The Vertex Action: The 4π-Rule as Regge Calculus of the Space-Shadowed Double, the Einstein Equation δ = f/φ², and Three Matters in the Ratio of Cone Numbers (INDYNA Research Note 31)

Abstract

The unified formula of the flag universe (Notes 24–30) superposes two worlds over one shell: the edge world (Regge action with Nambu threads) and the vertex world (the $4\pi$-rule: $4\pi$ minus the sum of radial dihedral angles at each shell node). We settle the structure of the vertex world. Definition. Its shadow ball is space-shadowed: base lengths scaled by $\varphi$, radial lengths unscaled — the shadow with compressed time (time/space $=1/\varphi$ of the visible); the edge world's shadow scales both. With a fully scaled shadow the vertex force would be $2\delta_{\rm rad}=(2/\varphi^2)f$ and the vertex home flat; the curved home $r^\ast=1.353$ exists only for the space-shadowed double, whose equilibrium is a midpoint rule: $\theta_v(r^\ast)+\theta_v(r^\ast/\varphi)=2\theta_v(1)$ for every node type, i.e. the home is the mirror point between the visible world and its $\varphi$-compressed copy about flatness. Conservativity and explicit action. The raw Jacobian of the $4\pi$-force is symmetric to $10^{-10}$; by Schläfli's identity the explicit action is $V_B=4\pi\sum_v r_v-\sum_{\rm cones}\sum_{e}\ell_e\theta_e$ over the cones of both balls with each ball's own edge lengths, whose gradient reproduces the $4\pi$-force to $10^{-8}$: the $4\pi$-rule is Regge calculus of the space-shadowed double with per-ball interface hinges, and $4\pi$ is two balls times $2\pi$. The unified formula is one Regge calculus with two shadows and two interface bookkeepings, hence a Lagrangian system. Gravity. Schläfli gives the edge force $(1+\varphi)\delta_{\rm rad}=\varphi^2\delta_{\rm rad}$, so the matter law of Note 25 is the Einstein equation $\delta_{\rm rad}=f/\varphi^2$ with coupling $\varphi^2=1+\varphi$ (visible plus shadow); Bianchi is Schläfli; the balance $\sum_\ell\ell\,\partial S/\partial\ell=(D-2)S$ holds exactly; all flags are congruent with circumradius $0.32082$, below which no state exists (no singularity). Three matters. At the exact home $f_E:f_M:f_F=1:0.71787:0.60000$, and $f_F/f_E=3/5$ to $5\cdot10^{-7}$: at equilibrium the radial dihedrals at vertices and face centres tilt by the same angle ($0.73926^\circ$), so the matter of face centres to that of vertices equals the cone-number ratio $6/10$ — and this holds in every Platonic flag universe (octahedron $3/4$, tetrahedron $1$, cube $4/3$), the golden house curving least ($r^\ast=1.353<1.363<1.381$). The mid-edge ratio ($0.71787$, dually symmetric across the Platonic pairs) has no closed form yet. The potential landscape along the radial axis has the home as global minimum, a soft rise outward, a wall of $+277$ at the space-shortage radius. 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.