Orbital carriers and inheritance in discrete-time quadratic dynamics

The paper explicitly defines carriers ψk(x) and auxiliaries Sk(σ), and shows how Sk(σ)=0 projects ψk to period‑k orbits or to clusters when Sk has nonlinear factors; this is illustrated in detail for k=4,5,6, with S4, S5, and S6 matching the factors and examples the model uses (e.g., S6(σ)=(σ+1)(σ−1)(σ^3−21σ+28)(σ^4+σ^3−24σ^2−4σ+16), and the monic sextics o6,1 and o6,2 obtained at σ=1 and σ=−1) . The paper also states and verifies the inheritance identity c6,1(x)=o6,2(x^3−3x), and explains how the six cubic preimages split into three period‑6 orbits, i.e., the degree‑18 cluster c6,1 . The model solution reproduces all these facts and adds a clean dynamical proof of exact period using the commutation φ∘g=g∘φ with g(x)=x^2−2 and φ(x)=x^3−3x, noting f=2−x^2=−g so f^6=g^6, thereby showing that preimages inherit exact period 6. This commutation-based argument is not spelled out in the paper but is correct and complementary. Hence both are correct, with different proof emphases.