Geometry from Information: Uniqueness of Relative Entropy as the Gravitational Potential and Canonical Realisation of the Informational Index Principle.

"This work establishes a unique axiomatic derivation of Einstein's field equations and the Lovelock gravity family from the principles of quantum information theory. By proving the uniqueness of relative entropy as a gravitational potential, we derive the Bogoliubov-Kubo-Mori (BKM) metric on the space of metrics. The central result provides the first canonical realization of the Informational Index Principle: identifying the Breuer-Fredholm index of an affine Dirac operator in the Type II$_{\infty}$ crossed-product factor as the Euler characteristic \chi(M)=2. This resolves a fundamental open problem in the operator-algebraic foundation of de Sitter space by uniquely determining the abstract spectral triple through modular data."