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Measure-Relational Algebra
A specification for data systems that carry measurement uncertainty. A stored quantity is a probability measure on a metrized domain, not a point; dependence is carried as data; computation is pushforward; and collapsing a measure to a single number is an explicit, audited operation rather than a silent default. The classical model of exact values is recovered inside it as a special case. The core is, in the manner of Codd’s 1970 paper, implementation-independent: it states what any conforming system must preserve, and provides the instruments to gauge what an implementation loses. The objective is correctness, prior to safety: a system that reasons over measured quantities can only be as right as its memory is honest about what it knows.
Collapse commutes with a numerical derivation, for every joint input law, if and only if the derivation is affine — and with every measurable derivation if and only if the values were exact to begin with.
Papers
A paper set in under another cites it. All are public drafts, revised as reviews come in. Latest revisions, September 28, 2026: the core has a DOI (Zenodo, one DOI for all versions), every paper and essay here cites it by that DOI, and one sentence of the core’s geophysical section is corrected to what the cited Methods state.
The specification: objects, operations, the non-commutation theorem and its instruments, the geophysical case, and eight open problems. Public draft revised September 28, 2026. Seventeen pages. DOI 10.5281/zenodo.22999911 (all versions). Cite as: M. P. Laing, Measure-Relational Algebra: The Core, Zenodo, 2026, doi:10.5281/zenodo.22999911.
Codd’s 1981 Turing lecture did not argue the relational model; it argued what the model bought. This article makes the same kind of case for replacing the exact-valued cell by a probability measure: the minimal capability a system must have to be called measure-relational, the two skeptical questions Codd answered in 1981, and what we do not have — installations. Written for the SIGMOD Record. Five pages.
Put a learned function downstream of a point-valued record and the collapse artifact becomes the learner’s exact target: a consistent learner trained on such records estimates the collapsed regression function. What a memory must preserve for reasoning over it to stay calibrated, and a benchmark design that would show the artifact. Nine pages.
A system that improves itself trains each generation on records its previous generation wrote. Three limits of that loop: with labels fixed, the first generation’s artifact is a fixed point and no generation improves on it; with each generation labeled by the last, the learned response contracts to a constant while the training loss goes to zero; with each reading stored as a conditioning event on a carried law, the same readings drive the posterior to the truth. Twelve pages. For non-specialists: Why a Self-Improving System Cannot Trust Its Own Score, below.
A cut of hallucination by cause rather than by symptom. One component is an artifact in a strict sense: structure manufactured by the point-collapse of uncertain quantities in the training record, then faithfully reported by a learner doing exactly what it should. More data, more capacity, and better optimization do not remove it. A separation criterion, a synthetic experiment with closed-form pass criteria, and the observational partition stated as open work. Fifteen pages.
When one model’s outputs are written into many records, every entry inherits a component of the model’s error, coupled through a cause none of them stores. Three consequences: averaging model-derived entries converges on the model, not the truth; the discarded shared covariance biases every consumer curved at the mean, and the persistent part is carried entirely by the off-diagonal entries; and institutions that never exchange a byte are coupled by a shared upstream model. The coupling as a first-class object a record layer must carry. Cites the two papers above as well. Ten pages.
Essays
Written for non-specialists. Each is a companion to one of the papers; nothing depends on an essay.
Every self-improvement loop judges each generation by a score, and a score is a measurement. Five steps from that observation to the conclusion: the loop’s own evaluator reports success at the moment the response has been forgotten, and the repair is a write convention. Companion to The Fixed Point of Self-Improvement; the entry point for readers arriving from the recursive-self-improvement discussion. Five pages.
The companion essay to the core: a ceiling that wasn’t there, the mechanism behind it, and why the repair is structural. Four pages.
A working engineer’s history of schema: how Codd moved correctness out of programs and into structure, how we tore that down and rebuilt most of it — and the floor it never had. Four pages.
One little term — half the curvature, times the uncertainty — discovered, renamed, and forgotten by field after field for three centuries. A short history, and the case for finally giving it a home. Four pages.
Galton’s ox retold for deployment scale: averaging works when errors are independent — and the record never says whether they were. Shared-cause coupling, the floor it puts under the crowd, and the record that could expose it. Companion to What a Memory Must Preserve. Five pages.
Two ways for a witness to be wrong: inventing on the stand, or faithfully reading a record that every archive copied from one clerk’s error. Companion to Which Hallucinations Are Artifacts?: why one class of model error is neither confabulation nor curable by scale. Four pages.
An airliner carries two engines because two can fail separately. What independence buys — in redundancy, in averaging, in a leaderboard — and what happens to the arithmetic when one upstream model feeds every entry. Companion to What a Deployment Correlates; sibling of The Committee That Never Disagrees. Three pages.