Physical labels and endpoint bounds
Abstract
Positive packet symmetry and rightward ear transport
A nonnegative packet can be reordered arbitrarily without changing its endpoint sum. Every intermediate height remains between its starting and ending heights. The same reasoning slides a positive ear right across a positive interval: the two local paths have identical endpoints, and the old physical word stays fixed.
For a valid old zero-sum word, a final positive step contradicts nonnegativity of the preceding prefix. An ear after all old labels exposes the enlarged old-label total -A, so it cannot be legal when A>0. These are the two boundary exclusions needed for four slots rather than a fifth tail slot.