Overlap-based obstruction to contraction maps with compressed coefficient representation

Prove or refute that, for a candidate holographic entropy inequality written with distinct terms and integer coefficients (compressed representation), if the set of left-hand-side terms has fewer mutual overlaps than the right-hand-side terms, then no contraction map of type (2) (defined on bit strings of length equal to the number of distinct terms with weighted Hamming distance by coefficients) can exist.

Background

The appendix surveys different formulations of contraction maps, distinguishing between maps defined on bit strings that either split repeated terms (type (1)) or respect compressed representations with integer coefficients (type (2)). It also distinguishes whether the domain includes only realizable bit strings (a) or the full domain (b).

Motivated by examples where sums of primitive inequalities fail to admit a type (2) contraction map due to overlapping structure, the authors propose Conjecture A1 as a natural structural obstruction related to overlap asymmetry between LHS and RHS terms.