On the Origin of General Relativity - A projection-based derivation from information geometry

We develop a projection-based framework in which spacetime geometry and gravitational dynamics arise as effective, stable features of an information-geometric description associated with a field Φ governed by a generative equation G[Φ] = 0 on a pre-representational domain Mprim. Local statistical families ρθ [Φ] induce an information metric that defines an emergent Lorentzian spacetime geometry, while a projected variational principle yields Einstein-type field equations with an effective cosmological constant. The purpose of the present work is not to introduce new gravitational dynamics but to clarify the structural conditions under which the familiar equations of general relativity arise as a stable and generic effective description. Under general and physically natural assumptions—locality, diffeomorphism invariance, regularity and boundedness of projected derivatives—we show that the resulting geometric structure is robust under variations of the underlying statistical description. In particular, (i) curvature invariants remain finite within the representational domain of validity as long as the statistical families remain regular, (ii) Einsteinian dynamics arise as the unique local, covariant effective equations at second order in derivatives and (iii) the effective cosmological constant Λeff decomposes into geometric and statistical contributions. The statistical component originates from normalization and information-geometric properties of the ensemble and is insensitive to vacuum fluctuations of projected quantum fields. A global arrow of time emerges from spectral asymmetry of the projected generator K+, reflecting the irreversibility inherent in projecting from the generative level to a coarse- grained macroscopic description. Together, these results suggest that the characteristic structures of general relativity are not fine-tuned but arise generically from a broad class of projection-based representations.