Real-rootedness of the Poincar\'e polynomials of $\overline{\mathcal M}_{0,n}$: an AI-assisted proof

ArXi:2605.29151v1 Announce Type: cross We prove real-rootedness for the Poincar\'e polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli. The proof starts from the Keel--Manin--Getzler recurrence, but its main new idea is a bivariate deformation $F_m(y,t)$ of the Poincar\'e polynomial. This deformation reveals a hidden interlacing structure not visible in the one-variable recurrence.