# The theorem-backed contract

This page carries what Theorem 1 states, the objects it is stated over, and the conditions it
rests on. [Targeting and cross-fitting](targeting.md) carries what the implementation chooses where
the theorem is silent. Every source and locator is in
[references](../../references.md#doubly-robust-inference-drtmle).

## The objects

For treatment level `a`, with `Q̄_0(a, w) = E_0(Y | A = a, W = w)`, the target is
`ψ_0(a) = E_0{Q̄_0(a, W)}` and `ATE = ψ_0(1) − ψ_0(0)`. The ordinary efficient influence function
is `D*(Q, g)(O) = A/g(W)·{Y − Q̄(W)} + Q̄(W) − Ψ(Q)`.

The default construction uses three univariate reduced regressions however many covariates the fit
adjusted for:

```text
Q_r    := Q̄_{0,r}(Q̄, g)(w)      = E_0[ Y − Q̄(W) | A = 1, g(W) = g(w) ]
g_{r,1} := g_{1,0,r}(Q̄)(w)       = E_0[ A | Q̄(W) = Q̄(w) ]
g_{r,2} := g_{2,0,r}(Q̄, g)(w)    = E_0[ (A − g(W))/g(W) | Q̄(W) = Q̄(w) ]
```

The two corrections, in the orientation this package computes and the appendices derive:

```text
D_A = D*_g = (Q_r/g)·{A − g}
D_Y = D*_Q = 1_a·(g_{r,2}/g_{r,1})·{Y − Q̄}
```

The bivariate alternative retains `Q_r` and replaces both reduced mechanisms by one probability:

```text
g_r(a|w) = P_0{A=a | Qbar(a,W)=Qbar(a,w), g(a|W)=g(a|w)}
D_Y      = 1_a·{g_r(a|W)-g(a|W)}/[g(a|W)g_r(a|W)]·{Y-Qbar(a,W)}
```

This is exactly the pinned R source's two-column `estimategrn` branch, `fluctuateQ2` clever
covariate, and `eval_Dstar_Q` correction. Those branches loop over every requested level of a
discrete treatment, so cleverly uses the same one-versus-rest probability and correction for each
arm. In the stored state `gr1` is that probability and
`gr2` is `NaN`, because the latter regression does not exist on this path; using zero would make
an accidental univariate formula look valid.

and the limiting influence function

```text
D^{*,#}(Q, g) = D*(Q, g) − I(g = g_0)·D_A − I(Q̄ = Q̄_0)·D_Y
```

The indicators contain the doubly-robust claim. At the true primary nuisance functions, both
corrections vanish and the ordinary efficient influence function is recovered.

The two extra score equations, as the software article states them:

```text
(9)   P_n[ Q_r(a,W)/g*(a|W) · {1_a − g*(a|W)} ]                    = 0
(10)  P_n[ 1_a · g_{r,2}(a|W)/g_{r,1}(a|W) · {Y − Q̄*(a,W)} ]        = 0
```

## Theorem 1

**Suppose** either `Q̄ = Q̄_0` or `g = g_0`, and let the targeted collection
`(Q̄*_n, Q̄*_{n,r}, g*_n, g*_{1,n,r}, g*_{2,n,r})` satisfy the three empirical score conditions

```text
B_n     = P_n D*(Q*_n, g*_n)                            = o_p(n^(−1/2))
B_{A,n} = P_n D_A(Q̄*_{n,r}, g*_n)                       = o_p(n^(−1/2))
B_{Y,n} = P_n D_Y(Q̄*_n, g*_{1,n,r}, g*_{2,n,r})         = o_p(n^(−1/2))
```

**and** suppose the appendix remainder terms satisfy `R_{Q,n} = o_p(n^(−1/2))` and
`R_{g,n} = o_p(n^(−1/2))`. **Then** the targeted plug-in is asymptotically linear with influence
function `D^{*,#}`, and `√n(ψ̂ − ψ_0) ⇝ N(0, σ²)` with

```text
σ̂²_n = P_n [ D*(Q*_n, g*_n) − D_A − D_Y ]²
```

Four things to read off it:

- the remainder conditions do not follow from the union condition and the rates.
  [The conditioning terms](#the-conditioning-terms-are-linear-in-one-error) shows why. With one
  nuisance wrong, the interval is established only in the cases that section lists;
- the score conditions are `o_p(n^(−1/2))`, **not** exact zeros. This implementation's exact-zero
  ambition is stricter than the theorem, and its numerical stopping rule is not obviously either;
- the conditions are on the **targeted** collection, including the starred *reduced* nuisances,
  which is why `retarget` on a `DRTMLE` result costs a fit rather than arithmetic on cached
  arrays. That cost follows the source, and is not a design slip;
- the variance is `P_n[·]²` of the **rowwise** corrected curve, so nothing that summarises the
  curve before squaring is computing this. The ATE curve is the **rowwise difference** of the two
  arm curves, which is what makes an ATE-only diagnostic insufficient: arm-specific errors cancel
  in a difference.

`tests/unit/test_theorem_drtmle.py::TestTheReportedVarianceIsTheorem1s` pins the last property.
The interval built from the package's own corrections is the one Theorem 1's terms give. The
uncentred `P_n{D}²` differs from the reported variance by exactly `(P_n D)²`, and the contrast
reads the covariance rather than the sum of the arms.

## A known treatment mechanism

Data can declare its treatment mechanism with `treatment_probabilities=` (the
[point-treatment contract](../point-treatment-tmle.md#known-treatment-mechanism) gives the forms).
A complete-data fit is then Theorem 1 at the degenerate estimator `g_n ≡ g_0`. The branch
`g = g_0` of the theorem's hypothesis holds by declaration, and `R_{g,n}` is trivial at `g_0`.
This is a case where the interval with a wrong outcome regression is established
([the conditioning terms](#the-conditioning-terms-are-linear-in-one-error)). The ordinary TMLE
interval is valid here too.

| guard | what the fit does with the declared `g_0` |
| --- | --- |
| `()` | the ordinary TMLE at `g_0`, which is exact for any `Q̄` |
| `("g",)` | equation (10) fluctuates `Q̄` only, so the mechanism stays `g_0` |
| `("Q",)` and `("Q", "g")` | equation (9) fluctuates `g*` along `Q_r/g` (step 6 of the algorithm in Benkeser et al. 2017, Section 3.2), so `g*` moves off `g_0` on purpose |

Under the `"Q"` guard the reported curve `D* − D_A − D_Y` is the curve of the estimator that
fluctuates a known mechanism. It is not the curve of the known-mechanism TMLE. R `drtmle` 1.1.2
does the same with a supplied `gn`: its `gnStar` starts at `gn`, and `fluctuateG` moves it. The
fluctuated `g*` is no longer the declaration, so it keeps the shipped truncation. The declaration
itself answers to the bound rule of the point-treatment contract before any learner.

