Information-Driven Gravity: Lorentzian Signature Emergence from Statistical Geometry

We establish that Lorentzian spacetime signature (−,+,+,+) emerges from the positive-definite Fisher-Rao statistical geometry of Information-Driven Gravity (IDG) without Wick rotation or analytic continuation. The central result is the Bridge Lemma: the ghost-freedom condition det(Φ) > 0 (Theorem 4.1, Paper VIII), combined with the Chentsov uniqueness theorem and the ADM decomposition of the Fisher tensor, forces the physical metric on the propagating degrees of freedom to carry exactly one negative eigenvalue. The sign flip is a direct consequence of the weak energy condition on matter sources, the negative sourcing of the trace mode, and the Schur complement constraint eliminating Φ₀₀ as an independent degree of freedom. This resolves the foundational question raised by the θμ → xμ identification: the statistical parameter space and physical spacetime carry the same unique geometric object, and Lorentzian signature is not imposed but derived.