Information Access across Scales of Resolution and Noise

How much an observation reveals about a specified physical event can be measured, in information theory, by Theil’s uncertainty coefficient (Theil’s U), a known statistical quantity. This fraction is called the operational degree of reality. 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. The event may concern a particle, a dynamical system or a sensor network, and its descriptor may be a position, a temperature or whether a threshold is crossed. An observer is any instrument or subsystem whose response can be modelled as a channel. For a finite-valued descriptor Xδ at resolution δ and observer output YO, the score is Rδ(E; O) = I(Xδ; YO) / H(Xδ). A zero score means statistical independence, not nonexistence. If a second observation is a stochastic degradation of the first, the score cannot increase, and the loss is exactly I(Xδ; YO1 ∣ YO2) / H(Xδ). A finer description need not raise the score; an exact decomposition states when it does. For Gaussian noise, an exact residual identity bounds the information left inside each descriptive cell. For bounded sources with positive information dimension, refining the resolution while the noise falls as σ = c δa gives R → min(a, 1). A cycle-consistency example shows that observers with equal total scores can differ completely on a specific task. In 61,617 cardiologist-labelled heartbeats from 25 records, the waveform gives an estimated predictive lower bound of about 0.87 on the within-record score. This bound changes little until the amplitude recording step is 32 times coarser than the recorder’s. Tests on air-quality and smartphone sensor data are reported in full, including the predictions that failed. The underlying principles are classical.