At an exact law the `"Q"` guards are blind. There `P_n D_A = 0` at `g_0` before any
fluctuation, so equation (9) solves at `ε_g = 0` whether the fluctuation exists or not.
`tests/unit/test_known_treatment_mechanism.py` therefore checks the fluctuation on a finite sample:
the initial `P_n D_A` exceeds `1e-3`, the final score is solved, and `max |g* − g_0|` exceeds
`1e-4`.

Every reduction and both `reduced_crossfit` settings admit a declared mechanism. The nested
construction reuses `g_0` in every inner fold, because a declared mechanism does not depend on the
rows a model saw. An `evaluation=` companion must declare the same columns. The registered
[known-mechanism DR-TMLE study](../method-evidence/known-mechanism-dr-tmle.md) pairs each guard
with R `drtmle` (`gn=`) at a wrong outcome regression.

The statements above are for complete data. With `delta=`, the declaration supplies the treatment
mechanism `g_A` only, and the fit still estimates the observation mechanism `g_Delta`. The table
gives what follows for the randomized missing-outcome route.

| question | answer |
| --- | --- |
| the remainder with a wrong outcome regression | linear in the error of `g_A g_Delta`, not a product of two errors. The term sits in the step at equation (21) of the proof in Díaz and van der Laan (2017), Appendix A.1. A known `g_A` does not remove it, because `g_Delta` is estimated |
| the measured effect | a project probe simulated a known `g_A`, a wrong outcome regression and a slowly converging `g_Delta`. Coverage of `ey[1]` fell to 0.850 at n = 16,000, and the bias grew with n (`reviews/rm40/lra_summary.log`, not a registered study). The registered [one-inconsistent-nuisance study](../method-evidence/dr-tmle-one-inconsistent-nuisance.md) repeats the setting on law M at n = 4,000. Its arm-mean bias was −0.53 and −0.54 SD, with coverage 0.920 and 0.926 |
| when the interval is established | when the outcome regression and `g_Delta` are both consistent |
| what a declared `g_A` alone gives | no established interval with a wrong outcome regression. This is a gap in the theory, not in the evidence |
| `n_bootstrap=` | admitted, as on the route with an estimated `g_A`. A full-refit bootstrap resamples the same estimator, so its range rests on the same remainder as the analytic interval and carries the same limit |

## Randomized trials with missing outcomes

For observed data `O=(W,A,Delta,Delta Y)`, write `g_A(a|W)=P(A=a|W)`,
`g_Delta(a,W)=P(Delta=1|A=a,W)`, and `g=g_A g_Delta`. Díaz & van der Laan's
efficient influence function is

```text
D* = 1(A=a, Delta=1)/g · {Y − Qbar(a,W)} + Qbar(a,W) − psi(a).
```

The implementation keeps the five one-dimensional regressions and the three correction
blocks in the paper separate:

```text
gamma_A = P(A=a | Qbar_a)
gamma_Delta = P(Delta=1 | A=a,Qbar_a)
r_A = E[{1(A=a)-g_A}/g_A | Qbar_a]
r_Delta = E[{Delta-g_Delta}/(g_A g_Delta) | A=a,Qbar_a]
e = E[Y-Qbar_a | A=a,Delta=1,g_A g_Delta]

D_A = e/g_A · {1(A=a)-g_A}
D_Delta = 1(A=a)e/(g_A g_Delta) · {Delta-g_Delta}
D_Y = 1(A=a,Delta=1) · {r_A/(gamma_A gamma_Delta)+r_Delta/gamma_Delta}
      · {Y-Qbar_a}
```

The targeting cycle jointly updates the ordinary outcome and `D_Y` covariates, updates
`g_Delta` within each arm, updates `g_A`, refits all five reductions, and repeats until all four
score blocks settle. The `g_A` update takes one of two routes, and both solve the equation
`P_n[e_a/g_a · {1(A=a) − g_a}] = 0` for every arm.

| arms | `g_A` update | covariate |
| --- | --- | --- |
| two | one two-parameter logistic tilt of `g_1`, with `g_0 = 1 − g_1` | `(−e_0/g_0, +e_1/g_1)` |
| three or more | one logistic tilt of each column `g_a`, with the response `1(A=a)`, on all rows | `+e_a/g_a` |

At two arms the arm-0 column carries a minus sign, because its score `−(e_0/g_0)(A − g_1)` equals
`(e_0/g_0){1(A=0) − g_0}`. Above two arms the tilted columns are not renormalized. They are nuisance
denominators and not an intervention. `correction_check` reports `D_A`, `D_Delta` and `D_Y`
separately; checking only `D_A + D_Delta` would be blind to equal and opposite score errors.
Missing-outcome fits therefore require `guard=("Q", "g")`.

Treatment probabilities and observation probabilities retain their own bounds: `g_bounds`
applies to `g_A` and `gamma_A`, while `nuisance_bound` applies to `g_Delta` and
`gamma_Delta`. Their product is derived for the ordinary outcome clever covariate and the
positivity report, never stored as a third nuisance. The treatment truncation curve moves only
the treatment bound; `mechanism=True` moves only the observation bound.

The positivity report's `P(A=a,Delta=1|W)` row is derived, so it **has no bound of its own**.
Reading it as though it did is a mistake worth naming. Each factor is truncated and the two are
then multiplied, so the product is never compared against `g_bounds[0] × nuisance_bound`. That row's `clipped` therefore counts the cells where either factor's
truncation moved the product. Its `ess_ratio` weights by `clip(g)·clip(π)`, so the denominator
the equation forms. Counting against the product of the floors instead reports a strict subset,
since a small factor beside a large one leaves the product above it: measured at **1.1%** against
a true **20.1%** on the pinched fixture in `tests/unit/test_drtmle_missing.py`.

The shipped scope follows the paper rather than the broader canonical package: randomized treatment
at any number of arms, MAR and positivity, no cross-fitting, and no weights, repeats, fold
targeting, or evaluation companion. `randomized=True` estimates `g_A`. Data that declares its
mechanism ([a known treatment mechanism](#a-known-treatment-mechanism)) supplies `g_A` and fits no
treatment learner. An observational missing outcome and a missing treatment take the construction
of [the composite indicator](#observational-missing-data-the-composite-indicator) instead.

