Structropy – Toward a Metric of Organization (github.com)

Authors present "Structropy," an exploratory proposal for an Organization Index (OI_H) as a formal complement to entropy: instead of measuring randomness, Structropy quantifies how efficiently elements can be located given a query distribution. Operationalized around Shannon entropy H(P), the index rates systems by expected retrieval cost — linear scan ≈ 0 (near perfect disorder), optimal comparison-based search (binary search / entropy bound) ≈ 1, while auxiliary structures (hashes, direct addressing) can exceed 1 unless a capped normalization maps them back into [0,1]. The draft positions OI_H as a practical, entropy-aware analogue to IR step-discounting metrics (NDCG/MRR), emphasizing monotonicity (fewer expected steps → higher OI) and smooth sensitivity (∆OI = O(1/n)). Technically, OI_H ties expected search depth to H(P) and distinguishes comparison-only limits (OI ≤ 1) from “super-organized” systems that exploit extra structure (OI scaling up to O(log n)). It borrows ideas from optimal search trees and Huffman-like code length bounds, accounts for skewed query distributions, multi-level caches and maintenance costs, and is explicitly an index (not a metric). Example: a 52-card scenario yields low OI for an unsorted pile (~0.21), OI ≈ 1 for binary-searchable order, and much higher values for hashed indexing unless capped. The notes sketch a framework useful to IR, databases, cognitive models and evolutionary biology for quantifying organization and access efficiency as a counterpart to information-theoretic measures of disorder.