Combinatorial periods of asymmetric billiard trajectories in the regular pentagon

We prove that every even integer other than $2$, $12$, $14$ and $18$ is the combinatorial period, the number of bounces before repeating, of a periodic billiard trajectory in the regular pentagon that does not have five-fold rotational symmetry. Davis and Lelièvre conjectured this in 2018 and checked it up to period $4000$, and Kontorovich and Zhang proved it for almost every even number. As they explain, a theorem of Davis, Fuchs and Tabachnikov turns the question into one about vectors. Each vector $(a+b\varphi,c+d\varphi)$ obtained from $(0,1)$ by products of four explicit $2\times2$ matrices, where $\varphi$ is the golden ratio and $a,b,c,d$ are integers, has the half-period $\ell=a+b+c+d$, and when a congruence modulo $5$ holds it gives such a trajectory of period $n=2\ell$. Varying the power of one of the two triangular matrices in such a product gives half-periods in an arithmetic progression. We give $379$ of these progressions, which together contain every half-period $\ell\ge2$ except nine values. Six of the nine come from explicit trajectories checked in exact arithmetic; the other three are halves of excluded periods. The covering argument is checked in Lean, assuming the cited theorem. Whether the four excluded periods occur is not treated. This work has not yet been reviewed by experts. The proofs, the Lean formalization and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction.

Written proof, with Lean and computer checks; not yet reviewed by experts. Produced by AI models under the author's direction.