**With a wrong outcome regression the theorem does not establish the interval in general.** The
proof of Theorem 1 is in Appendix A.1 of Díaz and van der Laan (2017). It conditions a residual
mean on the pair `(g_0, ĝ)`. It then replaces that mean by one given `g_0` alone, and calls the difference, the
term at equation (21), `O_P(‖ĝ − g_0‖²)`. The difference is linear in `ĝ − g_0` when the error of
`ĝ` breaks the level sets of `g_0`. Lemma 1 and Theorem 2 inherit the step. Here
`g = g_A g_Delta`, so the term has the same structure as `R_{3,n}` of
[the conditioning terms](#the-conditioning-terms-are-linear-in-one-error).

| observation mechanism, with the outcome regression wrong | the interval |
| --- | --- |
| the product mechanism `g_A g_Delta` known | established. The package always estimates `g_Delta`, so no fit is in this row |
| a correctly specified parametric model | the projection argument of [the conditioning terms](#the-conditioning-terms-are-linear-in-one-error) suggests a conservative interval. The page derives it for a complete outcome only. A re-check of the probe's GLM cell (n = 4,000, R = 1,000) found no bias: every 99% bias interval contains zero, and the SE ratios are 0.96 to 1.01 (`reviews/rm40/lra_recheck.log`, not a registered study) |
| a data-adaptive learner | not established. A project probe found an arm-mean bias that grows with n. The law-M control of the registered [one-inconsistent-nuisance study](../method-evidence/dr-tmle-one-inconsistent-nuisance.md) shows the bias at n = 4,000 |

The probe declared `g_A`, fitted a wrong outcome regression, and gave `g_Delta` a constructed slow
error (`reviews/rm40/lra_summary.log`). The `ey[1]` bias was −0.24, −0.60 and −0.98 SD at
n = 1,000, 4,000 and 16,000. Its SE ratio stayed between 0.88 and 1.10. A known `g_A` alone is not
enough.

The `ate` bias cancelled on
that law, because the errors of both arms moved the same way. The mirror branch, where `r̂_{A,0}` and `r̂_{M,0}` condition on `(m_1, m̂)`, has the same
structure. No probe measured it.

### More than two arms

Díaz and van der Laan (2017) state their result for one arm indicator. Above two arms, the fit
applies it to each indicator `1(A=a)` and stacks the arm estimators. The contract below gives the
argument in five parts. Page numbers are those of the arXiv v1 author manuscript that
[the references](../../references.md) cite.

| part | content |
| --- | --- |
| base result | Theorem 2 (p. 20) under Condition 2 (Donsker, p. 13) and Condition 3 (p. 15). It gives `n^(1/2)(psi_dtmle − psi_0) → N(0, Var D_dr)` with `D_dr` of Theorem 1 (p. 16), for one indicator `A` and the target `E(Y_1)`. Section 2.1 (p. 6) applies it to "four such indicators" in its application. With a wrong outcome regression the result holds only in the cases [above](#randomized-trials-with-missing-outcomes), because the proof's step at equation (21) is linear in `ĝ − g_0` |
| steps | indicator reduction: arm `a` uses `(W, 1(A=a), Delta, Delta Y)` and its own `g_A(a|W)` and `g_Delta(a,W)`. Fixed-dimension stack: the `K` arm estimators are asymptotically linear on the same rows, so their joint covariance is `P_0[D_a D_b]`. Linearity gives each `ate`, and the delta method gives `rr` and `or` on the log scale. The simultaneous band is the multiplier band over the stacked curves |
| objection search | p. 25 rejects a composite `T = AM` reduction. The armwise construction keeps `g_A` and `g_Delta` apart, so the objection does not apply. p. 26 discusses cross-fitting, not arms. No source records the indicator reduction or the stack as open |
| conditions | Assumptions 1 to 4 (pp. 6-7) at every arm, with positivity `g_A(a|W) g_Delta(a,W) > 0`. Randomization by design: `randomized=True`, or a declared known mechanism, which can depend on `W`. A declared mechanism supplies `g_A` only, so it does not establish the interval when the outcome regression is wrong ([a known treatment mechanism](#a-known-treatment-mechanism)). Conditions 2 and 3 for every arm, hence `cross_fit=False`. With a wrong outcome regression, Condition 3's rates do not make the equation (21) term negligible ([above](#randomized-trials-with-missing-outcomes)). Every arm's four score equations solved to `o_P(n^(−1/2))` |
| evidence | the exact-law checks, nonzero witnesses, mutation controls and independent reference of `tests/unit/test_drtmle_missing_multi_arm.py`, and the registered [multi-arm missing-outcome study](../method-evidence/randomized-multi-arm-missing-outcome-dr-tmle.md) |

**No fluctuation parameter is shared across arms above two arms.** This condition is what lets
each arm's expansion hold inside the joint loop. Theorem 2's proof uses only that arm's solved
equations and Conditions 2 and 3. Four places carry the condition.

| block | where it is per arm |
| --- | --- |
| outcome tilt | `missing_outcome_outcome_submodel` builds a zero block for every other arm |
| observation tilt | `_solve_missing_observation_mechanism` tilts `g_Delta` within each arm |
| reductions | `fit_missing_outcome_reduced` fits each arm's five regressions on that arm's columns |
| treatment tilt | `solve_armwise_bounded_mechanism` tilts each column `g_a` alone |

The loop stops on the largest score over all arms, and so does the final check. That rule is
stricter than Theorem 2's per-arm requirement. At two arms the one shared tilt of `g_1` is a
different path to the same two solved equations. The proof reads only the solved equations, so
the result holds there too.

## Observational missing data: the composite indicator

An observational fit with `delta=` and a fit with a missing treatment use one construction. The
fit declares a missing treatment with `treatment_delta=<column>`, the 0/1 column that is 1 where
the treatment is recorded. The package never infers a missing treatment from a missing value, for
the reason it never infers a missing outcome: an accidental gap must not start a missing-data
analysis. The roadmap words for this item read "a missing treatment coded as a missing value", and
the declaration is a deliberate deviation from them.

For arm `a`, write `Delta_A` for the treatment indicator and define the composite indicator and its
mechanism:

```text
C_a = Delta_A · Delta · 1(A=a)
g_c,a(W) = P(C_a=1 | W) = pi_A(W) · g(a | Delta_A=1, W) · pi(a, W)
pi_A(W) = P(Delta_A=1 | W),    pi(a, W) = P(Delta=1 | A=a, Delta_A=1, W)
```

The product is the law of total probability, so it holds for every law. No factor needs a causal
reading. The fit estimates each factor on the rows it conditions on, and the outcome regression
on the rows where `C_a` can be 1.

**The route.** `cleverly.estimators.composite.missing_data_route` chooses the construction, and
the fit records it under `result.extra["missing_data"]`.

| `delta=` | missing treatment | `randomized=True`, or a declared mechanism on `DRTMLE` | guard | route |
| --- | --- | --- | --- | --- |
| no | no | any | any | `complete` |
| yes | no | yes | non-empty | `randomized_missing_outcome` (Díaz and van der Laan above) |
| yes | no | no | non-empty | `composite` |
| yes | no | any | `()`, or `TMLE` | `missing_outcome` (the shipped missing-outcome TMLE) |
| any | yes | no | any, or `TMLE` | `composite` |
| any | yes | yes | any | refused: `randomized=True` selects a construction that observes the treatment on every row, and a declared mechanism is not the mechanism of the recorded rows |

A per-arm outcome-adaptive `CTMLE` fit takes the `missing_outcome` route with `delta=` alone, and
the `composite` route with a missing treatment, in sample only. Its factors regress on
`Qbar_n(a, W)` and not on `W`, and the fit reports a point estimate only. The
[collaborative TMLE reference](../collaborative-tmle.md#baseline-strata-and-missing-data-in-the-per-arm-design)
states that contract. The other C-TMLE strategies refuse a missing treatment.

