Learning the Count, Keeping the Unit: Task-Algebra Heads for Relational Regression

Relational prediction tasks define their targets through programs: a future sum, a count, or an event condition. These programs can specify how a model should predict, as well as what it should predict. We develop Task-Algebra, a compiler for a restricted aggregation-query fragment that extracts a query-time unit factor, transforms the training target, and reconstructs predictions with that factor kept exact. For a static entity-owned unit, a future sum becomes $Y=sN$; monetary absolute error becomes exactly $s$-weighted count error. A conditional-median approximation result explains why this parameterization can save finite model capacity even when both models receive identical inputs. On four chronological Amazon item-LTV development evaluations with matched 67-feature CatBoost heads, the factorized head reduces mean absolute error from 44.481 to 42.071, a 5.42% reduction, with a category-clustered gain interval of $[1.133,4.466]$. Larger trees, numerical loss controls, a history-selected population, and continuous decoders reveal how approximation and optimization contribute. The resulting account connects a task's aggregation semantics to an executable prediction head and to the regimes in which keeping a known unit outside the learner is useful.