Weak Simplex Conjecture

For every integer $n\ge 1$ , energy $E>0$ , and noise standard deviation $\sigma>0$ , let $x_0,\ldots,x_n\in\mathbb{R}^n$ satisfy $\lVert x_i\rVert^2=E$ for all $i$ , and let the signals have equal prior probabilities and be transmitted over an additive white Gaussian noise channel with maximum-likelihood decoding. If $P_c(x_0,\ldots,x_n;\sigma)$ denotes the probability of correct decoding, then $P_c$ is maximized by the vertices of a regular simplex centered at the origin, characterized by $\langle x_i,x_j\rangle=-E/n$ for all $i\ne j$ . Moreover, equality holds only when the signal set is a regular simplex, up to a permutation of the vertices and an orthogonal transformation of $\mathbb{R}^n$ .

References

Primary source

arXiv

Additional references

The Equality Cases of the Weak Simplex Conjecture — arXiv

Posted 1 month ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A July preprint claims the regular arrangement is optimal, and an August preprint claims it is the only optimum, but neither claim has independent verification.

The conjecture says that the regular simplex uniquely maximizes decoding success. Cover (1987) and Massey (1988) formulated versions; Landau and Slepian's 1966 proof was valid only in dimension three. Dunbridge established low-noise asymptotic and all-noise local optimality, while Steiner (1994) refuted the stronger conjecture without resolving the weak one.

July–August 2026 claimed resolution

A July 2026 preprint claims a complete proof of the maximizing inequality in every dimension and noise regime, while explicitly leaving uniqueness open. On August 19, 2026, The Equality Cases of the Weak Simplex Conjecture claimed the missing uniqueness result and reported a Lean formalization. Both mathematical claims remain unverified.

Community submission (unverified)

On September 9, 2026, a submitted proof claimed a stronger result and linked a Lean formalization, with substantial generative-AI assistance; this is not independent verification.

Current status (as of September 2026): the optimal-value statement and uniqueness are claimed in separate 2026 preprints, but the full conjecture remains unverified.

Sources

ProofPreprint of a stronger result I have obtained with the help of AI. A Lean formalization is included as well.See full solutionHide full solution

I have posted a preprint of a solution with a stronger result. I obtained it with the help of AI. A Lean formalization is included as well.