An explicit iid-uniform counterexample to scalar entropy-power supermodularity

Definitions and mathematical statement

For a real random variable $X$ with density $f_X$, use natural logarithms and differential entropy \[ h(X)=-\int_{{\mathbb{R}}}f_X(x)\log f_X(x)\,{\,\mathrm{d}} x,\qquad N(X)=e^{2h(X)}. \] No factor $(2\pi e)^{-1}$ is included. For independent real $X_1,X_2,X_3$ with densities, and with the four entropies finite so that every term is unambiguous, Madiman's question is whether \[ D:=N(X_1+X_2+X_3)+N(X_2)-N(X_1+X_2)-N(X_2+X_3)\geq0. \] The source calls its entropy $H$; the notation $h$ here emphasizes differential entropy. The source notes equality for independent Gaussian variables, including unequal positive variances.

The aid reports an explicit counterexample with independent identically distributed $U_i\sim\operatorname{Unif}[0,1]$: \[ h(U_i)=0,\qquad h(U_1+U_2)=\tfrac12,\qquad h(U_1+U_2+U_3)=\tfrac76+\log2-\tfrac{\sqrt3}{2}\log(2+\sqrt3). \] Its reported deficit is \[ D=4e^{7/3}(2+\sqrt3)^{-\sqrt3}+1-2e<0. \] The aid says the strict sign has a rational-series certificate, not reproduced in the supplied summary. It also reports robustness: for independent standard Gaussians $G_i$, independent of all $U_i$, there is $\sigma_0>0$ such that the triple $X_i=U_i+\sigma G_i$ has $D<0$ for every $0<\sigma<\sigma_0$. The resulting densities are smooth, strictly positive and log-concave. This is the reported entropy-continuity extension, not an independently checked numerical certificate.

The original remarks append a distinct general inequality. For independent $X_1,\ldots,X_n$ and nonempty $S\subseteq[n]$, put $X_S=\sum_{i\in S}X_i$. They define a fractional covering by \[ \beta_S\geq0,\qquad\sum_{S\ni i}\beta_S\geq1\quad(i\in[n]), \] and claim that the displayed three-variable inequality would imply \[ N(X_{[n]})\geq\sum_{S\subseteq[n]}\beta_SN(X_S). \] This claim is not valid with unrestricted covering weights as written: taking $\beta_{[n]}=2$ and all other weights zero would require $N(X_{[n]})\geq2N(X_{[n]})$ even for nondegenerate Gaussians. The empty-set term also has no stated entropy convention. The source wording is preserved here rather than silently changed to a fractional partition or packing condition.

The same remarks report known inequalities for all $(n-1)$-subsets, attributed to Artstein, Ball and Naor, and a regular-hypergraph generalization attributed to Barron and Madiman. A regular hypergraph has the same number of member sets through each index. These historical special-case assertions require the appropriate normalized weights, such as $1/(n-1)$ on all $(n-1)$-subsets; they cannot support the unrestricted covering formula above. Complete theorem statements and weight normalizations are absent from the source excerpt.

Short English statement

The supplied iid-uniform example makes the proposed entropy-power supermodularity deficit strictly negative, with failure persisting after small Gaussian smoothing. The remarks also contain a covering-weight formula whose literal quantifiers are inconsistent; that issue is kept separate from the reported normalized special cases.

Sources

• Local registry: permanent UnsolvedMath ID $20001293$, exact public code catalogue entry; aim-convex-geometry-notes.json, notes[24].
• AIM workshop "Algorithmic convex geometry", Problem 6, attributed to Mokshay Madiman: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf. The supplied verification identifies $H$ as differential entropy and confirms independent real variables with densities.
• Original remarks state Gaussian equality and name Artstein, Ball and Naor for the all-$(n-1)$-subsets inequality, and Barron and Madiman for regular hypergraphs. No full bibliographic entries are supplied; the literature field is empty.
• Research attempt 1: counterexample, status at run refuted, rigor label proved, novelty confidence low, classification SOLVED-IN-LITERATURE. It reports exact iid-uniform entropies, a rational strict-sign certificate and Gaussian-smoothing robustness. The current registry status is solved. No separate authoritative citation for this counterexample or certificate is supplied.