sympathic, Please try to accept this with an open mind:
There is no process here of any kind.
Any given infinite collection is simply an incomplete mathematical element, and this incompleteness is essential to any infinite complex, because of a very simply reason:
No complex can be atomic, were the atomic has two opposite qualitative aspects which are: Locality (total finite) and Non-locality (total infinity).
The linkage between these qualitative aspects is resulted by what is called Quantity.
There is a finite quantity, which is accurate and has a sum.
There is an infinite quantity, which is inaccurate and does not have a sum (it is called a fog under OM (
http://www.internationalskeptics.com/forums/showpost.php?p=5734631&postcount=9165)).
Since Standard Math does not understand the qualitative foundations of Complexity and Quantity, its reasoning is limited only the accurate aspect of Complexity.
OM is a reasoning that enables to deal with both inaccuracy and accuracy under a one comprehensive framework, which simply can't be comprehended from Standard Math Accuracy-only reasoning.
In other words sympathic, what you call process is a direct result of the limited understanding of Standard Math of the concept of Quantity and the concept of Complexity.
About your "fully loaded magic black box"; this time please to not ignore
http://www.internationalskeptics.com/forums/showpost.php?p=5735873&postcount=9169 .
Again, the set of oranges is not itself an orange (it is not similar to the property of its members).
The series of accurate values (2a+2b+2c+2d+...) does not itself have an accurate value.
This resoning is beyond the limited reasoning, which deals only with accurate values.