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)

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 4π-rule: 4π 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 φ, radial lengths unscaled — the shadow with compressed time (time/space =1/φ of the visible); the edge world's shadow scales both. With a fully scaled shadow the vertex force would be 2δrad=(2/φ2)f and the vertex home flat; the curved home r*=1.353 exists only for the space-shadowed double, whose equilibrium is a midpoint rule: θv(r*)+θv(r*/φ)=2θv(1) for every node type, i.e. the home is the mirror point between the visible world and its φ-compressed copy about flatness. Conservativity and explicit action. The raw Jacobian of the 4π-force is symmetric to 10-10; by Schläfli's identity the explicit action is VB=4πΣv rv-ΣconesΣeℓeθe over the cones of both balls with each ball's own edge lengths, whose gradient reproduces the 4π-force to 10-8: the 4π-rule is Regge calculus of the space-shadowed double with per-ball interface hinges, and 4π is two balls times 2π. 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+φ)δrad=φ2δrad, so the matter law of Note 25 is the Einstein equation δrad=f/φ2 with coupling φ2=1+φ (visible plus shadow); Bianchi is Schläfli; the balance Σ_ℓℓ ∂ S/∂ℓ=(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 fE:fM:fF=1:0.71787:0.60000, and fF/fE=3/5 to 5·10-7: at equilibrium the radial dihedrals at vertices and face centres tilt by the same angle (0.73926^°), 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*=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.