Exact physical arithmetic
Abstract
Actual paired chains for equal last three gaps
Let k=a+3s and A=1+2^a+2^(a+s)+2^(a+2s). The physical chains start at 0,s,2s,3s and have lengths s,s,s,a. With x=2^a,q=2^s,D=q^2+q+1, their old heads are A=xD+1,B=x(q^2+q)+q+1,L=xq^2+D,M=q^3+D. The enlarged heads are 2A,2B-1,2L-1-q,2M-D. The old successors are B,L,M,A; the enlarged successors are B’,L’,M’,A. Every nonterminal value is the corresponding head times its exact local power of two. Every doubled terminal is its successor minus the row modulus. The fresh value is A. These formulas are applications of the independent four-gap centered-source lemmas, with no orbit primitivity or distinct-value premise.