The Footballhedron: Information-Geometric Origin of Spacetime, Gravity, and Gauge Structure

We introduce an information-geometric framework in which spacetime, gravity, and gauge kinematics naturally emerge from quantum correlations. Central to this approach is the "footballhedron," a convex geometric structure encoding the global state's correlation patterns. Observers perceive spacetime as lower-dimensional projections of this object, resulting in an emergent information-theoretic metric consistent with classical geometry. Rotations of the footballhedron correspond to Lorentz transformations, while extremizing Fisher information yields Einstein's equations and an emergent correlation-defined stress-energy tensor. A finite Coxeter tessellation of the footballhedron provides an intrinsic ultraviolet cutoff at the Planck scale and naturally generates internal gauge symmetries through its automorphisms. We propose experimental tests using spin entanglement and optomechanical setups. Explicit holographic examples, including cosmological scenarios and tensor-network models, illustrate how this unified approach coherently integrates quantum correlations, gravitational dynamics, and gauge structure in an observer-dependent geometry.