Cancellation-aware order coherence: an exact decomposition of absolute concordance change into signed Systemic Tau and a cancellation index

Systemic Tau (τs), the mean pairwise Kendall concordance of module activity on a common event index, is a signed average: concordance gained on some pairs can offset concordance lost on others, so structural change can pass unseen. A cancellation-immune alarm is the mean absolute change D of pairwise concordance from its calibration baseline over a fixed edge set. Here we show that both statistics are instances of one averaging operator on the pairwise concordance matrix and that, on a common edge set, D decomposes exactly into the absolute change of the signed mean plus a non-negative cancellation term equal to twice the minority-sign mass of edge changes. The resulting index κ∈[0,1] is the fraction of total change that the signed mean cannot see. We derive bounds, invariances and the no-change behaviour of D and κ, neither of which is near zero without change, and give a conformal alarm rule with a finite-sample false-alarm guarantee. In synthetic event streams, balanced rewiring left τs at its false-alarm level while D detected it in 95.0% of windows. Files: the note (PDF), and a code archive with the Python package, unit tests, the script that regenerates every number, Table 1 and Fig. 1, and the resulting CSV/JSON outputs. All data are synthetic. The note text and figures are CC BY 4.0. The code in the archive is MIT. Companion to Padilla-Villanueva, J. Systemic Tau as a homeostatic trigger for structural self-repair in event-driven networks, preprint https://doi.org/10.5281/zenodo.23001549 (code https://doi.org/10.5281/zenodo.23001551). This note does not replace that article. Repository: https://github.com/johelpadilla/cancellation-aware-tau