INDYNA-Statik: The Unified Field Formula of the Golden Universe — A Closed Account of the Two-Shell Operator, the Flag Double, Derived Dynamics, Algebraic Time, the Metallic Roofs, the Observer, the Derivation of the Golden Ratio, and the Vertex Action, with Predictions and Open Problems

INDYNA-Statik is a single formula and a single geometry. The formula is the two-shell operator M(g)=A1+gA2 over ℚ(√5), where A1,A2 are the two angular shells of the icosahedral vertex set (neighbour cosines ±1/√5) and g runs over the golden ladder of units φ2k. The geometry is the flag double: the barycentric refinement of the icosahedron (12 vertices, 30 edge midpoints, 20 face centres, 120 congruent right flags) carried by two balls over one shell, a visible ball and a shadow ball scaled by φ. From these two objects the register derives, in order: a static home r*=1.353; a dynamics obtained from the statics by a staircase theorem, with a matter law f=φ2δ; a forced cosmology with a threshold at vc∈(0.07,0.08) and a Krein collision; an algebraic time (present = sum, time = difference of the Galois shells; true clocks √5 in three dimensions, 4 and 6 in four); the metallic roofs, in which only gold commutes and the silver, bronze and cubic guests are repetitions of the rational; an observer that must be gyroscopic, is the oriented σ-cycle, locks as a cube of two counter-rotating tetrahedra, and can be derived as the geometric magnetism of the locked inversion; the golden ratio itself, derived from two counter-twisted tetrahedra under a space-shortage constraint in three dimensions (cos t*=φ2/√8) and four (cos a*=φ√(3/8)); and the explicit vertex action, by which the 4π-rule is Regge calculus of the space-shadowed double, 4π being two balls times 2π, with the Einstein equation δ=f/φ2 whose coupling φ2=1+φ is visible plus shadow. The whole is one conservative Regge action with two shadows and two interface bookkeepings, homogeneous of degree one (the balance law), free of singularities below the space-shortage radius 0.32082, and anchored to physics by one length: the E8 line sequence of CoNb2O6, of which six lines are confirmed and m7≈1.95 meV, m8≈2.39 meV are predicted, together with a non-E8 line of the observer at 1.82 meV. Every statement carries a status; fallen candidates are kept and named. Capstone manuscript of the INDYNA-STATIK programme; companion to INDYNA Notes 16–34 and the law register (master data record, doi:10.5281/zenodo.21891088). Licence CC-BY-4.0.