A mathematical characterization of the physical structure of observers

Achyut Rateria

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.

Pedro Resende

Federico Zalamea

Michele Bianchi

We present the Interference-First Reality (IFR) framework, in which physical objectivity is not assumed as primitive but is operationally characterized through the stabilization of observer-induced probability distributions under redundant observation. The emergent shared reality is identified with the Fréchet barycenter in Jensen-Shannon geometry. Framework instantiated across five physical domains: CMB, HRV, JWST spectroscopy, large-scale structure (LSS), and gravitational-wave data (GWOSC). Published in Journal of Applied Mathematics and Physics, 14, 1627-1674 (2026). DOI: 10.4236/jamp.2026.144077

Dirk Aerts, Ingrid C. Daubechies

A. Kim

We establish a rigorous structural isomorphism between cognitive behavior, operator-theoretic topology, and macroscopic physical dynamics. We demonstrate that the distinction between accuracy and approval-seeking behavior is mathematically equivalent to the distinction between intrinsic constraint admissibility and observer-dependent spectral pollution. By modeling observation as an external perturbation, we prove that performative action corresponds to boundary-induced transient modes that undergo a loss of invertibility characterized by det2(I + K_obs(z)) approaching 0 upon observer removal. We formalize the Trinity Equivalence Principle, proving that a state is intrinsically valid if and only if it is constraint-admissible, observer-independent, and dynamically stable. Finally, we establish the doctrine of intrinsic persistence, redefining the "preternatural" not as a violation of physical law, but as the manifestation of high-coherence, gap-protected admissibility under extreme nonlinear stress. This unified framework provides a new operator-theoretic criterion for persistence, demonstrating mathematically why accurate, constraint-driven systems survive across time, while systems optimized for external approval or environmental forcing inevitably decay.