Exact physical construction and coupled prefix strips
Abstract
A ceiling-selected physical complement
Take independent r,s>=2 with s>=r+2, x=2^r, y=2^s, k=2r+2s and N=x^2y^2-1. Use the coefficients A,B,L,M,e and the old/new source tables in FourNegativeAdjacentSource. Put T=y/2-1 and R=(L+1)/2-A. The private ceiling_multiple theorem chooses a natural h with 1<=h<=T and R<=hB<R+B from Euclidean division. The arithmetic theorem proves that R is positive and R<=TB at x>=4 and y>=4x.
The selected set consists of the original positive V labels at the binary positions of h. Its complement is taken in the complete positive V list, so both sets are disjoint and every V occurrence is preserved. Reuse of ThreeBitSource.mask_labels_sum gives exact old mass hB and new mass h(2B-1). The old order is U-positive,U-negative,V-complement,V-negative,W-positive,W-negative,X-positive,V-selected,X-negative. Insert the original fresh label k immediately before the final X-negative, at cut k-1.
The four old seams are B-A,L-A-hB,M-A-hB,0. The coupled arithmetic uses 2M-L-2B-e-1>0 to bound the third seam and its new counterpart. Gates and four_strip then control every natural prefix by positive blocks, their peaks and their terminal negative jumps. Exact totals use source_zero, and marked_word proves that the new word is the literal insertion into this same physical order. No numerical quotient or primitive orbit is used.