The actual old and enlarged source
Abstract
Four independent centered ladders
For all independent a,b,c,d>=2, write R=2^a,T=2^b,Z=2^c,D=2^d, m=RTZ, C=RT+R+1, A=m+C and N=Dm-1. The internal source_table proves the centered reductions on the four actual label intervals starting at 0,d,d+c,d+c+b. The successive old ladder bases are A,DC+1,D(R+1)Z+Z+1,DTZ+TZ+T+1. The enlarged-source bases are twice these old bases minus 0,1,Z+1,TZ+T+1, respectively. Each final label has the exact negative centered value. The fresh physical label has enlarged-source value A.
The proof applies the existing centered_shift and centered_ladder at their actual types, proving positivity, odd modulus, base bounds and all four recurrences at these weaker fourth-gap hypotheses. It does not invoke the eventual construction or its stronger d>=a+b+c+3 premise.
This helper identifies source values on physical occurrences. It supplies no permutation or joint-order existence by itself; unrestricted Circle C3.1 and the complete max-plus-two construction remain separate obligations.