Physical labels and endpoint bounds
Abstract
Physical sources for four independent gaps
For all independent a,b,c,d>=2, the original source_table supplies the exact four centered old chains and four enlarged chains. The physical negative labels are d-1,d+c-1,d+c+b-1,k-1. The enlarged fourth chain ends with successor A, the fresh ear; it does not return to the enlarged first base. All other old labels have strictly positive values in both rows. The old total is zero, the enlarged total on old labels is -A, and the fresh label has value A. Numerical repetitions and nonprimitive seeds retain their physical labels.
A positive physical finite set has geometric mask in chain r equal to the sum of 2^i over its actual chain positions. Summing the exact centered formulas gives its old mass as the sum of F[r] times the masks, and its new mass as the sum of (2F[r]-eta[r]) times the same masks. No prefix legality is used to define or justify these masses.