Signatures of Relational Scaling: Consistent R∝αC^2 Patterns Across Bounded Systems

Closure, Reset, and the thermodynamic gap between Landauer and observed residue.In this framework, “closure” denotes the moment a system must compress or discard relational distinctions in order to return to a reusable state (a cyclic reset). Landauer’s bound kT\ln 2 provides a per-bit lower limit for erasure, but it does not specify the structure of what is erased. The empirical residue R observed across substrates is interpreted as the additional energetic cost of erasing correlation structure—the bookkeeping required to dissolve a high-complexity relational state back into a low-complexity, reusable one. In other words, R is not parameterized by raw state count alone, but by the density of pairwise distinctions that must be collapsed to complete the reset and restore cyclic reuse.