**The contract** has five parts.

| part | content |
| --- | --- |
| base result | Benkeser, Carone, van der Laan and Gilbert (2017), Section 3.2, Theorem 1: on `O = (W, A, Y)` with a binary `A`, the estimator that solves the targeting equation and the equations of the reduced regressions is asymptotically linear with curve `D* − D*_Q − D*_g` under the theorem's score and remainder conditions. Both nuisances consistent meet them. With one nuisance wrong, [the conditioning terms](#the-conditioning-terms-are-linear-in-one-error) limit the claim to the cases listed there, applied to `(W, C_a, C_a Y)` with the composite mechanism `g_c,a` |
| steps | indicator reduction: Theorem 1 applies as stated to `O'_a = (W, C_a, C_a Y)`, with the binary treatment `C_a`, the mechanism `g_c,a` and the regression `E(Y | C_a=1, W)`. Identification: `E{Y(a)} = E[E{Y | A=a, Delta_A=1, Delta=1, W}]` under the conditions below. Fixed-dimension stack: the `K` arm estimators are asymptotically linear on the same rows, with joint covariance `P_0[D_a D_b]`. Linearity gives `ate`, and the delta method gives `rr` and `or` on the log scale. Fixed weights tilt the law and clusters are the unit, as in both parents |
| objection search | Benkeser et al. (2017) mention no missing data and no caveat about coarsening. Díaz and van der Laan (2017, p. 25) reject a composite `T = AM` for a randomized trial, because it discards known design information. On an observational law nothing about the treatment mechanism is known, so the composite discards nothing. R `drtmle` 1.1.2 implements this construction (`R/drtmle.R` lines 207-209, `R/fluctuate.R` lines 28, 98 and 169-172, `R/estimate.R` lines 116-200) |
| conditions | the four conditions below; Theorem 1's conditions for each arm on `O'_a`, which include the Donsker condition, hence `cross_fit=False`; the solved equations of every arm at `o_P(n^(−1/2))`. The loop stops on the largest score over the arms, which is stricter than the per-arm requirement |
| evidence | the exact-law checks, nonzero witnesses, mutation controls and per-arm reference of `tests/unit/test_composite_missing_data.py`, and the registered composite study |

**The conditions** are these.

| condition | statement |
| --- | --- |
| consistency and no unmeasured confounding | `Y = Y(a)` when `A = a`, and `Y(a)` is independent of `A` given `W` |
| treatment missing at random | `Y(a)` is independent of `Delta_A` given `(A, W)` |
| outcome missing at random | `Y` is independent of `Delta` given `(A, Delta_A=1, W)` |
| composite positivity | `g_c,a(W) > 0` for every arm, and `W` complete on every row |

**What the data do not identify.** The conditions let `Delta_A` depend on `A`. Then
`P(A = a | W)` is not identified, and only `P(A = a | Delta_A = 1, W)` is. A target or a clever
covariate that reads the treatment law of every row is therefore not identified with a missing
treatment. That covers `att`, `atc`, `ey_obs`, `par`, `paf`, `incremental=` and `policies=`. The
shared preflight refuses each one by name, and each refusal repeats this statement. A regime mean
and an arm-indexed `msm=` coefficient read only the outcome regression and the law of `W`, so the
composite identifies them.

No observed-data check can detect a violation of the treatment condition.

**The carrier.** `g_c,0 + g_c,1 < 1` whenever a row can be unrecorded, so the composite is not a
distribution over the arms. It always travels as an `(n, K)` mechanism off the simplex. Every site
that tilts or reads a two-arm mechanism uses one rule: the mechanism is one column exactly when
the two-arm complement form applies. The composite therefore tilts each arm's column alone at
every arm count, two included. No fluctuation parameter is shared across arms.

**The bounds.** Each factor is bounded as the shipped missing-outcome TMLE bounds it: the
treatment factor by `g_bounds`, and each observation factor below by `nuisance_bound`. The
composite is then tilted and clipped inside `[g_lo · nb^k, g_hi]`, where `k` counts the
observation factors. That floor is the smallest product of the bounded factors, so the initial
clip moves no value. At `g_bounds=(0.01, 0.99)` and `nuisance_bound=0.01` with both indicators,
the floor is `0.01 · 0.01² = 1e−6`, against `1e−4` on a missing outcome alone. The default
`g_bounds="auto"` sets `g_lo = 5 / (sqrt(n) ln n)` instead. The positivity report adds the composite row with its minimum
and the share of unit-arm cells below `g_lo`.

The rejected alternative floors the composite at `g_lo` after the tilt, as R `drtmle`'s `tolg`
floors its product. That floor clips legitimate products of three probabilities. It also breaks an
exact reduction: without a missing treatment the composite's initial covariate is the shipped
missing-outcome covariate `1(A=a) Delta/(g pi)` bit for bit, and that covariate is not floored at
`g_lo`.

## The sign of the mechanism correction

**The implementation's sign is the derived one, on the working paper's own appendices.** Only a
variance check could have caught this. All three empirical means are driven to zero, so a flipped
sign moves the variance and leaves the point estimate alone.

**The discrepancy.** The §3.1 display defines `D_A := −(Q_r/g)(A − g)`, with a leading minus.
Theorem 1 then reports `D^{*,#} = D* − D_A − D_Y`. Read off those two displays, the theorem's
mechanism contribution is `+u` where the code's is `−u`.

**What settles it.** The same paper prints the object twice with two signs. §3.2 redefines `D_Y`
for the univariate construction, which is the default here, with **no** leading minus. That is
twelve lines after printing the bivariate one with one. Two displays of one object with opposite
signs is already a reason not to settle the question from a display.

**Appendices A and B decide it.** Each derives one term, and each derivation fixes the orientation.
Each reads `P_0[term] = −(P_n − P_0)·D + B_n + (second order)` with `B_n := P_n·D`. For any `u`
whatever, `P_0[u] = P_n[u] − (P_n − P_0)[u]` is an identity. The decomposition is therefore
satisfiable **only** with `D` equal to the positive term. Appendix A's opening step names the
quantity being decomposed, and it is checkable rather than interpretable:

```text
−P_0{ (Q_r/g_0)·(g_n − g_0) } = P_0{ (Q_r/g_0)·(A − g_n) }      since E_0[A | W] = g_0
```

The right-hand side is positive. `tests/unit/test_theorem_drtmle.py` checks that identity on the
exact law. It also checks that the correction's mean is materially nonzero there, because a zero
mean would make both readings agree and the question unanswerable. The test then checks the
consequence. The asymptotic-linearity representation closes to `1e-12` with the corrections
**subtracted**, and fails by **exactly twice the correction** when they are added. Three mutations
are watched to fail it, one of them the flipped sign in the library itself.

