Time, Cosmology and the Double Torus: Threshold, Krein Collision, Galois Time, and the Three Frame Laws (INDYNA Research Note 29)

Abstract

We collect the time-theoretic results of the flag universe (Notes 24–28). Dynamics. The homogeneous sector is real below a threshold velocity $v_c\in(0.070,0.072)$ that is a property of the inertia, not of the potential (edge, vertex and superposed worlds share it); above it two counter-flowing modes collide in a Krein collision and a spiral is forced. The cosmological sector consists exactly of the $\ell{=}1$ drift and $\ell{=}2$ shear harmonics; along the radial axis the drift changes sign exactly at flatness $r{=}1$ (inside the world stands, outside it wanders) while shear is unstable everywhere, strongest near flatness. The edge home $r^\ast{=}1.353$ is the tension world; at $\rho_g{=}0.689$ the geometry realises the shadow parameter $g{=}+1/\varphi$ with pressure threads: tension outside, nothing at flatness, pressure inside. Algebraic time. On commuting shell pairs each family is a Galois pair $a\pm b\sqrt5$: present $=a+\sigma a$ (rational), time $=a-\sigma a$; true clock $\sqrt{|N|}=\sqrt5$ in 3D, $4$ and $6$ in the 600-cell, $\sqrt5,1,\sqrt{11}$ on the commuting sector of the 120-cell; rational shells carry an antipodal inversion time. Geometry and algebra are bridged, not separated: after Schur elimination of the intermediate nodes the stiffness is exactly a shell operator with geometric weights (geometry $\times$ cut $=$ weights $\times$ algebra). Double torus. The time-periodic slab over the double is $S^3\times S^1$ ($\chi{=}0$ from three layers) and, with a second time, the 5-complex $S^3\times T^2$ ($\chi{=}0$); the 5D Regge action of the resting world is $T_1T_2\,S_3$ to $5\cdot10^{-9}$; the two times are isotropic and additive in the continuum (Euclidean time plane); the role swap visible$\leftrightarrow$shadow (the $\sigma$-cycle) costs a coexistence energy $E_\sigma$ paid entirely at the hinges of the first time, exerting a type-antagonistic force (corners out, midpoints in) fifty times the matter; the matter law holds on the double torus to four digits. Frame laws. (i) No chance: what looks equal but is not (equal angles, distinct relations) is structure. (ii) Breathing exists exactly where the sphere closes at finite radius with non-vanishing slope — in 3D, not in 4D (vertex convention). (iii) Balance: $\sum_\ell \ell\,\partial S/\partial\ell-(D{-}2)S=0$ exactly for every configuration (Euler homogeneity), verified in the $\sigma$-cycle. Finally, $\varphi$ is not selected by any energy minimum because it is the unit of energy: the shadow is the $\varphi$-scaled copy, so every shadow energy is $\varphi^k$ times the visible one ($\varphi$ for potentials, $\varphi^2$ for matter, $\varphi^3$ for inertia); radial action of the shadow ball $=1.618034\times$ the visible. 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.