Open problemChecked July 24, 2026

Hadamard Matrix Conjecture

An existence conjecture for highly orthogonal square matrices.

Combinatorics Matrix theory Design theory

Mathematical statement

For every positive integer $n$ divisible by $4$ , there is an $n\times n$ matrix $H$ with entries in $\{\pm1\}$ and $HH^{\mathsf T}=nI$ .

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

Hadamard.HadamardConjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem HadamardConjecture (k : ℕ) :    ∃ M, IsHadamard (n := 4 * k) M := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Hadamard matrix overviewbackground