Native Rotation Cuts and Their Occurrences
Abstract
Literal rotation displacement supplies every positive ordered native cut.
The cut proof follows the actual product rotation and its exact period. Conjugating equal-charge vertices at displacement d splits a component with r identity edges into d and r-d, for every 0<d<r. The proof excludes the other charge sheet and proves the reverse identity-partner orientation gives the same complete type. The rotation powers identify the r actual edge occurrences bijectively, so the two orientations give coefficient r/2 on the positive ordered cut sum. Equal cuts, adjacent cuts, repeated parts, partner degeneracies and the empty cut range at r=1 are all retained. The exact counted row feeds native_row in the canonical Row owner, where its public statement is described.