Freedom and hidden order in six-colourings of the triangular tiling
Aidan Quigley · Zenodo (CERN European Organization for Nuclear Research) · 2026
We study six-colourings of the triangular tiling in which every vertex sees all six colours. Five independent switch families give an entropy lower bound of 5 log(2)/288 per triangular face. Explicit quadratic controls are nonperiodic along six lattice line families, while retaining periodic phase fields. Exact finite results connect cyclic inventories, one- and two-ring seed censuses, annulus transfer and certified extension traps. A frozen periodic control has bounded phase domains, and ribbon and finite-interval circle obstructions restrict several routes to stronger phase disorder. The technical supplement includes additional written results on empirical component-tail measures, finite M3 guards, exact sealed-mass accounting, trimer congruence and necessary frontier-radius bounds. The infinite M3 existence problem remains open. Formal verification disclosure: Lean 4.24.0 verifies ten of the original eleven numbered results completely. Theorem 3 is partially verified: the one-ring count, joining-test partition, nonextension of all failed tests, geometric weighted-trace correspondence and completions for 50,720 supplied representative seeds are checked. The numerical weighted two-ring total and the final proof of exhaustive coverage of all 599,799 survivors remain unfinished. The full printed census retains its historical exact computer-assisted evidence. The additional written results in Section 8.1 and Supplement S.13 are not Lean verified. Verification does not establish novelty or priority. The deposit contains the main article, technical supplement and Lean proof archive. The archive contains only Lean proof sources and three supporting files: a short README, a proof index with source hashes and pinned toolchain versions, and a script for checking the proofs. Personal notes, correspondence, research drafts, data bundles and manuscript sources are excluded. AI-use disclosure: OpenAI's GPT-6 Astra (gpt-6-astra), used through Codex and referred to as Astra, assisted with mathematical exploration, proof drafts, programming, Lean formalisation, manuscript preparation and review. Anthropic's Claude Fable 5.1 (claude-fable-5-1), referred to as Fable, provided additional mathematical and manuscript review. The author is responsible for the final manuscript. AI review is not independent human peer review. Exploratory web companion: Freedom & Hidden Order, version 1.1.1, freedom-and-hidden-order.vercel.app. The app provides interactive raw-colour and phase views of selected examples and a separately labelled finite R100 preview of forthcoming work. The preview is outside the article's reported examples and does not establish M3 or infinite extendability.