Selection
Abstract
Selection
Eight boundary sectors use (x,p,q;v,m/h,M/h)=(8,4,2;3,9,9),(8,4,4;6,18,22),(8,8,2;9,12,15),(8,8,4;5,48,49),(8,16,2;15,29,31),(16,8,2;7,17,19),(16,8,4;6,49,51),(16,16,2;10,41,43), with h=2^H and before=true for every natural H. Their actual old and new endpoints and head-minus-one slacks are affine in h-1 with nonnegative coefficients. Binary multiplication by 2^H shifts support, preserving each literal submask. The original contract permits m=M, including the first family. These private constructions are consumed before the H split. The dyadic residue decomposition chooses a certified row for every q=8*2^s. Integer bounds are transported to natural masks without subtraction underflow. The unique late-fresh row preserves the source cut immediately after nX.