To solve this optimization problem using Gurobi, we first need to clarify and simplify the constraints given in the natural language description. The objective function is to minimize \(3x_0 + 7x_1\), where \(x_0\) represents the amount of corn cobs and \(x_1\) represents the quantity of chicken breasts.

Given resources/attributes are:
- 'r0': grams of fat, with upper bound 80, and each corn cob contains 17 grams of fat (\(x_0\)), each chicken breast contains 4 grams of fat (\(x_1\)).
- 'r1': grams of carbohydrates, with upper bound 227, and each corn cob contains 22 grams of carbohydrates (\(x_0\)), each chicken breast contains 18 grams of carbohydrates (\(x_1\)).

Constraints:
1. Fat from corn cobs and chicken breasts must be at least 16 grams.
2. Carbohydrates from both must be at least 52 grams.
3. \(-7x_0 + 10x_1 \geq 0\).
4. Total fat cannot exceed 69 grams.
5. Total carbohydrates cannot exceed 210 grams.
6. \(x_0\) and \(x_1\) must be integers.

Let's formulate these constraints mathematically:
1. \(17x_0 + 4x_1 \geq 16\)
2. \(22x_0 + 18x_1 \geq 52\)
3. \(-7x_0 + 10x_1 \geq 0\)
4. \(17x_0 + 4x_1 \leq 69\)
5. \(22x_0 + 18x_1 \leq 210\)

Now, let's write the Gurobi code for this problem:

```python
from gurobipy import *

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

# Define variables
x0 = m.addVar(vtype=GRB.INTEGER, name="corn_cobs")
x1 = m.addVar(vtype=GRB.INTEGER, name="chicken_breasts")

# Objective function: Minimize 3*x0 + 7*x1
m.setObjective(3*x0 + 7*x1, GRB.MINIMIZE)

# Constraints
m.addConstr(17*x0 + 4*x1 >= 16, "min_fat")
m.addConstr(22*x0 + 18*x1 >= 52, "min_carbohydrates")
m.addConstr(-7*x0 + 10*x1 >= 0, "custom_constraint")
m.addConstr(17*x0 + 4*x1 <= 69, "max_fat")
m.addConstr(22*x0 + 18*x1 <= 210, "max_carbohydrates")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print("Corn cobs:", x0.x)
    print("Chicken breasts:", x1.x)
    print("Objective function value:", m.objVal)
else:
    print("No optimal solution found. Status:", m.status)
```