# Assumption ledger

## Unit and learner ontology

At world level, \(U\) is the model-level unit-selection variable and
\(U=u^\star\) is one actual token/unit embedding. Record index \(i\) is only
bookkeeping. Pretreatment \(X=x\) is factual evidence; the learned
\(q_\phi(u\mid x)\) is epistemic uncertainty about that same token and does not
resample identity. All factual and alternative doses reuse this abduction result
and change only \(a\) in \(Y_u(a)=f_\theta(a,E;u)\).

Here *unit selection* names the choice of token identity. Dose assignment \(A\),
and any sample-inclusion mechanism, are separate objects. Full support of
\(q_\phi(u\mid x)\) or a positive Cauchy scale is not dose positivity and does
not repair sample-selection bias or hidden confounding.

The general abduction family includes point, Gaussian, and Cauchy results. This
paper implements the latter two. A Cauchy distribution has location and scale,
but a non-degenerate one has no finite mean or variance. Heavy tails express a
globally open set of candidate unit embeddings; they do not imply universal
outcome support.

## Causal identification

For each dose $a$ in the target support, the paper uses consistency, no interference, weak conditional exchangeability $Y(a)\perp A\mid X$, and conditional density positivity $f_{A\mid X}(a\mid x)>0$. Abduction does not establish these assumptions and does not repair hidden confounding.

## Support

Identification is pointwise on the conditional dose support. Stable estimation requires a stronger lower density bound on the reported region. Smoothness permits interpolation; it does not turn unsupported extrapolation into identified causal evidence.

## Distributional family

Conditional on $X=x$, candidate coordinates under the learner's abduction
distribution are modeled as independent Cauchy variables. Event noise is unit
Cauchy and independent of the candidate representation at each queried dose.
This is a tractable robust family, not a claim that physical units are
literally Cauchy.

## Dose-indexed mechanism

The response mechanism is $Y(a)=c(a)+w(a)^\top U+\sigma(a)E_a$, with continuous differentiable $c,w,\sigma$ and $\sigma(a)>0$. The implementation uses a shared neural dose mechanism with smooth Fourier features.

## Cross-dose coupling

Identified marginal response kernels do not require a joint law across doses. Pairwise individual-contrast laws are model-implied only after choosing how $E_a$ and $E_{a'}$ are coupled. A collection of unrelated pairwise choices is not automatically a coherent stochastic process over all doses.

## Mean boundary

A non-degenerate Cauchy response or contrast has no ordinary mean. Primary targets are location and quantile dose-response objects. In official benchmarks, the reported "average dose-response" for HCGM-Dose is the test-population average of its conditional location. This is comparable to the benchmark oracle because those generators add symmetric zero-location Gaussian outcome noise, making their conditional mean and median surfaces coincide. Elsewhere, average-response language is used only for finite-mean baselines or explicitly bounded utilities.