The Two Anchors of a Measure: The Input and Output Grains Common Analytics Leaves Uncaptured

A large and common class of analytical measures — averages, rates, ratios, indices, distinct counts, even minimums and maximums — is governed not by one grain but by two: the familiar output anchor (the grain a result is reported at) and a usually-unspoken input anchor (the grain the measure was reduced over). The input anchor is substantive — it is why "average order value" can yield several different, individually correct numbers over the same small dataset — yet current data practice has no systematic way to capture it, reason about it, or use it when a measure is served. This paper makes the two anchors precise. It gives a criterion (path-independence) for exactly when the input anchor is "live"; shows that the average and the weighted average hide it for two structurally distinct reasons, unified under a single commutation law; and separates path-independence from a second property — whether a stored value is interpretable without its input grain — into a four-cell taxonomy with a single lossless cell. A consequence is that operators which pass every additivity and summarizability check, including MIN and MAX, can remain uninterpretable once materialized. The paper observes that the one mechanism the field uses for the anchors today — the measure's name (ARPU vs ARPPU, gross vs net, DAU vs MAU) — is the field independently discovering the problem and patching it in a medium that can neither record the anchors reliably nor check them, and closes by deriving, rather than asserting, the properties any structured account of the two anchors must have. The treatment is operator-level and tool-neutral. This is a companion to The Silent Failure Atlas (Zenodo v1.3, DOI: 10.5281/zenodo.20762839); the extremum/count result appears there as failure mode F1.11.