Information-Theoretic Selection of Lorentz Signature

Relativistic kinematics is usually assumed in information-based approaches to gravity. We show that, already in the minimal nontrivial case d = 4, Jaynes' maximum-entropy inference can instead select the Lorentzian real form from a finite-dimensional density operator, without assuming a background manifold, metric, or causal order. In the intrinsic reference basis that diagonalizes the real symmetric part of the state, the density operator splits into ordered intrinsic populations and a real antisymmetric coherence bivector. If the Jaynes constraint is isotropic in the bivector sector - depending only on the unique positive-definite SO(4)-invariant quadratic norm - then the sign of the Jaynes multiplier selects the real form: beta > 0 yields the compact Euclidean branch, whereas the population-inverted branch beta < 0 (bounded spectra) forces imaginary boosts and yields the Lorentz algebra so(1,3). The same minimal isotropy singles out three-plus-one as the minimal dimension in which rotations and boosts can be paired without additional tensorial structure (Supplemental Material). Lorentz signature thus appears as a pre-geometric output of maximum-entropy inference.