Finite-Horizon Emergence of Curved Lorentzian Readout Geometry in the Balance-Field Equation

Abstract

This 20-page preprint establishes a constructive finite-horizon geometry theorem for a four-mode, full-persistence sector of the Balance-Field Equation (BFG). An invariant family reduces the complete operator update to a rational eigenvalue recursion with endogenous weights. Four spectral moments give local coordinates on the simple-spectrum quotient; their full-rank Euclidean pullback is necessarily locally flat. An explicitly declared update-dependent Lorentzian completion admits nonzero curvature certified by exact rational arithmetic. For every prescribed finite horizon, one nonempty open class simultaneously supports spectral regularity, nonzero Riemann curvature at every sampled step, a common temporal function and endpoint-exact future timelike interpolation. Complete proofs, assumptions, countercases, four numbered 3D simulation figures, ten external scholarly references and the full executable curvature certificate are included. Simulation source and numerical results are embedded in this single PDF. The theorem concerns conditional readout geometry; physical dimension, a unique Lorentz rule, a calibrated physical clock and gravitational field equations are not derived from the finite kernel. AI-assisted drafting and computational checking are disclosed in the manuscript.