Cubic Polynomial Maps with Periodic Critical Orbit, Part III: Tessellations and Orbit Portraits

The paper’s Landing Theorem (Theorem 2.7) states that every rational-angle parameter ray in an escape region Eh (in the cubic slice Sp with a marked periodic critical point) lands uniquely; landing parameters are parabolic when the co-critical angle is co-periodic and Misiurewicz otherwise, with the special periodic-angle clause for 2a being repelling. The paper proves this via a preliminary landing lemma (rational rays accumulate only at parabolic or Misiurewicz maps), a countability-plus-connectedness argument for uniqueness, a contradiction argument for the co-periodic→parabolic direction, and a detailed analysis (Lemma 2.10) of the parabolic alternative and the free critical orbit along the ray. These ingredients are explicitly laid out in the provided PDF (definitions of parameter rays and co-critical angles; Eh ≅ punctured disk; Lemma 2.8; proof of Theorem 2.7) . The candidate model’s solution proves the same statement for multiplicity μ=1, using the co-critical Böttcher coordinate Φ(F)=BF(2aF), continuity of dynamic rays in the basin of infinity, a combinatorial classification by the finite ray-graph generated from θ and θ±1/3, and a uniqueness claim by observing that the cluster set is connected and the target loci are (claimed) discrete. Substantively, this is a different proof strategy than the paper’s: the model’s Step 3 uses a “two distinct rays co-map under F^q ⇒ not repelling ⇒ parabolic” access argument, whereas the paper’s Step 1 uses a direct contradiction via the periodic image 3θ landing on the critical value; the model appeals to (post)critically finite rigidity to assert discreteness, while the paper only needs countability and connectedness. The model’s conclusions match the paper’s landing classification and periodic-angle clause. One minor flaw in the model’s write-up is calling the parabolic and Misiurewicz loci “discrete” in Sp; the paper only claims these loci are countable and uses that (together with connectedness of the accumulation set) to deduce uniqueness of landing . Despite this overstatement, the model’s uniqueness still follows if one replaces “discrete” by “countable”. Overall, both are correct, with materially different proof routes.