The Equality Cases of the Weak Simplex Conjecture

Abstract: Among $n+1$ equiprobable equal-energy signals in $\R<sup>n$ under additive white Gaussian noise with maximum-likelihood decoding, which arrangement maximizes the probability of correct decoding? The question is Shannon's, recorded by Rice in 1950. Mulgund proved in 2026 that the regular-simplex value bounds the correct-decoding probability of every signal set at every signal-to-noise ratio, leaving open whether the simplex is the only maximizer. This paper determines the equality cases in a form stronger than uniqueness. A signal set other than a regular simplex falls strictly below the bound at every positive signal-to-noise ratio. Hence a code meeting the bound at one positive operating point is already a regular simplex, up to vertex relabeling and an orthogonal map. In probabilistic form, among the correlation matrices that signal sets induce, any matrix other than the identity gives a lower-orthant probability strictly above its independent counterpart at every finite threshold, leaving no room for a nontrivial equality. No code of ambient dimension below $n$ attains the bound. Under an energy budget $E$ with unrestricted blocklength the optimal codebook is uniquely the regular simplex of circumradius $\sqrt{E}$. Every optimal codeword therefore exhausts its allowance. Equality in the Simplex Mean Width Conjecture likewise occurs only at the regular simplex. The proof strengthens the first self-convolution step of Mulgund's argument with Royen's correlation theorem. The single-parameter rigidity is machine-checked in Lean 4.

• The paper proves that equality in the Weak Simplex Conjecture at even one positive SNR forces the signals to form a centered regular simplex, up to relabeling and orthogonal transformation.
• The proof strengthens Mulgund’s stochastic-domination argument using strict Gaussian rectangle and truncated-exponential inequalities, covering singular covariance matrices without approximation.
• The result extends to moment-generating functions, finite-energy infinite-blocklength AWGN coding, and equality in the Simplex Mean Width Conjecture, while leaving quantitative stability and cases with other numbers of signals open.

Overview

This paper resolves Open Problem 8.1 of Mulgund [(2608.19093), citing Mulgund 2026], which asked for the equality cases of the stochastic-domination theorem that settled the Weak Simplex Conjecture. The Weak Simplex Conjecture, originating with an observation of Shannon reported by Rice in 1950 and named by Massey in his 1988 Shannon Lecture, asserts that $n+1$ equiprobable equal-energy signals in $\mathbb{R}^n$ under additive white Gaussian noise (AWGN) with maximum-likelihood decoding maximize the probability of correct decoding precisely when arranged as a regular simplex inscribed in the energy sphere. Mulgund proved the optimality claim at every signal-to-noise ratio (SNR) via a stochastic-domination argument but left open whether the regular simplex is the unique maximizer.

The paper answers this question in a form stronger than mere uniqueness. Its central finding is that any signal set other than a regular simplex falls strictly below the simplex bound at every positive SNR; consequently, equality at a single positive operating point already forces the code to be a centered regular simplex up to vertex relabeling and orthogonal transformation. The result extends to the moment-generating function (MGF) comparison, the finite-energy infinite-blocklength model, and the equality case of the Simplex Mean Width Conjecture (SMWC) in convex geometry.

Background and problem structure

The coding problem admits two equivalent translations, both restated for notational fixity. Completing the square in the Gaussian likelihood converts correct-decoding probability into

$$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$

where $G$ is the Gram matrix of the signals. The code thus enters only through $G$, and design becomes an extremal problem on correlated Gaussian maxima: the regular simplex corresponds to $G_\Delta = \frac{n+1}{n}(I - \frac{1}{n+1}J)$, whose associated Gaussian vector has independent coordinates after centering. Second, via Kabluchko–Litvak–Zaporozhets, $\mathbb{E}\max_i \langle g,y_i\rangle$ equals the mean width of the simplex spanned by the vertices up to a dimensional constant, so the SMWC is the $\lambda\to 0$ face of the coding problem — meaning any statement proved for every $\lambda>0$ covers it automatically, though not conversely.

Historical context matters here: the Strong (average-energy) Simplex Conjecture is false by Steiner's counterexample, so the rigidity established here is specific to the equal-energy formulation. Two earlier announced proofs (Goldsmith for mean width; Pastore for coding) contain documented gaps and are cited only as prior announcements.

Main results

The paper establishes four statements:

• Single-threshold strictness: For any correlation matrix $R$ with $\mathbb{R}^n$0, possibly singular, if $\mathbb{R}^n$1 then $\mathbb{R}^n$2 for every finite $\mathbb{R}^n$3. Equality at even one finite threshold forces $\mathbb{R}^n$4.
• MGF rigidity: $\mathbb{R}^n$5 with equality at some (equivalently every) $\mathbb{R}^n$6 if and only if $\mathbb{R}^n$7.
• Coding consequences: (a) A code attaining the simplex value at one positive SNR is a centered regular simplex; no code of ambient dimension $\mathbb{R}^n$8 attains the bound at all. (b) In the deterministic no-feedback infinite-sequence AWGN model with per-codeword energy budget $\mathbb{R}^n$9 and unrestricted channel uses, the optimal codebook is uniquely the regular simplex of circumradius $$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$0 — notably, full energy expenditure per codeword emerges as a consequence rather than an assumption.
• SMWC equality: Among simplices in the unit ball, mean-width equality holds only for the inscribed regular simplex.