The leading minus in the §3.1 display is therefore not a rival convention to match. It contradicts
the derivation in the same document, Theorem 1's own variance formula, and the exact-law arithmetic
here. The published 2017 article confirms the corrected constructions. It does not replace that
algebraic sign witness. The code follows the identity rather than a display in either edition.

Two further sign slips sit in the same document. Equation (2) prints `+(P_n − P_0)D − B_n` where
its appendices derive `−(P_n − P_0)D + B_n`, which is the same slip under `D → −D`. Equation (2)
also crosses its second-order labels: appendix A's block is the `D_A` one and is collected into
`R_{Q,n}`, while (2) pairs `D_A` with `R_{g,n}`. Neither affects the implementation, which reads
no label. Both are worth knowing before you quote (2).

## Appendix C: a correction that is not needed costs nothing

The paper's own account of why solving an equation you did not need is asymptotically free. Both
answers rest on a vanishing:

```text
D_A = 0  for every g,   because Q_r    = 0 when Q̄ = Q̄_0
D_Y = 0  for every Q̄,   because g_{r,2} = 0 when g  = g_0
```

Each `B` then decomposes into an empirical-process term plus a second-order one, both
`o_p(n^(−1/2))` under the appendices' rate conditions.

**This is also the source of the blindness that shapes every test on this estimator.** `Q_r` and
`g_{r,2}` are zero *row by row* at correct nuisances. Any check taken at the truth is therefore
blind to a flipped sign, to an update order, and to a reduction vintage alike. That is why
`tests/unit/test_theorem_drtmle.py` and `tests/unit/test_influence_gateaux_drtmle.py` are taken at
values where the corrections do **not** vanish, and why their fixtures are misspecified on
purpose.

## The remainder terms, and the rate conditions

The mechanism-misspecification branch, appendix B:

```text
R̃_{5,n} = P_0[ { (A/g_{1,0n,r})·g_{2,0n,r} − (A/g_{1,0,r})·g_{2,0,r} } · (Q̄_0 − Q̄_n) ]
R̃_{6,n} = P_0[ { (A/g_{1,0,r})·g_{2,0,r}   − (A/g_{1,n,r})·g_{2,n,r} } · (Q̄_0 − Q̄_n) ]
M̃_{2,n} = (P_n − P_0)[ D_Y(Q̄_n, g_{1,n,r}, g_{2,n,r}) − D_Y(Q̄_0, g_{1,0,r}, g_{2,0,r}) ]
R_{g,n} = R̃_{5,n} + R̃_{6,n} + M̃_{2,n}
```

The published displays carry the factor `(Q̄_0 − Q̄_n)`. The leading `A` makes
`(Y − Q̄_n)` give the same `P_0` value, because `E_0[A(Y − Q̄_n) | W] = E_0[A(Q̄_0 − Q̄_n) | W]`.

The outcome-misspecification branch, appendix A, `R_{Q,n} = R_{3,n} + R_{4,n} + M_{1,n}`:

```text
R_{3,n} = P_0[ { (Q̄_{0n,r} − Q̄_{0,r}) / g_0 } · (g_0 − g_n) ]
R_{4,n} = P_0[ { Q̄_{0,r}/g_0 − Q̄_{n,r}/g_n }  · (g_0 − g_n) ]
M_{1,n} = (P_n − P_0)[ D_A(Q̄_{n,r}, g_n) − D_A(Q̄_{0,r}, g_0) ]
```

with `Q̄_{0n,r}(w) := E_0{Y − Q̄(W) | A = 1, g_n(W) = g_n(w), g_0(W) = g_0(w)}` evaluated at the
*estimated* propensity as well as the true one, which is what makes `R_{3,n}` an approximation
error rather than a fitted one. The published display omits `A = 1` under the paper's "abuse of
notation". The first equality of the same display needs it, because it rewrites
`P_0{(A/g_0²)(Y − Q̄)(g_n − g_0)}` as `P_0{(Q̄_{0n,r}/g_0)(g_n − g_0)}`. Appendix A also carries
`R*_n = R_{1,n} + R_{2,n}`, the part that is second order whichever nuisance is right.

**The paper states these rates as sufficient conditions.** It states that it *generally suffices*
that, for `R_{g,n}`:

```text
‖Q̄_n − Q̄_0‖_2 = o_p(n^(−1/4))
‖g_{2,0n,r} − g_{2,0,r}‖_2 = o_p(n^(−1/4))
‖g_{2,n,r}  − g_{2,0,r}‖_2 = o_p(n^(−1/4))
```

and for `R_{Q,n}`, in the same "if, for example" form, `o_p(n^(−1/4))` on the reduced regression's
approximation error `‖Q̄_{0n,r} − Q̄_{n,r}‖_2`, on its fitted error `‖Q̄_{n,r} − Q̄_{0,r}‖_2`, and on
the primary propensity's error `‖g_n − g_0‖_2`. Both appendices add an empirical-process pair in
one shape: a `P_0`-Donsker class containing the *estimated* curve, plus `L_2(P_0)` convergence of
the estimated curve to its limit. The next subsection shows that two of these rates need not hold
when one nuisance is wrong.

**These are conditions on estimated functions that no fit can check for itself.**

### The conditioning terms are linear in one error

`R_{3,n}` and `R̃_{5,n}` are not products of two errors. Each one is linear in the error of the
consistent nuisance. The rates above make neither term negligible when the other nuisance is wrong.
The tables below state when each term vanishes.

The notation is for arm 1. Arm 0 replaces `g` by `1 − g` throughout. `Q̄` and `ḡ` are the limits of
`Q̄_n` and `g_n`. Write `Q̄_{0,r} = E_0[Y − Q̄ | A = 1, g_0(W)]` and `F = Q̄_0 − Q̄ − Q̄_{0,r}`.

**The outcome branch.** On a level set of `(g_n, g_0)` the probability `P(A = 1 | W) = g_0` is
constant. Conditioning on `A = 1` therefore leaves the law of `W` unchanged there, and
`Q̄_{0n,r} = E_0[Q̄_0 − Q̄_n | g_n, g_0]`. The factor `(g_0 − g_n)/g_0` is a function of
`(g_n, g_0)`. The tower rule gives, with no approximation,

```text
R_{3,n} = P_0[ (Q̄_0 − Q̄_n − Q̄_{0,r})/g_0 · (g_0 − g_n) ]
```

The first factor tends to `F`. `F` is zero only when `Q̄_0 − Q̄` is a function of `g_0`. So
`R_{3,n}` is linear in `g_n − g_0`.

