Systemic Tau and Hierarchical Ordinal Conjunctions: A Relational Theory of Critical Transitions

Version 1.1.2 adds an erratum. The window-13 excess3 figures in v1.1.1 used nested-recd ≤ 0.2.2, which pooled symbol values instead of counting joint tuples. With nested-recd 0.2.3 the six-seed means are 2.210 at r=3.30 and 2.802 at r=3.85, and the 32-point grid maximum is at r=3.95 (2.897). highL3 above 1.75 is saturated. Res_pair is unchanged. The v1.1.1 PDF is still in this record and is the file under review at AIP Advances; it is not rewritten here. Erratum: 10.5281/zenodo.22970130. Software: 10.5281/zenodo.22970079. Foundations paper (English, ~30 pp.) for a dual relational account of critical transitions in complex systems. Systemic Tau (τs) is an ordinal, multivariate measure of cross-variable coupling: sliding-window Kendall-type rank concordance tracks relational coherence rather than the amplitude of any single observable. Hierarchical ordinal conjunctions are operationalized through the Discrete Extramental Clock (RECD): three levels of structural claim—coincidence (Φ1), persistent relational structure (Φ2), and the irreducible residual beyond pairwise models (Respair)—plus the scalable hybrid proxy excess3 (not identical to Respair). Weighted advances use admissible aggregation weights αℓ(λ); aggregation share is distinct from abundance. Early warning is cast as a change in hierarchical ordinal structure, not merely a rise in fluctuation size. Claims, estimators, and aggregation policy are kept on separate planes. Includes synthetic validation on coupled logistic maps and a working hypothesis for the excess3 peak near the Feigenbaum accumulation point. v1.1.1 micro-release: title-page footnote with version DOI, concept DOI, and source repository. Concept DOI remains stable across versions. Contents: PDF + TeX source + bibliography; key figures.