Complex and Projective Billiards

The uploaded paper proves exactly the two statements under audit (Theorem 3.51): in the plane, every C∞ 3‑reflective local projective billiard is right‑spherical; in dimensions d ≥ 3, no C∞ 3‑pseudo‑reflective local projective billiard exists. The planar part is obtained by first proving in the complex analytic category that 3‑reflectivity forces the classical boundaries to be lines and hence the billiard to be right‑spherical (Proposition 3.56 and Theorem 3.71), and then passing from C∞ to analytic via Pfaffian systems (Section 3.3.5) . The higher‑dimensional non‑existence is proved via a complex version (Theorem 3.82) and planarization arguments (Lemma 3.84), implying hyperplanarity (Lemma 3.83) and a central‑pencil rigidity that leads to contradiction (Lemma 3.86) . The candidate solution outlines a different, pencil‑identity based proof sketch in the plane, arriving at the same classification, and gives a qualitative dimension‑counting argument for d ≥ 3; it cites Fierobe for full details and is consistent with the paper’s conclusions. The converse (right‑spherical is 3‑reflective) is also established in the paper (Proposition 3.10) .