Symmetric unimodality under free additive convolution

Definitions and mathematical statement

The corrupted source asks whether the convolution of two symmetric unimodal distributions is unimodal, but its binary-operation glyph is lost. The aid reconstructs free additive convolution $\boxplus$, citing agreement with Hasebe--Ueda's Conjecture 3.5. The following "Invariant Subspaces for an Operator," Haagerup, is an absorbed section heading, not part of this question. The registry's clean-statement field is explanatory verification prose rather than a replacement mathematical statement.

For centered Borel probabilities on ${\mathbb{R}}$, use weak symmetric unimodality in the form $\mu=c\delta_0+f(x)\,{\,\mathrm{d}} x$, where $c\ge0$, $f$ is even and nonincreasing on $(0,\infty)$, and total mass is one. Translation handles a specified nonzero symmetry center. Free additive convolution is the law of the sum of freely independent self-adjoint variables with the given laws. The reconstructed question is \[ \mu,\nu\text{ symmetric and weakly unimodal} \quad\Longrightarrow\quad \mu\boxplus\nu\text{ unimodal}? \] The aid describes this arbitrary-pair question as open in its checked literature.

Its special case takes the centered semicircle $S(0,t)$ of variance $t>0$, with density $(2\pi t)^{-1}\sqrt{4t-x^2}$ on $[-2\sqrt t,2\sqrt t]$. For every symmetric weakly unimodal $\mu$, the aid reports that the density $p_t$ of $\mu\boxplus S(0,t)$ has a unique global maximum at zero. The exact height certificate is \[ p_t(0)=\frac{v}{\pi t},\qquad \int_{\mathbb{R}}\frac{{\,\mathrm{d}}\mu(x)}{x^2+v^2}=\frac1t, \] where $v>0$ is the unique solution. If $\mu$ has finite variance $\sigma^2$, the stated bounds are \[ \frac{\sqrt{(t-\sigma^2)_+}}{\pi t} \le p_t(0)\le\frac1{\pi\sqrt t},\qquad r_+=\max\{r,0\}. \] The lower bound can be zero and is not then a positive lower estimate.

To delimit the input hypothesis, the aid reports that for the symmetric two-point law $\mu_a=(\delta_{-a}+\delta_a)/2$, $a>0$, the semicircular convolution is unimodal exactly at and above $t=4a^2$. This input is not weakly unimodal, so the example is an obstruction to dropping input unimodality, not a counterexample to the reconstructed arbitrary-unimodal-pair conjecture. The exact scalar criterion and variance bounds concern the semicircular special case only.

Short English statement

The missing operation is reconstructed as free additive convolution. The supplied result strengthens a semicircular special case to a unique central mode and gives an exact height equation and bounds. The general convolution of two symmetric unimodal inputs remains unresolved in the aid's assessment.

Sources

• Local UnsolvedMath registry, permanent ID 20002676, source code catalogue entry; created and updated 14 August 2026. Canonical aim-probability-notes.json, zero-based notes[117], tag question, item 16.
• AIM "Free Analysis": https://aimath.org/WWN/freeanalysis/freeanalysis.pdf. The statement is reconstructed_unverified; the verification records the missing operation and absorbed following Haagerup section heading. Original remarks and literature field are empty.
• The supplied clean-field/aid context names Hasebe--Ueda, Conjecture 3.5, as independently confirming the free-additive question, but gives no full title, date, or bibliographic link. Research attempt 1: partial_result, problem status at run open, rigor label proved, novelty confidence low. Full references for the semicircular theorem and refinement are absent.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.