Full rotation orbits in triangulations of 3L-gons: a Catalan structure theorem, and the maximum-weight conjecture for sparse triangulation shifts

Author: Reynout Vos, independent researcher https://www.raiv.nl (MSc Computer Science, Eindhoven University of Technology, 1999-2005) This record deposits a combinatorial structure theorem about rotation orbits in polygon triangulations, together with the code and data that verify the implementation. It accompanies G. Alkauskas, Regular triangle unions with maximal number of sides (arXiv:2510.22584v5). It is a new record, distinct from the author's exact-certificates record (doi:10.5281/zenodo.21536545), which deposits certified geometric configurations. Theorem (proven). Let m = 3L with L ≥ 4 and let ρ denote rotation by −3 acting on the diagonals of a convex m-gon. A triangulation of the m-gon contains at most two full ρ-orbits, and contains exactly two if and only if it is the union of an “ear” orbit {(v, v+2) : v ≡ re (mod 3)}, a compatible “inner” orbit (the boundary of the L-gon on the vertices ≡ ri (mod 3), with ri ∈ {re, re−1}), and an arbitrary triangulation of that inner L-gon. There are exactly 6·CL−2 such triangulations, Ck the k-th Catalan number: 12, 30, 84, 252 for L = 4, 5, 6, 7. The proof is elementary and uniform in L, and requires no enumeration. Conjecture (not proven). These triangulations arise in Alkauskas's triangulation-shift model, where each triangulation of the (n+1)-gon carries a weight ν bounded by an explicit U(n). For n ≡ 2 (mod 3), so m = n+1 = 3L, we conjecture that the triangulations classified above are exactly the maximizers of ν (Conjecture 4.1), and verify this computationally at n = 11, 14, 17 and 20. The theorem is about ρ-orbit containment; the identification with the maximum-weight triangulations is conjectural. The easier direction of the conjecture is reduced to two local counting lemmas, with the closed form ν = 35L − 25 = U(n) closing exactly; the harder direction, working from the equality cases of Alkauskas's upper-bound argument, is open. Attribution. The shift model, the bound U(n), its attainment, and the brute-force determination of maximizers for 5 ≤ n ≤ 16 are Alkauskas's; section 1.3 of the report states the boundary precisely. New here are the classification, the count, the enumeration-free generator, and the reduction. Geometric realizability – the relation between r(n) and e(n) – is outside the scope of this record. Contents and reproduction. sparse_structure_theorem.md (the report; cite this), README.md, SUPPLEMENT.md (research archive: an implementation of Alkauskas's extraction map, analyses of certified geometric configurations, compiler forensics, other residue classes – not part of the argument and not fully reproducible from this deposit alone), research_log.md, the scripts shifts.py, sparse_constructor.py, sparse_theorem_verify.py, union_to_shift_core.py, sparse_defect_enumerate.py, sparse_defect_bounded.py, sparse_anatomy.py, make_figure.py, verify_all.py, the figure figure1.svg, the champion sets census11_regenerated.json, census14.json, census17.json, census20.json, the constructed set champions23_constructed.json, and SHA256SUMS.txt. Running python3 verify_all.py executes Tables A–F. These are regression tests of the implementation and of the local lemmas, not evidence for the theorem, which has a deductive proof; section 5.3 of the report tabulates provenance claim by claim, distinguishing what is reproduced from this deposit from what is compared against an archived earlier computation (in particular the n = 20 census of 1.77 × 109 triangulations is compared against, not reproduced). SHA-256 of the complete archive: 5af0a74871bc8ebbfe6f3155a3a02646c98bd2d0b659ce5148b31b5356cfcf15. Notes. Produced with AI assistance (Anthropic Claude, OpenAI ChatGPT) for code, analysis and drafting under the author's direction; all machine claims are regenerated by the included scripts. Exhaustive enumerations were run on the author's hardware. An earlier draft of this record described the classification as a characterization of the maximum-weight triangulations; that identification is Conjecture 4.1 and is not proven. The present text states it as a conjecture throughout. The revision history and the expert review prompting it are archived with the project.