Third terminal caps and the inserted source value
Abstract
Five gates for the third paired seams and fresh prefix
For integer x>=4 and R,Z,lambda>=1, set t=2xR, z=2xtZ and y=2zlambda. With D=y(z+1)+1, A=xD+1, B=xt(y+1)+t+1, L=xzt+zt+z+1, Bn=2B-1, Ln=2L-z-1 and N=xyzt-1, the five private results prove B-lambdaL>=0, Bn-lambdaLn>=0, 2lambdaL-B-1>=0, 2lambdaLn-Bn-1>=0 and N-A>=0.
The doubled third old seam is 2(B-lambdaL), with successor B; the first and third inequalities give its range [0,B-1]. The doubled third enlarged seam is 2(Bn-lambdaLn), with successor Bn; the second and fourth inequalities give its range [0,Bn-1]. The remaining gate supplies the upper bound A<=N for the actual fresh prefix at literal cut zero.
The proofs use exact shifted polynomial expansions and nonnegativity at x=4+u, R=1+v, Z=1+w and lambda=1+q. Selection includes every one of these five gates. Schedule uses both third seam bounds in the four-strip construction and installs the fresh-prefix bound before lifting to the enlarged physical order. The final bank is controlled using source totals and terminal bounds in Schedule.