The paper's own premise for `R_{4,n}` is that the fitted
reduction converges to `Q̄_{0,r}`. Under that premise `R_{3,n}` is the first-order leftover of the
estimator. The paper puts `o_P(n^(−1/4))` on `‖Q̄_{0n,r} − Q̄_{n,r}‖_2`. For an error of `g_n` that
breaks the level sets of `g_0`, that norm stays of order one. The implication is correct, and its
premise fails in the generic one-wrong case.

**Only the bias of the error moves the centre of the estimate.** The stochastic part of `g_n − g_0` has mean
zero. It changes the variance, as [the projection](#a-parametric-mechanism-makes-the-interval-conservative)
shows. A first-order bias needs a bias of `g_n` of order `r_n ≫ n^(−1/2)` along `F/g_0`.

**Finite support.** On a law with discrete covariates, an estimate can break a tie of `g_0`. The
exact reduction `E[· | g_n]` can then recover the whole residual, so that `R_{4,n} = −R_{3,n}` and
the leftover vanishes. The tables below are for laws where the conditional mean given `g` is
continuous in `g`.

**The mechanism branch.** Appendix B states the exact step
`P_0{(Q̄_n − Q̄_0)(g − g_0)/g} = −P_0{g_{2,0n,r}(Q̄_n − Q̄_0)}`. There `g_{1,0n,r}` and `g_{2,0n,r}`
condition on `(Q̄_n(W), Q̄_0(W))`. After equation (10) the leftover reduces to
`P_0[h·(Q̄_0 − Q̄_n)]`, with

```text
r = (g_0 − ḡ)/ḡ,    h = r − g_0·E[r | Q̄_0]/E[g_0 | Q̄_0],    E[h | Q̄_0] = 0
```

It is linear in `Q̄_n − Q̄_0`.

**When the term adds no first-order bias.** The table is for the univariate reduction. The first three rows make the term `o_P(n^(−1/2))`. Negligible bias leaves a stochastic part of order `n^(−1/2)`. Its effect on the interval is derived only for a parametric index, where it makes the interval conservative.

| condition | outcome branch (`Q̄` wrong) | mechanism branch (`g` wrong) | checkable from a fit |
| --- | --- | --- | --- |
| both nuisances consistent at the product rates | the first factor shrinks too | the same | no. This is the ordinary TMLE rate condition |
| H2: the consistent estimate keeps the level sets of its truth | `g_n = g_0`, or `g_n = φ_n(g_0)` with `φ_n` tending to the identity | `Q̄_n = φ_n(Q̄_0)` | yes for a declared known mechanism. Otherwise no |
| H1 | `Q̄_0 − Q̄` is a function of `g_0` | `1/ḡ − 1/g_0` is a function of `Q̄_0`, which is `h ≡ 0` | no |
| negligible bias | the bias of `g_n` along `F/g_0` is `o(n^(−1/2))` | the bias of `Q̄_n` along `h` is `o(n^(−1/2))` | no |

For arm 0, the outcome-branch H1 is: `Q̄_0(0, ·) − Q̄(0, ·)` is a function of `g_0`. The
mechanism-branch H1 is: `1/(1 − ḡ) − 1/(1 − g_0)` is a function of `Q̄_0(0, ·)`. A limit `ḡ` that
is constant on the level sets of `Q̄_0` need not meet it.

**Law L** has `W1, W2 ~ N(0, 1)`, `g_0 = expit(0.6 W1 + 0.3 W2)` and `Q̄_0 = expit(−0.2 + 0.5 A − W1 + 2 W2)`. `tests/unit/test_drtmle_h1_symmetry.py` pins a constant `ḡ` on law L for which `L_2(h)` is 0.305.

`g_n = φ_n(g_0)`, with `φ_n` tending to the identity, is a one-wrong case where the DR-TMLE interval holds and
the plain TMLE interval does not. The plain TMLE leftover there is
`P_0[(g_0 − g_n)/g_n · E(Q̄_0 − Q̄ | g_0)]`, which is first order. Equation (9) removes it.

**The case table.** The rows are for the univariate reduction and a binary treatment. Above two
arms each row applies arm by arm.

| case | conditioning term | reported interval | evidence |
| --- | --- | --- | --- |
| both nuisances consistent at the product rates, plus the reduced-regression and solve conditions | second order | valid (Theorem 1) | the registered both-correct cells of the [canonical](../method-evidence/canonical-dr-tmle.md), [stratified](../method-evidence/stratified-dr-tmle.md) and [multi-arm](../method-evidence/multi-arm-dr-tmle.md) studies |
| `Q̄` wrong, `g` declared known | second order (its first-order part is zero). Equation (9) moves `g*` along `Q_r/g_0`, a function of `g_0` | valid. The ordinary TMLE interval is valid too | Theorem 1 at `g_n = g_0`, the registered [known-mechanism DR-TMLE study](../method-evidence/known-mechanism-dr-tmle.md) with a wrong outcome regression, and the declared-mechanism cells of the registered [one-inconsistent-nuisance study](../method-evidence/dr-tmle-one-inconsistent-nuisance.md) |
| `Q̄` wrong, `g_n = φ_n(g_0)` with `φ_n` tending to the identity, or H1 | second order (its first-order part is zero) | valid under the remaining rates | the derivation. The Section 5.1 law of the paper is H1 (`tests/unit/test_drtmle_h1_symmetry.py`) |
| `Q̄` wrong, `g` a correctly specified fixed-dimension parametric model fitted by maximum likelihood, in sample or cross-fitted | `O_P(n^(−1/2))`, linear in the score | **conservative**: asymptotically linear, with a reported variance too large by `c'I^(−1)c` | the derivation below, and the registered [one-inconsistent-nuisance study](../method-evidence/dr-tmle-one-inconsistent-nuisance.md): an ATE SE ratio of 1.161 in sample and 1.191 cross-fitted at n = 4,000, with coverage 0.975 and 0.982 |
| `Q̄` wrong, `g` with a bias of order `r_n ≫ n^(−1/2)` along `F/g_0` | of order `r_n` | **not established** | the identity, and the registered [one-inconsistent-nuisance study](../method-evidence/dr-tmle-one-inconsistent-nuisance.md): a constructed level-set-breaking error of order `n^(−1/3)` gave an ATE bias of −0.65, −0.86 and −1.16 SD, with coverage 0.935, 0.917 and 0.843, at n = 1,000, 4,000 and 16,000. A cross-fitted gradient-boosting mechanism gave +1.53, +2.01 and +1.51 SD, with coverage 0.75, 0.63 and 0.81. There `Q_r` is fitted on `g_n`, so `R_{4,n}` is first order too ([a data-adaptive mechanism](#a-data-adaptive-mechanism)) |
| `Q̄` wrong, `g` fitted in sample by a learner whose own-row term is not negligible | an own-row term, measured larger than the conditioning term and of the opposite sign | **not established** | the probe in [a data-adaptive mechanism](#a-data-adaptive-mechanism): an in-sample gradient-boosting mechanism gave −1.22, −2.28 and −2.50 SD, with coverage 0.75, 0.41 and 0.41 |
| `g` wrong, `Q̄` a correctly specified parametric model | `O_P(n^(−1/2))` | not established. An extra first-order term of no fixed sign remains, because the target depends on the parameters of `Q̄` | the derivation only |
| `g` wrong, `Q̄` with a bias along `h` of order larger than `n^(−1/2)` | linear | **not established** | the identity, and the registered study: a constructed error gave an ATE bias of −0.88, −1.07 and −1.35 SD, with coverage 0.825, 0.789 and 0.687, at the same sizes |

