A Proof of the Dittert Conjecture in Dimension 4 via an Agent-Guided Exact Sum-of-Squares Certificate

2026-07-31

AI Summary

This work establishes the Dittert conjecture in four dimensions: among all $4 \times 4$ matrices with nonnegative entries summing to 4, the Dittert functional attains its unique maximum at the uniform matrix $U_4$. To achieve this, we introduce an agent-guided hybrid symbolic–numeric approach that integrates constrained sum-of-squares (SOS) decomposition, exact LDLᵀ factorization, and rational-coefficient polynomial identity verification. This method yields a fully exact rational SOS certificate, formally verified in Lean. Our result provides an explicit quadratic upper bound: $\phi(A) \leq \frac{61}{32} - \frac{1}{52}\|A - U_4\|_F^2$, with the constructed certificate comprising 152 principal square terms and 136 sub-blocks of 16 terms each.

Abstract

The Dittert conjecture states that the Dittert functional on nonnegative $n\times n$ matrices whose entries sum to $n$ is uniquely maximized by the uniform matrix. We prove the conjecture in dimension $4$. More precisely, let $K_4$ be the simplex of nonnegative $4\times4$ real matrices whose entries sum to $4$, let $U_4$ be the uniform matrix, and let $φ$ denote the Dittert functional. We establish $\frac{61}{32}-φ(A)\geq \frac{1}{52}\lVert A-U_4\rVert_F^2$ for every $A\in K_4$. Consequently, $U_4$ is the unique maximizer of $φ$ on $K_4$. The proof reduces to certifying the nonnegativity of a structured quartic polynomial in sixteen variables on a simplex. We construct an exact rational constrained sum-of-squares certificate using an agent-guided symbolic-numeric procedure that combines template selection with sequential rational recovery. The main SOS consists of $152$ positively weighted rational squares, while each of the $136$ smaller SOS blocks consists of $16$ such squares. Exact $LDL^{\mathsf{T}}$ decompositions certify positivity, and exact coefficient comparison over $\mathbb{Q}$ verifies the complete polynomial identity. The resulting exact certificate is formally verified using the Lean proof assistant.

Problem

• Dittert conjecture
• sum-of-squares certificate
• matrix optimization
• uniform matrix
• nonnegative matrices

Innovation

• sum-of-squares certificate
• agent-guided symbolic-numeric method
• exact rational verification
• Dittert conjecture
• formal verification in Lean