The selector range and the old second seam
Abstract
Three exact gates for the signed ceiling
Let t=2xR, z=2xtZ and y=2zlambda, with integer x>=4 and R,Z,lambda>=1. Set D=y(z+1)+1, A=xD+1, L=xzt+zt+z+1, M=tyz+D, N=xyzt-1 and E=2N-M-(t-1)A. The three private polynomial results establish E>=0, E<=2lambdaL(z-1) and 2lambdaL<=A-1.
For the positive denominator den=2lambdaL, Selection uses the signed integer ceiling H=-floor(-E/den). The first two gates allow its exact division argument to prove 0<=H<=z-1 before any conversion to a natural mask. That argument also proves 0<=delta<den for delta=den H-E. The third gate then supplies the old second successor cap delta<=A-1.
The gate proofs use explicit shifted polynomial identities at x=4+u, R=1+v, Z=1+w and lambda=1+q, followed by nonnegativity of every term. Integer division and the Euclidean remainder are handled in Selection; the present module supplies the polynomial inequalities they require. No real-valued relaxation replaces the ceiling or its native mask.