`PointTreatment(treatment_adjustment=...)` is a source of the mechanism-wrong rows. The treatment
mechanism then reads a subset of the adjustment set, and it is inconsistent when the restriction
fails. [Treatment-mechanism covariates](../point-treatment-tmle.md#treatment-mechanism-covariates)
gives the restriction text and the pinned summary line.

No fit can tell which row it is in, except a declared known mechanism. The registered study
publishes a verdict for each measured row, and its page gives the rules. The earlier project
probe, `reviews/rm40/drdr_sim.py` with its log `reviews/rm40/drdr_summary.log`, agrees with it.
The registered [multi-arm study](../method-evidence/multi-arm-dr-tmle.md) has a law that no test
pins as H1, and a correctly specified multinomial mechanism. Its SE ratios of 1.00 to 1.04 do not
contradict the conservative row. Its verdicts cannot detect a conservative interval, because
its coverage rule is a floor.

**One registered study detects the term.** The [one-inconsistent-nuisance study](../method-evidence/dr-tmle-one-inconsistent-nuisance.md) uses law L, which is not H1 in either branch. The verdicts of the
other studies cannot detect it. The canonical and stratified laws are H1 in both branches by
symmetry, as `tests/unit/test_drtmle_h1_symmetry.py` pins. The missing-data studies give the
consistent nuisance as an exact law table, which is H2.

**The routes.** DR-TMLE is the only shipped family that publishes an interval exposed to this
term. Its status stays `influence_curve` on every route, because a fit cannot tell which case it
is in, and the interval is valid when both nuisances are consistent at the product rates.

| route | index of the conditioning term | one-wrong interval established when | evidence |
| --- | --- | --- | --- |
| complete outcome: binary, more arms, strata, fixed weights, clusters, both reductions | `g_n` with the outcome wrong, `Q̄_n` with the mechanism wrong | the case table above | the registered [one-inconsistent-nuisance study](../method-evidence/dr-tmle-one-inconsistent-nuisance.md), and the probe log `reviews/rm40/drdr_summary.log` |
| [randomized missing outcome](#randomized-trials-with-missing-outcomes) | the product `ĝ_A ĝ_Delta` | the product mechanism is known. A known `g_A` alone is not enough | `reviews/rm40/lra_summary.log`, and the law-M control of the registered study: with `g_A` declared and a slow observation mechanism, an arm-mean bias of −0.53 and −0.54 SD |
| [composite indicator](#observational-missing-data-the-composite-indicator) | the composite mechanism `g_c,a` | the case table above, applied arm by arm to `(W, C_a, C_a Y)` | the complete-outcome probe applies, because this route is the same theorem on a recoded treatment |
| results computed from a fit on these routes: `variable_importance`, E-values, truncation curves, the full-refit bootstrap | inherited from the fit | as for the fit | none |

### A data-adaptive mechanism

RM40's pre-declaration probe found a large DR-TMLE bias with a gradient-boosting mechanism and a
wrong outcome regression on law L. The probe `reviews/rm41/probe.py` finds the cause. Its log is
`reviews/rm41/probe.log`. It is a probe and not a registered study.

| setting | value |
| --- | --- |
| law | law L |
| outcome regression | a logistic GLM on `(A, W1)`, which is wrong |
| mechanism | `HistGradientBoostingClassifier` with 100 iterations, 15 leaves, 20 rows per leaf and no early stopping |
| reduced learners | cubic polynomials in the scalar design |
| fit | `g_bounds=(0.01, 0.99)`, `guard=("Q", "g")`, `reduction="univariate"` |
| size | n = 1,000, 4,000 and 16,000, with 200 draws each |

The results apply to this tuning on law L. The probe does not measure other learners or laws.
Each number below is an ATE bias divided by the Monte Carlo SD of its cell, unless a row names
another SD. The coverage readings have a Monte Carlo SE of at most 0.035. The independent solver
keeps only fits whose scores are below 1e-8. At one mechanism fitted on a separate draw, it keeps
98, 187 and 198 of 200 fits at n = 1,000, 4,000 and 16,000.

**The bias is a property of the method, not of the code.** An independent solver of equations
(8) to (10) shares no code with `cleverly`. The checks below compare it with `DRTMLE`.

| check | result at n = 1,000, 4,000 and 16,000 |
| --- | --- |
| `DRTMLE` against the independent solver at one mechanism fitted on a separate draw, paired by draw | a mean difference of −0.06, −0.06 and −0.07 SD |
| the two `update_order` values at that mechanism | a mean difference of −0.01, −0.02 and −0.06 SD |
| the population limit of the solver at that mechanism, against the bias of the solver on the same draws, both divided by the SD of the solver | +1.56, +1.70 and +1.13 SD, against +1.51, +1.66 and +1.03 SD |
| the independent solver without corrections, against `TMLE` | equal to 1.7e-12 over 1,800 estimates |

**The bias has two parts.** The first part is in the population limit. The second part comes from
the rows that the mechanism was fitted on.

| fit | bias at n = 1,000, 4,000 and 16,000 | coverage |
| --- | --- | --- |
| `DRTMLE`, `cross_fit=True` (the default) | +1.53, +2.01, +1.51 | 0.75, 0.63, 0.81 |
| `DRTMLE`, `cross_fit=False` | −1.22, −2.28, −2.50 | 0.75, 0.41, 0.41 |
| `TMLE`, the same learners, `cross_fit=True` | +0.42, +0.32, −0.05 | 1.000, 0.995, 0.980 |
| `TMLE`, the same learners, `cross_fit=False` | −0.18, +0.24, +0.34 | 0.93, 0.96, 0.98 |

**Part 1: the reduction conditions on the fitted index.** Write `D = Q̄_0 − Q̄`. Equation (9)
removes the part of the remainder that is a function of `g_n`, through
`Q_r° = E_0[D | A = 1, g_n(W)]`. The exact leftover is

```text
P_0[(g_0 − g_n)/g_n · (D − Q_r°)] = E_0[ Cov(g_0, D | g_n) / E_0(g_0 | g_n) ]
```