The practical implication for signal design is stated plainly: the regular simplex is not one optimum among several; every departure costs at every positive SNR, so any numerical search reporting a tie against the simplex is exhibiting a numerical artifact.

A closed-form worked example illustrates the mechanism within equicorrelated families ($$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$1 recovers the simplex), where $$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$2 is manifestly strictly maximized at the smallest feasible $$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$3. The authors correctly note this reduction does not generalize, since arbitrary Gram matrices admit no one-dimensional sufficient statistic.

Proof architecture

The engine is a strict version of the first normalized self-convolution step in Mulgund's adaptive-tilting argument, supported by two new strict inequalities:

1. Strict symmetric rectangle inequality: For centered Gaussian vectors with unit marginals and $$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$4, possibly singular, $$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$5 at all finite radii. The proof isolates a genuine two-dimensional strict gap — via a covariance monotonicity argument showing the conditional interval probability $$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$6 is strictly decreasing on $$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$7 — then propagates it through remaining coordinates using Royen's Gaussian correlation theorem [Royen 2014]. Singular matrices are handled by preimage under a factor map, deliberately avoiding positive-definite approximation, since limits of strict inequalities need not remain strict.
2. Strict product inequality for truncated exponentials: For Mulgund's tilted factors $$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$8 with vanishing first moment, self-convolution converts each factor into a symmetric interval indicator (the exponential cancels in $$P_c(x_1,\dots,x_m;\lambda)=\frac{e^{-\lambda^2/2}}{m}\,\mathbb{E}\exp\Bigl\{\lambda\max_{1\le i\le m}\xi_i\Bigr\},\qquad \xi\sim N(0,G),$$9). On the event where all radii are positive, Lemma 1 injects a strictly positive deficit $G$0; off that event both sides vanish or the integrand dies identically, so the deficit survives integration. Chaining through Mulgund's independently proved non-strict product theorem gives $G$1 without circularity — strictness enters only at the first convolution step.

Theorem assembly then follows: the tilting alignment condition supplies $G$2, $G$3; the change-of-measure identity (valid for singular $G$4) transports the strict product gap onto the lower-orthant probability; MGF rigidity follows from pointwise strictness of distribution functions via the tail integral representation plus the transfer construction $G$5 mapping general $G$6 to admissible $G$7, with $G$8. The finite-energy result reduces the $G$9 sequence experiment exactly to a finite-dimensional Gaussian shift via a sufficient statistic built from an orthonormal basis of $G$0, with likelihood ratios converging in $G$1 under a square-integrability bound; augmentation of codewords to full energy, followed by monotonicity of $G$2 in the circumradius, pins the optimum and forces every $G$3 to zero.

Machine verification

The single-parameter MGF rigidity is formalized in Lean 4 (v4.31.0, pinned mathlib revision, public repository with axiom audit restricted to the three standard Lean axioms), building on the formalization accompanying Mulgund's proof. The route taken is independent: stochastic domination combined with equality of one strictly increasing moment forces equality of laws, and a correlated Gaussian maximum carrying the independent distribution at all thresholds must be independent. Notably, Theorem 1's single-threshold statement rests on Royen's theorem, for which no formalization exists to date — the authors identify its grouping step's reduction to the slab-against-convex Khatri–Šidák case as a plausible formalization target. This is the one component of the main line of argument not machine-checked.

Limitations and open questions

The paper is candid about scope. Steiner's counterexample delimits everything: under average-energy constraints no such rigidity can hold, so the results are specific to equal-energy formulations. Boundary conditions are sharp and cannot be dropped — the threshold must be finite, the MGF parameter strictly positive, the energy strictly positive — because each formal endpoint yields trivial equality. The rank-constrained problem for $G$4 signals remains open, including the biorthogonal conjecture at $G$5. Finally, the pair-tail structure of the strict rectangle lemma suggests, but does not yet deliver, a quantitative stability estimate bounding the decoding deficit in terms of distance from $G$6; the paper proves qualitative strictness only.

Conclusion

This paper completes the program begun by Mulgund by determining the full equality cases of the Weak Simplex Conjecture: the regular simplex is the unique maximizer at every positive SNR, with strictness holding at every finite threshold, every positive moment-generating parameter, and under any positive energy budget. The methodological contribution — a strict symmetric rectangle inequality sharpening Šidák–Khatri via Royen's theorem, deployed inside the self-convolution step of the tilting argument — is reusable wherever weak Gaussian correlation inequalities appear, and the finite-dimensional sufficiency reduction cleanly extends the coding answer to unrestricted-blocklength models used in non-asymptotic short-code analysis.