The task you've outlined is essentially to find the greatest common divisor (GCD) of two positive integers using the Euclidean algorithm. Below is a Python implementation of these steps:

```python
def gcd(a, b):
    # Step 1: Verify if a and b are positive integers
    if not (isinstance(a, int) and isinstance(b, int) and a > 0 and b > 0):
        return "Both numbers must be positive integers."
    
    # Ensure a is the larger number
    if a < b:
        a, b = b, a
    
    # Step 2 and beyond: Perform Euclidean division until remainder is 0
    while b != 0:
        a, b = b, a % b
    
    # Step 3 and 4: Return the last non-zero remainder which is now stored in 'a'
    return a

# Example usage:
x = 252
y = 105
result = gcd(x, y)
print(f"The GCD of {x} and {y} is {result}")
```

Here's a breakdown of the code:
1. **Function Definition**: `gcd(a, b)` takes two parameters.
2. **Step 1**: It first checks if both `a` and `b` are positive integers.
3. **Swap if necessary**: If `b` is greater than `a`, they are swapped to make `a` the larger number.
4. **Euclidean Algorithm**: The loop continues until the remainder `b` is zero. Within the loop, the remainder of `a` divided by `b` is calculated and becomes the new `b`, while the old `b` becomes the new `a`.
5. **Return Value**: Once `b` is zero, the last non-zero remainder before `b` reached zero (which has been assigned to `a` in the process) will be the greatest common divisor (GCD).

This code correctly identifies the GCD of two numbers by following the algorithmic steps you've specified.