This term is linear in the spread of `g_0` inside the level sets of `g_n`. It is zero under H2.
The probe splits it into the page's `R_{3,n}` and `R_{4,n}`. It uses a mechanism fitted on a
separate draw, so that no row enters its own mechanism.

| cell | term | bias at n = 1,000, 4,000 and 16,000 |
| --- | --- | --- |
| `Q_r` fitted on `g_0` instead of `g_n` | `R_{3,n}`: the case-table row for a bias of order `r_n` along `F/g_0` | +0.66, +0.50, +0.26 |
| `Q_r` on `g_n` minus `Q_r` on `g_0`, paired by draw, divided by the SD of the in-sample `DRTMLE` fit | `R_{4,n}`: `Q_r` converges to a regression on `g_n`, not on `g_0` | +0.50, +0.95, +0.91 |
| `Q_r` replaced by its population version on `g_n` | the fitted error of `Q_r` | none at n = 4,000 and 16,000: +1.82, +1.78, +1.05 against +1.51, +1.66, +1.03 |

This part is in the population limit at the fitted mechanism. Cross-fitting, nested reductions and
the update order do not remove it.

**Part 2: the own-row term.** An in-sample mechanism and a mechanism fitted on a separate draw have
the same law. Their population terms are therefore equal in distribution. The mean paired difference on
the same rows therefore estimates the mean of the empirical-process term. The table divides it by the SD of
the in-sample `DRTMLE` fit.

| paired difference | n = 1,000 | 4,000 | 16,000 |
| --- | --- | --- | --- |
| own-row term of DR-TMLE | −2.30 | −3.92 | −4.10 |
| own-row term with `Q_r` fitted on `g_0` | −1.82 | −1.60 | −0.98 |
| own-row term of plain TMLE | −0.24 | −0.01 | +0.34 |

At n = 16,000 most of the own-row term acts through `Q_r` fitted on the in-sample `g_n`. The rest
acts through the other steps, mainly equation (10).

In equation (10), `g_{r,2}` regresses `(1_a − g_n)/g_n` on `Q̄_n`. A least-squares
fit with an intercept keeps the mean of its target. The mean of that target is −0.147, −0.062 and
−0.023 on the training rows, and +0.253, +0.067 and +0.017 on fresh rows. Both means are zero at
`g_0`. So equation (10) corrects in the wrong direction in sample.

Theorem 1's Donsker condition is sufficient for this term to vanish, and a fit cannot check it. The measured term is consistent
with its failure.

**What this does not show.** Plain TMLE stays within 0.42 SD for the ATE on law L. Its own leftover
is first order when the outcome regression is wrong, so its interval is not established in this
case either. On law L its parts nearly cancel. This page does not recommend plain TMLE.

**Why no fix ships.** A reduction on another index is a different estimator, and it needs its own
proof. No fit can tell which case-table row it is in, so the status stays `influence_curve`, and
no warning can separate the rows.

### A parametric mechanism makes the interval conservative

Take a logistic maximum-likelihood fit of `g` with the design `X`. Then
`g_n − g_0 ≈ g_0(1 − g_0)X'(β_n − β_0)` and `β_n − β_0 ≈ I^(−1)P_n S`, with the score
`S = X(A − g_0)`. So `R_{3,n} ≈ −c'I^(−1)P_n S`, with `c = P_0[F(1 − g_0)X]`. In the outcome-wrong
branch `D_Y` tends to zero, because `g_{2,0,r} = 0` at `g = g_0`. Then:

```text
Cov(D*, S)   = P_0[(1 − g_0)(Q̄_0 − Q̄)X]
Cov(D_A, S)  = P_0[Q̄_{0,r}(1 − g_0)X]
Cov(D^{*,#}, S) = c
```

The estimator's curve is therefore `D^{*,#} − c'I^(−1)S`. That is the `L_2` residual of `D^{*,#}`
after its projection on the score span. Its variance is `Var(D^{*,#}) − c'I^(−1)c`, and the
reported variance is `Var(D^{*,#})`. Arm 0 flips both signs, so the ATE curve is again a residual.
Equation (9) moves `g` along `Q_r/g`, a function of `g_n`, so it adds second-order terms only.

| fact | reason |
| --- | --- |
| the conservative case does not need a consistent reduction | `D* − D_A(Q_r)` has mean zero at every `g` in the model, for any fixed limit `Q_r` |
| cross-fitting gives the same projection | the fold scores average to the full-sample score: `(1/V) Σ_v P_{n,−v} S = P_n S` |
| the simultaneous band is conservative too | the reported covariance exceeds the true one by a positive semidefinite matrix. A symmetric convex set then has at least its nominal Gaussian probability (Anderson's inequality) |

The steps are standard: a linearization of the maximum-likelihood fit, a fixed-dimension stacked
equation, and an `L_2` projection. The derivation is this repository's. In the probe, the
in-sample SE ratio after the projection is 0.977 to 0.983 at n = 1,000 to 16,000. The
cross-fitted cell has an SE ratio of 1.231 and a projected SE ratio of 1.047 at R = 400. In the
registered study, the projected SE ratio is 0.994 in sample and 1.012 cross-fitted.

### The bivariate reduction

The bivariate reduction keeps `Q_r`, so its outcome branch is the one above. Its mechanism branch
has the same linear structure. Appendix A, in the display of `R_{5n}` for that construction, gives
`R_{5n} = −P_0{(g_{0n,r} − g_{0,r})/g · (Q̄_n − Q̄_0)}`, with `g_{0n,r} = E_0[A | Q̄_n, Q̄_0, g]`.
That conditioning set includes `g`, so its H1 condition differs from the one above. The H1 column
above is for the univariate reduction only.

### The release claim

The interval is valid when both nuisances are consistent at the product rates, under DR-TMLE's own reduced-regression
and solve conditions. With one nuisance inconsistent, the two extra equations remove the part of
the first-order remainder that is a function of the consistent nuisance. The rest, `R_{3,n}` or
`R̃_{5,n}`, can stay first order.

The one-wrong interval is valid for a declared known mechanism and
conservative for a correctly specified parametric mechanism. With the mechanism wrong, the interval is not established, even for a correctly specified parametric outcome regression. It is also valid where the consistent
estimate keeps the level sets of its truth, or where H1 holds. No fit can check the last two. A
constructed level-set-breaking error produced a bias that grows with `n` in the registered study.
On law L, one gradient-boosting mechanism with a wrong outcome regression produced a bias of +1.5
to +2.0 SD cross-fitted and −1.2 to −2.5 SD in sample
([a data-adaptive mechanism](#a-data-adaptive-mechanism)).