * x1 >= 20, name="Mary_George_productivity")
m.addConstr(25 * x0**2 + 144 * x2**2 >= 27, name="Mary_Paul_productivity_squared")
m.addConstr(8 * x1 + 12 * x2 >= 15, name="George_Paul_productivity")
m.addConstr(5 * x0 + 8 * x1 + 12 * x2 >= 15, name="all_productivity")

m.addConstr(169 * x1**2 + 100 * x2**2 >= 8, name="George_Paul_paperwork_squared")
m.addConstr(10 * x0 + 10 * x2 >= 26, name="Mary_Paul_paperwork")
m.addConstr(10 * x0 + 13 * x1 + 10 * x2 >= 13, name="all_paperwork")

m.addConstr(-x1**2 + 4 * x2**2 >= 0, name="George_Paul_work")
m.addConstr(5 * x0 + 12 * x2 <= 85, name="Mary_Paul_productivity_limit")
m.addConstr(5 * x0 + 8 * x1 <= 85, name="Mary_George_productivity_limit")
m.addConstr(5 * x0 + 8 * x1 + 12 * x2 <= 79, name="all_productivity_limit")
m.addConstr(100 * x0**2 + 169 * x1**2 <= 47, name="Mary_George_paperwork_squared")

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic Representation
```json
{
    'sym_variables': [('x0', 'hours worked by Mary'), ('x1', 'hours worked by George'), ('x2', 'hours worked by Paul')],
    'objective_function': '1.6*x0^2 + 2.23*x0*x1 + 5.54*x0*x2 + 6.78*x1^2 + 7.89*x1*x2 + 6.63*x2^2 + 8.77*x0 + 1.15*x2',
    'constraints': [
        '5*x0 <= 113',
        '10*x0 <= 78',
        '8*x1 <= 113',
        '13*x1 <= 78',
        '12*x2 <= 113',
        '10*x2 <= 78',
        '5*x0 + 8*x1 >= 20',
        '25*x0^2 + 144*x2^2 >= 27',
        '8*x1 + 12*x2 >= 15',
        '5*x0 + 8*x1 + 12*x2 >= 15',
        '169*x1^2 + 100*x2^2 >= 8',
        '10*x0 + 10*x2 >= 26',
        '10*x0 + 13*x1 + 10*x2 >= 13',
        '-x1^2 + 4*x2^2 >= 0',
        '5*x0 + 12*x2 <= 85',
        '5*x0 + 8*x1 <= 85',
        '5*x0 + 8*x1 + 12*x2 <= 79',
        '100*x0^2 + 169*x1^2 <= 47'
    ]
}
```