The Metallic Roof: tan² Rung Law, Only Gold Dwells Finitely, and the Tower-in-the-Roof Theorem (INDYNA Research Note 26)

Rung law. In the 2m-gon ring with the conjugate shell pair (± 2cos(π/m)), the family crossing with the alternating family sits exactly at g*=tan2(π/2m): silver (1+√2)-2 for m=4, twelvefold (2+√3)-2 for m=6 — the metallic units are the tangents of the half-corner. Only gold dwells. The crystallographic restriction grants finite non-crystallographic reflection groups only to fivefoldness (H2,H3,H4): silver and the twelvefold possess the ladder arithmetic (Pisot) but no finite house; on Ammann–Beenker approximants the near-rung of the 7/5 cell dissolves at 17/12 (1417-vertex torus, no kink in the covering window) — the silver ground ladder does not exist even asymptotically. Tower in the roof. Odd powers of every metal are themselves metallic means (n = odd Lucas 1,4,11,29,… giving φ,φ3,φ5,…; Pell–Lucas 2,14,82 giving (1+√2)1,3,5), proof: N(μ)=-1⇒ μodd-μ-odd∈ℤ. Even powers form the sister family x2=nx-1 (n = even Lucas; 2+√3=((1+√3)/√2)2 included). In particular the fourth mean — mercury — equals φ3 exactly: mercury lives in gold's house (the chemistry of amalgam, echoed by arithmetic) and conducts exactly gold's frequencies in steps of three. Licence CC-BY-4.0. Continues: Note 25. Supplement to: INDYNA Law Register (L149–L154, L180). Part of the INDYNA Notes series (Notes 16–34). All numbers reproduce from the INDYNA law register (master data record, doi:10.5281/zenodo.21891088). Licence CC-BY-4.0.