Emergent Lorentzian Spacetime from Informational Geometry — Unifying the QFIM Spatial Metric with Tomita-Takesaki Temporal Flow

Abstract: The Algorithmic Theory of Reality (ATR) derives space and time from independent informational principles: the Quantum Fisher Information Metric (QFIM) generates spatial geometry from observer parameter sensitivity, while the Tomita-Takesaki modular flow generates temporal evolution from the observer's restricted data access. However, combining a positive-definite Riemannian metric (space) with a one-parameter flow (time) does not automatically produce the Lorentzian signature (-,+,+,+) required by relativity. This paper resolves this unification problem. We show that the QFIM generates compact (elliptic) spatial transformations, while the modular flow generates non-compact (hyperbolic) temporal automorphisms. Marrying compact and non-compact generators algebraically demands the pseudo-Riemannian signature (-,+,+,+), which emerges as a structural necessity rather than a postulate. Key Findings & Mechanisms: The Origin of the Minus Sign: The Lorentzian signature is rigorously derived from the algebraic mismatch between the finite-dimensional, compact projective space of the QFIM and the non-compact nature of thermal time. Informational Refraction: The spatial dependence of the modular Hamiltonian generates the complete Christoffel structure, identifying gravitational time dilation and geodesic acceleration simply as informational refraction across a variable clock rate. The Z-Scale (Zα) and Jacobson's Derivation: Following Jacobson's thermodynamic derivation, we recover the Einstein field equations as a strict resource balancing constraint. Local geometric curvature must emerge to prevent the algorithmic data density from breaching the observer's finite Zeno Threshold (Zα). Algorithmic Hawking Radiation: A discrete computational framework natively demonstrates that information crossing the observer's causal horizon triggers a Born Rule state truncation, dissipating Landauer heat at the local KMS temperature. Computational Verification: These theoretical results are explicitly verified via a suite of 2D sparse Hermitian lattice simulations. Utilizing strictly unitary Krylov subspace evolution, the simulator natively reproduces spatial geodesic lensing, gravitational time dilation, and the thermodynamic emergence of black hole horizons without hardcoding any Newtonian forces or Riemannian curvature tensors. The full Python computational verification suite and interactive HTML playgrounds are available at: github.com/srdrymn/atr-zeno-emergent-lorentzian-spacetime