Native Matching Surgery and Ordered Joins
Abstract
Native surgery supplies the exact internal and ordered external occurrence counts.
This source owns the actual conjugation surgery used by the canonical native row. An opposite-charge internal choice preserves the complete partition. A choice in distinct components joins their actual identity-edge sets, preserving all untouched components. Counting uses the finite component partition before mapping component cardinalities into the part multiset. Thus components with equal sizes remain distinct occurrences, and every ordered join erases two actual occurrences. The internal count is the sum of choose(r,2); the external coefficient is (1+b)/2 times the ordered r*s sum. These results are live inputs to the Row and Cuts owners. The public native-row statement and its complete occurrence formula are described in the canonical Row document.