The task described in your steps outlines an adaptation of the Sieve of Eratosthenes, which is a classic algorithm used to find all prime numbers up to a specified integer. The steps provided require creating a list where each value signifies whether the index is prime or not. The indices which end up being marked as `True` represent prime numbers. Here's how you might implement this in Python:

```python
import math

def sieve_of_eratosthenes(n):
    # Step 2: Initialize an array `x` with all entries set to True
    if n < 2:
        return []  # Handling case when n is less than 2
  
    x = [True] * (n+1)  # +1 because list is zero indexed

    # Step 3: Set the value of the first two positions to False
    x[0] = x[1] = False

    # Step 4: iterate from 2 to the nearest integer to the square root of n
    limit = int(math.sqrt(n)) + 1
    for j in range(2, limit):
        if x[j]:  # Step 4.1: Check if x[j] is True
            # Update the values to False for all multiples of j
            for multiple in range(j*j, n+1, j):
                x[multiple] = False

    # Step 4.2: Return the indices that are still True
    primes = [i for i in range(n+1) if x[i]]
    return primes

# Example use case
n = 30
prime_numbers = sieve_of_eratosthenes(n)
print("Prime numbers up to", n, ":", prime_numbers)
```

### Explanation:
1. **n+1 Array Initialization**: We initialize an array `x` with `n+1` entries set to `True` to accommodate all numbers from 0 to n (inclusive).
2. **Set First Two False**: As 0 and 1 are not prime numbers, these are explicitly set to `False`.
3. **Square Root Limitation**: The loop iterates up to the square root of `n`, because any factor larger than the square root would have been found as a multiple of a smaller number already.
4. **Marking Multiples as Non-prime**: For each valid `j` starting from 2, mark its multiples as `False` starting from `j*j` (because smaller multiples of `j` would have been marked by smaller primes).
5. **Extract Primes**: Finally, list comprehension is used to extract all indices that have a value of `True`. These indices represent prime numbers.

This code will provide you with a list of all prime numbers up to `n`. It operates efficiently by adhering to the principles of the Sieve of Eratosthenes.