The Theory of Data — Particles, Atoms, Anchors, Universes, and Lawful Transformation

The full technical construction of the theory of data. A value becomes data when it is typed and located: the location is an anchor point in a typed coordinate structure built from dimensions, axes with declared coarsening maps, coordinates, and anchors — every axis terminating in a singleton level, so an omitted axis is contracted, not absent. A particle is one typed value at one anchor point — locally meaningful but analytically simple. An atom (a column) is the homogeneous, functionally consistent binding of particles over an anchor inside a universe — equivalently a typed partial function. A universe is not a Cartesian product but a typed population with a law of existence (event vs. spine), which gives the missingness anchor its formal jurisdiction. A governed atom carries a V-anchor (where values live), an M-anchor (where absence speaks), and operator-indexed B-anchors (where motion stops) — with stock/flow shown to be domain explanations of B-anchor rules, not primitives. Two primitive value-bearing operations (mapper, reducer with aggregation fibers) combine with a layered anchor–atom–frame–universe calculus (restriction, reindexing, lag, windows, allocation, alignment, expansion, crossing): anchor operations determine which particles meet; mappers and reducers determine what their values may become. Lawful analytical movement is defined as a decidable conjunction relative to declared premises; a generated grammar maps expressions into the contract, adjudication tests it against data, and an honesty layer determines serve, disclose, clarify, or refuse. Companion foundations note: DOI 10.5281/zenodo.21696104. Companion system: Columna (Apache-2.0), pip install columna.