Time as the Irrational Part (INDYNA Research Note 19)

Abstract

Clock theorem. In the $\mathbb{Q}(\sqrt5)$ dynamics of INDYNA every spectral datum splits as $a+b\sqrt5$; the rational part is geometry, the $\sqrt5$-part is the clock: invisible to lengths, audible only as frequency. Pisot clock. For unit-valued squared frequencies the shadow frequency is the falling Galois conjugate and the clock product is the fixed norm, $\omega\,\omega'=\sqrt{|N|}$ (verified exactly on the cone modes $2\varphi^2\mapsto2$ and $6\varphi/\sqrt5\mapsto 6/\sqrt5$). Time is the falling Galois direction: the same mechanism that sharpens quasicrystal diffraction (Pisot property of the metallic means) renders the clock invisible to space. Later strengthened dynamically: unheld, this clock spirals; held by three observers, it ticks (Note 25). Licence CC-BY-4.0. Continues: Note 18. Supplement to: INDYNA Law Register (L84, L153a). 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.