The Operator Algebra Scanner: Grounding Recursion, Scale, Flow, and Entropy in Measured Telemetry

Four telemetry-derived quantities — self-call ratio (r), normalized concurrency (s), external-admission ratio (f), and state-hash entropy (e) — are computed per time-step from a running system and compared tick-to-tick. Each transition's sign pattern (+, 0, or −, per axis) is matched against the nine fixed-sign generators from the Operator Algebra's Table 2.1. A match is reported as a named, human-readable diagnosis (e.g., "Hydra → BOTTLENECK: scale expanding while flow is constrained"). Tested against four synthetic telemetry traces engineered to represent a sealed subsystem, a growing bottleneck, a self-reference collapse, and a stable baseline, the scanner correctly identified the intended pattern in three cases and, in the fourth (the stable baseline), initially threw a false positive traced to an entropy-window measurement artifact, unrelated to the algebra itself. That artifact was found, diagnosed, and fixed in two rounds; after the second fix, all four traces — including the baseline — matched the intended prediction exactly, with zero false positives. All code, the exact bug, both fixes, and the exact reproducible run output are included below.