Measure-Relational Algebra: The Core

Codd's relational model takes a cell to be an exact value and equality to be the primitive comparison. We replace the cell by a probability measure on a metrized domain, carry dependence among cells as an explicit coupling — one joint law per database state — and derive operations by pushforward. Selection, projection, join, derivation, and aggregation are closed under this semantics and reduce to their classical counterparts, under bag semantics, in the Dirac, discrete-metric limit. The object is close kin to a continuous possible-worlds semantics; what is claimed is the discipline imposed on it. One theorem gives the generalization its content: mean collapse commutes with a numerical derivation for every joint input law if and only if the derivation is affine, and commutes with every measurable derivation if and only if the cells are already Dirac. Point-valued practice — collapse first, combine afterwards — is therefore not an approximation of the uncertainty-preserving computation but a different one. The discrepancy is a bias, not noise: of definite sign for convex or concave derivations, of second-order magnitude set by curvature and covariance, and not removed by any increase in the number of sampled cases while the error mechanism is unchanged. The same reduction applied at observation — recording a noisy proxy as though it were the quantity — is a second and distinct loss. In a minimal closed-form model — linear response, multiplicative measurement error, lognormal prior — it manufactures a concave power law: a saturation the underlying system does not have, the curvature of the derivation contributing only a constant factor. A 2026 Nature result exhibits this at scale: an apparent saturation of Earth's geomagnetic response, long modelled as physics, is reproduced by observational uncertainty alone. Read constructively, the theorem is an optimizer criterion — an affine frontier above which collapse is exact and below which it carries an estimable price — and the paper closes with eight open problems — representation, normal forms, completeness, conditional cells, cost, decision, replication, and retrieval — that separate this specification from a system. What is claimed is the specification — what a conforming system must preserve, and the instruments to gauge what an implementation loses; the mathematics is elementary and long known, and the closed-form exhibit is regression dilution seen through the exponential. Not claimed: a system, a benchmark result, or how often the discrepancy is large. Public site draft 12 (September 28, 2026), as served at https://realql.org. Seventeen pages. Site draft 12 adds the author's ORCID, a colophon carrying this concept DOI, and corrects one sentence of Section 4.2 to what the cited Methods state; no numbered statement moved. Companion papers – on what a memory must preserve, on iterated learning, on deployment at scale, on an experimental programme – and a vision article stand beside it at the same address.