Real-rootedness of the Poincar'e polynomials of overlinemathcal M0,n: an AI-assisted proof

We prove real-rootedness for the Poincar'e polynomial [ Pn(t)=sumi=0n-3 dim H2i(overlinemathcal M0,n;mathbbQ)ti ] of the Deligne--Mumford moduli space overlinemathcal M0,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 Fm(y,t) of the Poincar'e polynomial. This deformation reveals a hidden interlacing structure not visible in the one-variable recurrence. For fixed t<0, the zero set of Fm in the y-direction is controlled by a Sturm--Rolle argument on the interval 0<y<1-t. The original polynomial is recovered on the slice y=1, and the ordered crossings of the moving roots through this slice give both real-rootedness and strict interlacing. Consequently, the Betti numbers of overlinemathcal M0,n form an ultra-log-concave sequence. We further prove real-rootedness and ultra-log-concavity for the Poincar'e polynomial of the Fulton--MacPherson space mathbbP1[n] of n ordered points in degenerations of the complex projective line. The proof for overlinemathcal M0,n was obtained through an iterative AI-assisted workflow with Co-Mathematician, an agentic frontier-model system developed by Google DeepMind. Our role was to formulate the problem, evaluate the proposed proof attempts, identify gaps and request corrections, compare the developing argument with the literature, and refine the presentation of the final proof. Our additional human contribution was to observe that a similar residual deformation strategy applies to the Fulton--MacPherson spaces mathbb P1[n], yielding the corresponding real-rootedness theorem.