Scale Partitioning by Incremental Nested Entropy: A Measure-Oriented Theory of Multiscale Structure

Abstract

Across complex systems science, networks, spatial structures, populations, spectra, and dynamical flows are often represented by object-level measures such as connectivity, frequency, geometric isolation, spectral strength, or deformation. Determining whether these values contain distinct scales commonly requires a chosen cutoff, a prescribed number of groups, or an assumed prevalence. We introduce Scale Partitioning by Incremental Nested Entropy (SPINE), a deterministic framework that identifies scale structure without these inputs. The central theory reduces an arbitrary positive prefix of ordered measure values to two entropy-effective descriptors: an effective number of contributing components and a characteristic measure scale. This reduction yields an exact finite-size criterion for when the next value produces an entropy-supported scale transition. As the effective size grows, the critical ratio converges to \(e\), which emerges from the balance between increasing diversity and increasing dominance rather than from threshold calibration. The same critical relation also determines the exact aligned perturbation margin of a detected boundary. Successive local transitions produce a data-supported number of scale strata, while a measure with no resolvable separation returns a single stratum. Numerical experiments verify the critical and stability relations to numerical precision across heterogeneous prefixes and recover two, three, and four generated scales without being supplied their number once separation is sufficient. In a double-gyre flow, SPINE identifies a high-expansion structure occupying \(12.85\%\) of the domain without prescribing a retained percentile. The framework applies to geometric, categorical, network, spectral, and dynamical measures.