General argument for extensive multiplicity in non-holographic blocks

Develop a general proof or argument establishing that combinatorial mechanical metamaterials constructed from non-holographic building blocks exhibit extensive multiplicity of compatible configurations—i.e., multiplicity that grows exponentially with the number of blocks—beyond case-specific constructions demonstrated for particular block types in square, honeycomb, and cubic lattices.

Background

The authors show, by explicit constructions and bounds, that for all non-holographic block types studied, the number of compatible metamaterials scales extensively with system size. They provide super-block tilings and parity-based compatibility arguments to obtain lower and upper bounds that both scale extensively.

Despite these results, they note the absence of a general theoretical argument covering all non-holographic cases, indicating a gap between empirical and constructive evidence and a unifying proof that would apply to the broader class of non-holographic blocks.