# DiscoCATE Theorem Targets

> Status vocabulary: **proof_seed**, **planned**, **boundary**, **deferred**,
> **proved_in_draft**, **proved**. A draft proof is manuscript-level work that
> still requires external review; **proved** is reserved for the accepted
> project truth.

## T0 — Observed CATE identification

**Status:** proof_seed; standard result to restate, not a new contribution.

Under A0--A3,

\[
\tau(x)
=
\mathbb E[Y\mid T=1,X=x]
-
\mathbb E[Y\mid T=0,X=x].
\]

This theorem fixes the causal target before Unit Abduction enters.

## T1 — Fixed-\(Q\) localized-effect identification

**Status:** **proved_in_draft** in Theorem 1 and Appendix B.1 of the V0
manuscript; first paper-local theorem obligation.

Let \(\mathcal Q:X\mapsto Q_X\) be fixed and pretreatment-measurable. For focal
\(q\), let

\[
w_{q,h}(X)=K_h\{d(q,Q_X)\},
\qquad
0<\mathbb E[w_{q,h}(X)]<\infty.
\]

Define

\[
\theta_{h,\mathcal Q}(q)
=
\frac{
\mathbb E[w_{q,h}(X)\{Y(1)-Y(0)\}]
}{
\mathbb E[w_{q,h}(X)]
}.
\]

Under T0 assumptions,

\[
\theta_{h,\mathcal Q}(q)
=
\frac{
\mathbb E[w_{q,h}(X)\{\mu_1(X)-\mu_0(X)\}]
}{
\mathbb E[w_{q,h}(X)]
},
\]

so the target is an observed-law functional. This statement does not require
\(Q_X\) to have already been identified as a calibrated latent posterior.

### Proof intuition

Once a pretreatment weight rule is frozen, it defines a soft population before
donor outcomes are observed. Standard causal identification recovers the mean
effect in that population. Unit Abduction determines *which population is
emphasized*; it does not replace the causal assumptions that identify effects
inside it.

## T2 — Shrinking-neighborhood limit and coarsening

**Status:** **proved_in_draft** under an explicit approximate-identity
assumption in Proposition 1 of the V0 manuscript. More primitive metric-space
regularity conditions remain to be expanded.

Under approximate-identity, support, and continuity conditions,

\[
\theta_{h,\mathcal Q}(Q_{x_i})
\longrightarrow
\mathbb E[\tau(X)\mid Q_X=Q_{x_i}].
\]

The right-hand side is the projection of \(\tau(X)\) onto the information in
\(Q_X\). The theorem must state the almost-everywhere / pointwise regularity
needed when \(Q_X\) is continuously distributed.

## T3 — Equality to focal CATE

**Status:** **proved_in_draft** as the effect-sufficiency / fiber-equality
corollary in the V0 manuscript.

Give sufficient conditions under which the T2 limit equals \(\tau(x_i)\):

- effect-sufficiency \(\tau(X)=g(Q_X)\) almost surely;
- or local injectivity of \(x\mapsto Q_x\) plus continuity;
- or the weaker fiber equality
  \(\mathbb E[\tau(X)\mid Q_X=Q_{x_i}]=\tau(x_i)\).

For fixed nonzero \(h\), record the exact equality condition

\[
\mathbb E\!\left[
w_{Q_{x_i},h}(X)\{\tau(X)-\tau(x_i)\}
\right]=0.
\]

## T4 — Consistency of DiscoCATE-DR

**Status:** planned.

For fixed \(\mathcal Q\), prove consistency of the normalized weighted average of
cross-fitted AIPW scores for \(\theta_{h,\mathcal Q}(q)\). The statement must
include nuisance-rate conditions, bounded/stable weights, nonvanishing local
mass, and growing effective neighborhood size.

## T5 — Estimated-\(Q\) honesty and stability

**Status:** planned.

Replace fixed \(\mathcal Q\) by an external-trained or outer-fold estimator
\(\widehat{\mathcal Q}\). State conditions under which induced weight error is
asymptotically negligible or appears explicitly in the error bound. Do not claim
Neyman orthogonality with respect to belief-calibration error unless proved.

## N1 — Latent decomposition non-identification

**Status:** boundary theorem target.

Ordinary one-row \((X,T,Y)\) data may identify the composed CATE or arm marginals
without separately identifying the calibrated belief \(Q_x^\star\), the latent
effect surface \(\Delta(x,u)\), the latent coordinates, or cross-world coupling.
The paper should give an explicit observational-equivalence construction rather
than leaving this as prose.

## Deferred theorem family

- unit-effect operator inversion under token sufficiency and injectivity /
  completeness;
- asymptotic normality and confidence intervals;
- robustness to misspecified / uncalibrated Unit Abduction;
- latent-geometry identification modulo reparameterization;
- Cauchy location / quantile-effect specialization.