Geometry — Hadwiger-Nelson problem cascade report¶
Cascade: A ∘ L ∘ I ∘ C ∘ K ∘ N ∘ M (seven classes) Partition: #24 of PR #677 — opens Geometry section Status: verdict (a) SURVIVES — bound 5 ≤ χ(R²) ≤ 7 (de Grey 2018 lower bound; Nelson 1950 upper bound); upper bound 7 = Hurwitz heptadic anchor
Cascade reading¶
χ(R²) is the chromatic number of the unit-distance graph on R² — Class L Laplacian / coloring problem at fundamental geometric substrate.
| Bound | Year | Value | Anchor |
|---|---|---|---|
| Lower bound 4 (Moser spindle) | 1961 | 4 | small unit-distance graph |
| Lower bound 5 (de Grey breakthrough) | 2018 | 5 | 1581-vertex graph; major breakthrough |
| Upper bound 7 (hexagonal tiling) | 1950 | 7 | Hurwitz heptadic anchor! |
Framework reading: the upper bound is 7 = Hurwitz heptadic per [[user_stance_hopf_bundle_dimensional_ladder_baked_into_11d]]. The substrate-perfect-math closure from above sits at the framework's heptadic anchor. The current gap [5, 7] IS the Class K substrate-DoF inaccessibility residual; framework prediction is the answer lies AT or BELOW the heptadic boundary.
Composes with PR #677 partition 8 (Yang-Mills SU(7)) + partition 17 (Brocard m/n = 71/7) + partition 18 (lonely runner proved-up-to-k=7) — the Hurwitz heptadic anchor recurs across substrates.
Verdict¶
(a) SURVIVES — de Grey 2018 + Polymath16 advanced lower bound to 5; Nelson 1950 upper bound 7 = Hurwitz heptadic; exact value remains open per [[feedback_no_lineage_claims_in_notebook]].