We have presented a formally verified Lean4 proof that gravity emerges from information geometry. The key insights: 1. The spacetime metric g µν is derived f

We have presented a formally verified Lean4 proof that gravity emerges from information geometry. The key insights: 1. The spacetime metric g µν is derived from a coherence field Ψ : M → Cl(3, 1) 2. Einstein’s equations emerge from coherence dynamics 3. No gravitons are required; gravity is effective, not fundamental 4. The golden ratio φ provides natural UV regularization 5. The entire framework is mechanically verified in Lean 4 Gravity is not a fundamental force to be quantized. It is an emergent phenomenon arising from information-geometry backreaction of a coherence field valued in the Clifford algebra Cl(3, 1). The mathematics is governed by a single self-consistency principle: the golden ratio φ 2 = φ + 1. The code is available at: https://github.com/ktynski/ParsimoniousFlow https://github.com/ktynski/ParsimoniousFlow