Hadamard matrix existence for all orders divisible by four

Determine whether for every integer m that is a multiple of four, there exists a Hadamard matrix of order m. This resolves the Hadamard matrix conjecture, which directly governs the existence of regular n-simplices inscribed in the n-dimensional cube with vertices at cube vertices for n+1=m.

Background

The paper uses Hadamard numbers (orders for which Hadamard matrices exist) to characterize when an n-dimensional regular simplex can be inscribed into the n-cube with vertices at cube vertices. This equivalence is central to several bounds and constructions for interpolation projectors and absorption indices.

The longstanding Hadamard matrix conjecture asserts existence of Hadamard matrices for all orders m divisible by 4. Resolving it would settle, among other things, the existence of a large class of regular inscribed simplices used throughout the paper.