Operational Reality across Scales of Resolution and Noise

Abstract

In information theory, how much an observation reveals about a specified physical event in a system such as a particle, a dynamical system or a sensor network can be measured by Theil's uncertainty coefficient (Theil's U), a known statistical quantity. This fraction is called the operational degree of reality. It is interpreted operationally and its dependence on resolution, noise and the event characterisitics are analyzed. These can include, position, temperature or whether a threshold is crossed, for example. Formally, it is the fraction of the information in a specified event descriptor that an observation makes accessible, at a finite resolution and relative to an observer. An observer is any instrument or subsystem whose response can be modelled as a channel. The measure is useful wherever instruments, simplified descriptions or data compression limit what can be known. For a finite-valued descriptor Xδ at resolution δ and observer output Yₒ, the score is Rδ(E;O) = I(Xδ;Yₒ) / H(Xδ). The score concerns access to information about an event, not whether the event exists: a zero score means statistical independence, not nonexistence. If a second observation is a stochastic degradation of the first, the score cannot increase, Rδ(E;O₂) ≤ Rδ(E;O₁), and the loss is exactly I(Xδ;Yₒ₁ | Yₒ₂) / H(Xδ). A finer description need not yield a larger accessible fraction, and an exact decomposition states when it does. For Gaussian noise, an exact residual identity bounds the information remaining within each descriptive cell. For bounded sources with positive information dimension, including singular measures such as the Cantor distribution, refining the resolution while the noise falls as σ = cδᵃ gives R → min(a, 1). A cycle-consistency example shows why the choice of feature matters. Two observers with equal total scores and equal singular spectra differ completely in detecting a global inconsistency, and scoring the feature that defines the task separates them exactly. Exact integrals, reproducible calculations and a validated estimator support the results. The measure and underlying principles remain classical.