A support obstruction and a Wasserstein repair of Conjecture 42

Definitions and mathematical statement

In the surrounding eigenfunction setting, let $-\Delta\varphi_i=\lambda_i\varphi_i$, with $\|\varphi_i\|_2=1$, and let $g^t$ be geodesic flow on $S^*X$ with normalized Liouville measure $\lambda_{\mathrm L}$. For real $a\in C_c^\infty(S^*X)$ with $\int a\,{\,\mathrm{d}}\lambda_{\mathrm L}=0$, the printed Conjecture 42 of J. Marklof, after Feingold--Peres, defines \[ \xi_i(a)=\lambda_i^{1/4}|\langle\operatorname{Op}(a)\varphi_i,\varphi_i\rangle|, \qquad V(a)=\int_{-\infty}^{\infty}\int_{S^*X}a(xg^t)a(x)\,{\,\mathrm{d}}\lambda_{\mathrm L}(x)\,{\,\mathrm{d}} t. \] It supposes that $X$ is "generic", without defining that class, and asserts \[ \frac1E\sum_{\lambda_i\le E}\xi_i(a)^2\longrightarrow V(a), \] \[ \frac1E\#\{i:\lambda_i\le E,\ V(a)^{-1/2}\xi_i(a)\in I\} \longrightarrow\frac1{\sqrt{2\pi}}\int_Ie^{-t^2/2}\,{\,\mathrm{d}} t \quad\text{for every interval }I\subset{\mathbb{R}}. \] The absolute value is in the source itself, according to the supplied verification. If $V(a)>0$, all normalized statistics are nonnegative, so the left side for $I=[-2,-1]$ is identically zero while the claimed Gaussian mass is positive. If $V(a)=0$, the displayed normalization is undefined; $V(a)<0$ is incompatible with the nonnegative second-moment limit. Thus the literal assertion is obstructed independently of any choice of "generic" surface. Convergence and positivity of the autocorrelation integral also require hypotheses absent from the extract.

The aid discusses two explicit repairs, neither silently substituted for the printed conjecture. For a self-adjoint quantization and $0<V(a)<\infty$, retain the sign and put \[ Z_i=\frac{\lambda_i^{1/4}\langle\operatorname{Op}(a)\varphi_i,\varphi_i\rangle}{\sqrt{V(a)}},\qquad N(E)=\#\{i:\lambda_i\le E\},\qquad \rho_E=\frac1{N(E)}\sum_{\lambda_i\le E}\delta_{Z_i}. \] For these probability empirical measures, a weak standard-Gaussian limit together with second moment tending to one is equivalent to quadratic-Wasserstein convergence \[ W_2(\rho_E,\mathcal N(0,1))\longrightarrow0, \qquad W_2(\rho,\sigma)^2= \inf_{\pi\in\Pi(\rho,\sigma)}\int|x-y|^2\,{\,\mathrm{d}}\pi(x,y). \] Equivalently, weak Gaussian convergence must be supplemented by uniform integrability of $Z_i^2$ in the empirical measures. The aid reports an equal-weight Gaussian-quantile construction with a zero-density outlier tail showing that a weak Gaussian law alone does not force the second-moment limit. If the absolute value is retained instead, its natural Gaussian-derived target is half-normal, with density $\sqrt{2/\pi}e^{-t^2/2}$ for $t\ge0$ and zero for $t<0$.

The source normalizes its sums by $E$, not by $N(E)$. Equating its formulas with probability statements needs the appropriate Weyl counting constant or a convention ensuring $N(E)/E\to1$; this cannot be assumed from normalized Liouville volume. The Wasserstein formulation above explicitly uses $N(E)$ and is a proposed normalized repair.

The source further reports Luo--Sarnak's modular-surface asymptotic \[ \sum_{\lambda_i\le E}|\langle\operatorname{Op}(a)\varphi_i,\varphi_i\rangle|^2 \sim\sqrt E\,B(a), \] where $B$ on $C_c^\infty(X)$ is related to but distinct from $V$ [169]. It therefore excludes the modular and other arithmetic surfaces from its intended "generic" examples. The following polygonal-billiard suggestion of A. Katok begins section 6.3 and is separate context.

Short English statement

The printed absolute-value statistic cannot have a centered Gaussian law because it never takes negative values. A signed, correctly counted version has a meaningful Gaussian-plus-second-moment formulation equivalent to Wasserstein convergence; keeping absolute values instead gives a half-normal target.

Sources

• J. Marklof, after Feingold--Peres [55], Conjecture 42 in American Institute of Mathematics, Emerging applications of measure rigidity, section 6.2, https://aimath.org/WWN/measrigid/measrigid.pdf. The source attributes the arithmetic variance comparison to W. Luo and P. Sarnak and cites [169]. Full bibliography entries are not supplied.
• Local UnsolvedMath registry, permanent ID $20003335$, source code catalogue entry, canonical aim-other-notes.json, notes[141], dated 14 August 2026. Supplied verification says the absolute value is present in the official source. Research attempt 1: counterexample, refuted at run, proved rigor, low novelty confidence; machine-generated and not peer reviewed. The literal support obstruction is distinct from any unresolved repaired conjecture.