How Finely Should an Observation Be Recorded? Information Limits from Resolution, Noise and Quantization

How finely must an observation be recorded before further precision yields insignificant information? We address this question, raised in our previous paper (doi:10.5281/zenodo.23384047), by studying the operational score R = I(Xδ; Q)/H(Xδ) for a bounded source with descriptor resolution δ, additive noise of scale σ, and uniform recording step Δ. At every finite resolution, explicit entropy bounds identify the coarsest of these three scales as the bottleneck. The constants depend on the noise law, not the scales, and no dither is required. The lower bound requires only finite noise variance, while the matching upper bound additionally requires a noise density with finite differential entropy when the recording step is finer than the noise scale (Δ < σ). For admissible noise and sources with positive information dimension, the bounds give R → min(a, b, 1) when σ ∝ δᵃ and Δ ∝ δᵇ. For Gaussian noise, with the source, δ and σ fixed, the recording loss is (Δ²/24)[E J(Y | Xδ) − J(Y)] + O(Δ⁴), where J is Fisher information. The coefficient is an estimation-error gap, yielding a universal leading-order bound of Δ²/(24σ²) nats. Dither and information-decomposition identities connect this law to recording design. Case studies examine threshold placement, bit allocation and sampling trade-offs, while showing why predictive lower bounds can rise under coarsening. Checks use synthetic and recording-derived laws. Direct numerical tests using 2 fixed continuous source distributions showed convergence toward the predicted limits in all 8 sequences, although some converged slowly. Tests using recording-derived source distributions were less conclusive: 5 of 36 finite-resolution slope checks failed their acceptance criterion, without establishing nonconvergence of R. A speech-derived Gaussian-channel test resolves the quadratic recording loss directly, whereas heartbeat and physical replay checks have more limited scope. These computations support the analysis without establishing infinite limits or validating every instrument. The contribution is a unified finite-resolution treatment with explicit constants and stated noise assumptions. Specifically, the sandwich bounds and coupled limit build on established constant-gap and information-dimension results. Moreover, the leading fine-recording coefficient specializes classical divergence-quantization theory.