Private arithmetic of the strict physical chamber
Abstract
The complete adjacent two-chain certificate
For r=2t,s>=8 and q=t/s a power of two, the four ranges are q<=s/2,q=s,q=2s,q>=4s. They select W masks zero, three, four, and twice an exact integer ceiling. The U mask is chosen by Euclidean division. The head remainder obeys R>=0 and 2R+(s+1)w<=st+s/2-1; in the large-ceiling range its intermediate integer bound is 4sq+2s^2+2s. The tail is t-3 when q=2 and t-q otherwise. Low-bit separation follows from w<q and q dividing t-q, with w=0 in the q=2 exception. Tail estimates and the eight seams certify the beta,alpha,gamma order and the literal ear before Q. These private unbounded certificates are consumed by the strict-chamber theorem, which reuses the original physical realizer.