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\title{Declarative Outcome-Conformant Synthesis: Exact, Closed-Form Specification Satisfaction and a Conformance Benchmark}
\author{Muhammed Rasin\\ Independent Researcher\\ \texttt{rasinbinabdulla@gmail.com}}
\date{}
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\begin{abstract}
We study a capability the dominant paradigm in synthetic tabular data does not
provide: exact satisfaction of a declared analytical outcome with no source data. Imitation
methods (copulas, GANs, diffusion, the Synthetic Data Vault) learn a real distribution and
sample from it, and they are judged on fidelity to real data. A large and practical class of
needs is different. The goal is to generate data with no source data, a cold start, that
reproduces a declared outcome (a revenue curve, a churn rate, a group-wise distribution)
across a relational schema. Off-the-shelf imitation tools offer no interface for such targets,
and, the structural point, no sampler can hit an exact aggregate, because sampling has
variance. We make this concrete: on a real public dataset, off-the-shelf learned synthesizers
trained on that very data miss the declared monthly aggregate by 74 to 86 percent; the
strongest steelman, a GaussianCopula trained per period, cuts the miss to about 19 percent and
still cannot reach 0. A closed-form generator reaches exactly 0. We name this task
outcome-conformant synthesis, argue its evaluation axis is conformance rather than fidelity,
and show the two axes are orthogonal: a generator's standing on one says nothing about its
standing on the other. We contribute three things. First, a formal account: a widely-used
family of exact-aggregate generators is exactly conditional-sum sampling of a Gamma
population (via Lukacs' characterization), which gives closed-form aggregate exactness, a
closed-form marginal coefficient of variation, an exact distortion bound, and a
scale-invariance property. The same identity lets us map the boundary the method does not
cross. We show, with the control whose absence invalidated an earlier draft, that enforcing
the exact aggregate costs at most 0.006 in normalized 1-Wasserstein distance to an arbitrary
external target marginal, anywhere from light to heavy tails; the entire residual gap to that
target, up to 0.89 for a heavy Pareto, is shape-family mismatch that an unconstrained draw
incurs identically. The open problem is therefore a shape-family problem, not a constraint
problem, and we say which is which. Second, SpecBench. Query-aware test-data work already
measures conformance of a sort, to declared query-output cardinalities; to our knowledge
SpecBench is the first benchmark to measure conformance to analytical outcomes (aggregate
curves, rates, group-wise distributions) for cold-start relational synthesis, with each metric
operationalizing a result from the analysis. Third, a closed-form, deterministic reference system. We state the scope up
front: exact aggregation alone is trivial (a rescale ties it given a hand-built schema); the
contribution is exact conformance jointly with closed-form marginals, foreign-key and temporal
integrity, determinism, and zero source data, plus a benchmark that measures them together. The
natural-language capability is a bounded convenience: a rule-based parser covers a curated set
of domains, and two language-model front-ends (an open-weights model and a recent OpenAI
flagship) each cover all 18 with full conformance once a deterministic bridge repair is in
place. The exactness lives in the engine downstream, not in any model. We concede fidelity to
imitation where real data exists.
\end{abstract}
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