The Flag Universe: A Closed Regge Double over the Icosahedral Flag Complex, its String Matter, and the Conjugation Tower (INDYNA Research Note 24)

Abstract

The common refinement of the icosahedral edge shell and its Galois shadow (the great-icosahedron chords) on the sphere is computed exactly: each visible edge crosses exactly one shadow edge, the shadow edges cross each other twenty times, and the refined sphere is the barycentric subdivision of the icosahedron — $V=62$ ($12+30+20$: vertices, edge midpoints, face points), $E=180$, $F=120$ triangles, one triangle per group element of $H_3$. The double of two cones over this sphere is a closed Regge 3-manifold of 240 tetrahedra (the $E_8$ number) which is an exact vacuum solution at unit radii: both balls are flat and the entire curvature is concentrated on the interface, with three exact class deficits $(0.702532,\,0.067165,\,1.089679)$, all positive. Read as $3{+}1$ gravity, the deficits are cosmic strings: the double is an exact static string universe, its matter a string network on the flag complex (model A, one metric). In the bimetric reading (model B, shadow edges $\varphi$-scaled, shared radials) the matter moves into the volumes as pairs of opposite string tension $\pm\mu$ with exact nodal cancellation: empty and static from outside, two oppositely filled worlds inside — the balance principle as a matter statement. Compression transmits through the interface with a saturating counter-pull (linear response $-1.9$ at the vertices: the shadow expands where the visible is squeezed), and no bottleneck occurs down to $20\%$. Behind the construction stands the conjugation tower: the stiffness blocks of all cones are exact golden expressions ($2\sqrt5\,\varphi^{-3}/a$, $6\varphi^{-3}/a$, $-4\varphi^{-5}$, $2\sqrt5\,\varphi^{-6}$, $1/10$), the shadow block is the Galois conjugate of the visible one, and the tower exponents $\varphi^{4},\varphi^{5},\varphi^{10},\varphi^{12}$ are pure conjugation — a parameter-free hierarchy mechanism. An observer tower coupled through balanced channels is exactly self-similar: each level carries its golden ladder untouched (machine precision). Honest boundaries are recorded: the rectified 600-cell leaves the field $\mathbb{Q}(\sqrt5)$ (minimal polynomial $x^4-6x^3-31x^2+60x+100$), delimiting the $\sigma$-order conjecture. 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.