An Icosahedral Six-Well Construction for the Max-Mid Strain Constraint: A Candidate Counterexample via Convex Integration

Abstract

Miller and Sawyer introduced the max-mid class of square-integrable trace-free symmetric matrix fields and conjectured that its intersection with the strain constraint space L^2_st is trivial. This preprint considers six trace-free symmetric matrices associated with the six unoriented axes of a regular icosahedron. Each matrix has eigenvalues (-2,1,1); every pair is symmetrized-rank-one compatible; the six matrices are affinely independent in the five-dimensional space of trace-free symmetric 3x3 matrices; and their barycenter is zero. Ruland, Zillinger and Zwicknagl state that the properties of their geometrically linearized convex-integration construction persist, with minor modifications, for pairwise compatible wells whose convex hull has the full dimension allowed by the trace constraint. If that general statement applies directly to the six wells constructed here, it yields a nonzero divergence-free Lipschitz vector field whose symmetric gradient is max-mid almost everywhere on a bounded domain. This would produce a nonzero element of L^2_st intersect L^2_maxmid, contrary to Conjecture 1.14 of Miller-Sawyer. The note isolates the explicit finite-dimensional geometry and the exact convex-integration transfer requiring independent specialist verification. This is a candidate argument and has not been peer reviewed