Construction and exact scope
Abstract
Physical shadow and charge
The physical label i records multiplication of the seed A by 2^i in the residue group modulo N. The enlarged action uses seed 2A modulo 2N+1. Both moduli are odd, so multiplication by two is a group automorphism. Physical labels retain repeated numeric values from short periods and noncoprime seeds. Shadow and charge compare the two actual centered actions at the same inherited occurrence. The internal source_shadow_charge derives q=2f-g and z=g-f from the accepted SourceTable.source_table. For the four physical positive chains U,V,W,X, the shadow bases are A,DC,D(R+1)Z,DTZ and the charge bases are 0,1,Z+1,TZ+T+1. The old f bases equal the shadow base plus the charge base. At the four negative physical labels, z is -B+DC/2,-B+D(R+1)Z/2,-B+DTZ/2,-B and q is 1,Z/2+1,T(Z+1)/2+1,(A+1)/2. The fresh enlarged-source label has value A. This source identification imposes no coprimality or distinct-numeric-value assumption.