Comprehensive connection between holography and multiplicity in combinatorial metamaterials

Establish a comprehensive theoretical connection between holographic order in combinatorial mechanical metamaterials—where boundary displacement textures determine bulk deformations—and the scaling behavior of the multiplicity of compatible metamaterials across square, honeycomb, and cubic lattices with single-mode anisotropic blocks; specifically, determine conditions under which holographic order implies sub-extensive multiplicity and characterize how this relationship generalizes across block types and lattice geometries.

Background

The paper introduces a unified framework for combinatorial mechanical metamaterials in square, honeycomb, and cubic lattices, distinguishing block types that induce holographic order (boundary determines bulk) from those that do not. Prior work showed sub-extensive multiplicity for specific holographic cases (e.g., cubic Block C2 and honeycomb Block H2), but a general theory linking holography to multiplicity was not previously consolidated.

Within this work, the authors systematically classify block behaviors (alternating vs persistent holography vs non-holographic) and derive multiplicity scaling results case-by-case. They highlight the need for a comprehensive theoretical connection explaining why holographic order leads to multiplicity that scales with boundary size and how this principle extends across different lattices and block symmetries.