Observer Categories, Inverse Limits, and the Reconstruction of Objective Reality

This paper develops a mathematical framework in which finite observation, rather than spacetime or physical fields, serves as the primitive structural element. We model observers as finite persistent systems possessing bounded discrimination, hysteresis, and dwell-time scales. Observers are organized into a directed refinement category whose objects generate finite observational partitions. Reality is then defined as the inverse limit of all compatible observer partitions. We prove the existence of a non-empty inverse-limit reality space, construct a unique projective limit measure on that space, and define observational gauge equivalence classes of histories. Objective physical quantities are characterized as gauge-invariant functionals on validated trajectories. The resulting framework establishes a mathematical foundation for reconstructing objective reality from compatibility among finite observers.