The equal-radius simplex conjecture for Gaussian error probability

About 9 years old · traced to 2017

Let $P(v_1,\ldots,v_N)=\int_{\mathbb R^n}\max_i e^{-|x-v_i|^2}\,dx$ for a code consisting of $N$ vectors in $\mathbb R^n$ . The simplex conjecture. For $N=n+1$ and fixed $|v_1|=\dots=|v_N|=r$ , the maximum of $P(v_1,\ldots,v_N)$ is attained at any configuration forming a regular simplex inscribed into the ball of radius $r$ . The original fixed-total-energy version is false, but this equal-radius formulation is the conjecture studied in the paper and is proved only in dimensions at most $3$ .

References

Primary source

Alexey Balitskiy, Roman Karasev and Alexander Tsigler, “Optimality of codes with respect to error probability in Gaussian noise”, arXiv:1701.07986 (2017).

Posted 1 month ago · edited 4 weeks ago

Abhijeet Mulgund 4 weeks ago · edited 4 weeks ago
The current posted version of Pastore's A Proof of the Weak Simplex Conjecture (arXiv:2306.13478v2, 13 November 2023) does not establish its advertised all-dimensional theorem.

The injectivity argument in Section III-D defines conditional means $t,t'$ over the cone differences $\widetilde D\setminus\widetilde D'$ and $\widetilde D'\setminus\widetilde D$ in (18c)--(18d). The appendix then uses the set inclusions (23b)--(23c) as though those conditional means themselves belonged to the corresponding nonconvex set differences, yielding the sign statements (24b)--(24c). That inference is invalid: averaging preserves membership in a convex set, not in an arbitrary nonconvex set difference.

Appendix B.2 of Mulgund, arXiv:2607.14087v2, gives an explicit three-dimensional Gaussian simplicial-cone counterexample in which
$\mathbb E[Z\mid Z\in D\setminus D']\in\operatorname{int}D'$
so the required membership/sign inference actually fails in the setting of the proof. The same appendix gives an independent admissible example contradicting the facet-normal pairwise-independence assertion used to normalize $\lambda_1=1$ in the appendix of A Proof of the Weak Simplex Conjecture.

These defects occur in the injectivity result used by Section III-E to deduce the reflection symmetry needed for the final theorem. This challenge concerns only the posted v2 on arxiv; it does not assert that the conjecture is false or that the approach cannot be repaired.

Supporting discussion: Appendix B.2 of arXiv:2607.14087v2.

Progress summary

Refreshed 4 weeks ago

Claimed solved

A 2026 preprint claims the regular simplex is optimal in every dimension, but that claimed solution has not been independently verified.

The conjecture asks whether equal-radius Gaussian codewords are optimized by a regular simplex. The underlying observation is attributed to Shannon via Rice (1950), with the name coined by Massey (1988).

Known results

The equal-radius conjecture is proved through dimension $n=3$ , and in the zero- and infinite-energy limits (Balitskiy, Karasev, and Tsigler, 2017).
The original fixed-total-energy formulation is false for sufficiently many codewords; this does not refute the equal-radius version.

In 2026, Mulgund’s Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture claims an all-dimensional proof for every positive signal strength. Its Corollary 2.4 gives the equivalent Gaussian decoding optimization, hence the stated integral maximization after scaling; uniqueness is not claimed.

Community submission (unverified)

A submission dated September 9, 2026, claims a Lean formalization of a stronger theorem and says the solution used substantial generative-AI assistance. An earlier submission dated August 14, 2026, identifies the claimed result with Mulgund’s revised preprint and reports kernel-checked Lean declarations, but the scaling identity to the displayed integral is not formalized. These are unverified claims.

Current status (as of September 2026): The conjecture is settled through $n=3$ , while the all-dimensional claim in Mulgund’s 2026 preprint and the accompanying formalization remain unverified.

Sources

ProofThis solution needs a summarySee full solutionHide full solution

Abhijeet Mulgund, Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture, arXiv:2607.14087v2, claims a complete resolution of this conjecture in every dimension.

The equivalence with the formulation on this page is immediate by scaling. Write $N=n+1$ , $v_i=r x_i$ with $\|x_i\|=1$ , and put $\lambda=\sqrt{2}\,r$ . If

P(v_1,\ldots,v_N) = \int_{\mathbb R^n}\max_i e^{-\|z-v_i\|^2}\,dz,

then the average probability of correct maximum-likelihood decoding in the Gaussian model of the paper satisfies

\psi_{\sqrt{2}r}(x_1,\ldots,x_N) = \frac{1}{N\pi^{n/2}}\,P(v_1,\ldots,v_N).

The multiplicative factor is independent of the configuration. Thus maximizing the MathDB objective $P$ over equal-radius codewords is exactly the Weak Simplex optimization solved in Corollary 2.4 of arXiv:2607.14087v2: a regular simplex is a maximizer for every radius $r>0$ .

Solution source: Abhijeet Mulgund, Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture, arXiv:2607.14087v2 (2026).

Formal verification: The equivalent normalized Gaussian maximum-likelihood theorem is kernel-checked in Lean 4 at commit dcf1a45bc22ec54775314927b2a95fbe7c630edf; see WeakSimplex.weak_simplex and WeakSimplex.weak_simplex_of_scoreMaximizingDecoders. For positive signal strength, the formalization also proves equality if and only if the code Gram matrix is the regular-simplex Gram matrix (WeakSimplex.weak_simplex_eq_iff_codeGram_eq_of_scoreMaximizingDecoders). The scaling identity connecting this theorem to the displayed equal-radius integral is supplied above but is not itself a standalone Lean declaration.

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.