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

We collect the time-theoretic results of the flag universe (Notes 24–28). Dynamics. The homogeneous sector is real below a threshold velocity vc∈(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 ℓ=1 drift and ℓ=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*=1.353 is the tension world; at ρg=0.689 the geometry realises the shadow parameter g=+1/φ with pressure threads: tension outside, nothing at flatness, pressure inside. Algebraic time. On commuting shell pairs each family is a Galois pair a± b√5: present =a+σ a (rational), time =a-σ a; true clock √(|N|)=√5 in 3D, 4 and 6 in the 600-cell, √5,1,√(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 × cut = weights × algebra). Double torus. The time-periodic slab over the double is S3× S1 (χ=0 from three layers) and, with a second time, the 5-complex S3× T2 (χ=0); the 5D Regge action of the resting world is T1T2 S3 to 5·10-9; the two times are isotropic and additive in the continuum (Euclidean time plane); the role swap visible↔shadow (the σ-cycle) costs a coexistence energy E_σ 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: Σ_ℓ ℓ ∂ S/∂ℓ-(D-2)S=0 exactly for every configuration (Euler homogeneity), verified in the σ-cycle. Finally, φ is not selected by any energy minimum because it is the unit of energy: the shadow is the φ-scaled copy, so every shadow energy is φk times the visible one (φ for potentials, φ2 for matter, φ3 for inertia); radial action of the shadow ball =1.618034× 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.