Analytical proof of greedy minimality

Establish analytically that the iterative greedy strain-construction procedure produces a minimum set of uncoupled strain vectors for the symmetry-reduced elastic stiffness matrices considered by the algorithm, rather than relying solely on exhaustive enumeration for the finite set of present cases.

Background

The paper formulates the selection of uncoupled strain patterns as a finite covering problem over the sparsity structure of the symmetry-reduced elastic stiffness matrix. The iterative procedure is a deterministic greedy construction that generates a compact candidate set by successively removing strain components and rows whose inclusion would produce coupled stress equations.

For every elastic-symmetry type treated in the paper, exhaustive enumeration of the at most 63 non-empty strain patterns verifies that the set produced by the iterative procedure has the exact minimum cardinality. However, this verification is computational and case-specific; the paper does not provide an analytical proof that the greedy procedure is generally minimal. Establishing such a proof would explain why the algorithm attains the exact minimum and would extend confidence in its minimality beyond the explicitly enumerated symmetry classes.