Imagine the broadcasting system as a package sorting facility. Inside it is a sorter, and the ranks are the input belts. Each round, the sorter looks at the current lowest rank across all arguments and accepts all elements with that rank.

Once the sorter establishes a unified traversal route, an inner function takes over to execute the actual data extraction. It navigates through the nested lists strictly following the predefined route. At each dimension step:

Arguments that are active for the current dimension advance one level deeper into their nested lists.
Arguments that are inactive at this dimension remain untouched, effectively broadcasting their current value to every element extracted from the active arguments.

The recursion continues until all predefined dimensions are exhausted. At the very bottom level, the target function is finally applied to the collected scalar values. The collected results are assembled back into a new nested list, and returned.

Let us trace through a concrete example:

[[1 2 3] [4 5 6]] \\!+\ [10 20 30]

Left operand [[1 2 3] [4 5 6]] has dIterator rank [1 0] (from \\!). Right operand [10 20 30] has rank [0].

The sorter sees ranks [0] and [1]. The lower rank is 0, so the right operand is active at this level — it advances one level, yielding elements 10, 20, 30. The left operand is inactive at rank 1 — it broadcasts its entire value [1 2 3] (then [4 5 6]) to each.

But the left operand's rank is now [0], so at the next level it becomes active and descends into its elements. Now we have scalars on both sides, and + applies.

Each scalar in the right operand is added to the corresponding scalar in both rows of the left operand. The result is a 3D matrix where each row is the original row plus the vector.