# Natural-extension verdicts

This page records the source audit of thirty-one compositions that the package refused or did not
implement. The audit applies the [Eligibility](../roadmap.md#eligibility) rule of the roadmap. A
part qualifies when each step is a standard step and no source records an objection. An objection
is a source that records a step as open or as a defect. A step that needs a new kind of limit
theorem is an objection too.

A verdict is not a capability. "Qualifies for implementation" means that the owning roadmap row may
implement the part. The part ships only when that row delivers its contract, its nonzero witnesses,
and its registered study. The verdict column marks each part that ships now.

The roadmap held this review as one row until the review was complete.
[The roadmap at commit `6a55f038`](https://github.com/esbraun/cleverly-tmle/blob/6a55f0388b703765eab79c78efec82474e67ab8e/docs/roadmap.md#x18-natural-extension-reviews)
records the questions as they were asked. Each owning row now holds the full contract of its part.
A row that delivers a part updates the last column of the table below.

## The terms

| term | meaning |
| --- | --- |
| refusal | what the package does today for the composition |
| base result | the published result that the extension starts from, with its exact locator. [References](../references.md) records each first-hand read |
| step | an argument that carries the base result to the composition. Each step names its row in the standard-step table of [Eligibility](../roadmap.md#eligibility) |
| objection search | the sources read for a recorded objection, and what they say |
| owning row | the roadmap row that implements the part, or the page that states the shipped contract |

## The verdicts

| part | refusal | base result | steps | objection search | verdict | owning row |
| --- | --- | --- | --- | --- | --- | --- |
| (a) observational treatment with a missing outcome, `DRTMLE` | none; the refusal is removed | Benkeser, Carone, van der Laan and Gilbert (2017), Section 3.2, Theorem 1 | an indicator reduction to $C_a = \Delta_A \Delta_Y 1\{A=a\}$. The mechanism is the composite $P(C_a = 1 \mid W)$, one binary regression for each arm | Benkeser et al. (2017) and Díaz and van der Laan (2017) record no objection. `drtmle` 1.1.2 uses the same indicator | shipped | [the composite contract](dr-tmle/theorem.md#observational-missing-data-the-composite-indicator) |
| (b) a missing treatment | none; `treatment_delta=` declares it | as part (a) | the indicator reduction of part (a). The missingness of $A$ may depend on $A$ and $W$, because the composite mechanism is regressed directly | none found | shipped, for targets that read the arm indicator | [the composite contract](dr-tmle/theorem.md#observational-missing-data-the-composite-indicator) |
| (c) a known stochastic categorical policy at a longitudinal node | none; a `Stochastic` node declared `density_kind="known"` expresses it | Díaz, Williams, Hoffman and Schenck (2023), Section 2, journal page 849, and Theorem 3, page 853 | none for a recorded randomizer, which Section 2 places in $L_t$. For the integrated estimator, the chain rule with a known weight: the curve is the conditional expectation of the Theorem 3 curve given the data without the randomizer | page 850 says multiply robust estimation "is not generally possible" for a regime that depends on P. That objection does not apply to a known policy density | shipped, for a known policy density that does not read the natural value $A_t$ | [known stochastic policies](longitudinal-tmle.md#known-stochastic-policies) |
| (d) a missing outcome with three or more arms, `DRTMLE` | none; the refusal is removed | Díaz and van der Laan (2017), Theorem 2 | an indicator reduction to $1\{A=a\}$ for each arm, a fixed-dimension stack, linearity, and the delta method | none. Each arm's Theorem 2 reads only its own column, so the tilts of the arms need no shared mechanism | shipped. Two arms keep the shipped binary estimator | [DR-TMLE, more than two arms](dr-tmle/theorem.md#more-than-two-arms) |
| (e) PAR and PAF with a missing outcome | none; the refusal is removed for ordinary TMLE and, for point estimates, for the per-arm outcome-adaptive `CTMLE` (part (o)). `DRTMLE`, the C-TMLE selector strategies and the shared outcome-adaptive design keep theirs ([X50](../roadmap.md#x50-further-dr-tmle-compositions) part (c) and [X51](../roadmap.md#x51-c-tmle-point-estimates-for-further-targets) part (a)) | Díaz, Carone and van der Laan (2016) for the natural-course mean, and the shipped missing-outcome arm mean | a fixed-dimension stack, linearity for PAR, and the delta method for PAF | the earlier roadmap sentence was a policy sentence and not a recorded objection | shipped | [PAR and PAF with missing outcomes](point-treatment-tmle.md#par-and-paf-with-missing-outcomes) |
| (f) cross-fitted clustered `LTMLE` | none; the refusal is removed | Díaz et al. (2023), Section 5.2, page 852, and Theorem 3, page 853 | a cluster as the unit, with whole-cluster folds and a cluster-summed curve | F22 recorded an absence of a source, not a defect | shipped, with the rules of part (g). Cross-fitted `msm=` with `id=` ships with part (l) | [longitudinal clusters](longitudinal-tmle.md#clusters) |
| (g) clustered intervals at unequal cluster sizes, and at fewer than 40 clusters | none; the `unequal_cluster_plugin` status is deleted. `few_cluster_plugin` applies below 10 positive-mass clusters, or below 20 for in-sample and cross-fitted `LTMLE` | Wang, Park, Small and Li (2024), Assumption 1(b) and Theorem 4(b). Nugent et al. (2024), Section 2.2, for the $t$ reference with $J-2$ degrees of freedom | a cluster as the unit, with the cluster size a bounded random cluster attribute, and the delta method for the ratio of cluster sums. The $t$ reference adds no limit theorem | Wang et al. (2024), Remark 3, recommend machine learning at about 100 clusters and parsimonious parametric models at about 20. That is a finite-sample warning. The registered few-cluster evidence uses parametric nuisance learners only | shipped. Two refinements of the contract: a derived estimate takes the smallest degrees of freedom of its inputs, and `plugin_interval` follows the reference of `ci` on an inferential fit and stays normal on a `few_cluster_plugin` fit. The interval floors match the smallest counts each fit's registered study measures. [X54](../roadmap.md#x54-small-sample-evidence-for-clustered-intervals-and-boundary-nodes) holds 4 to 9 clusters, or 4 to 19 for `LTMLE`. For a fold-evaluated point-treatment report, the cluster variance is centered within each fold, and each validation fold needs 2 positive-mass clusters | [clusters](inference.md#clusters) |
| (h) plug-in omitted-variable limits with a declared parametric mechanism | [F26](../roadmap.md#f26-confidence-limits-of-the-plug-in-omitted-variable-bound) | Chernozhukov, Cinelli, Newey, Sharma and Syrgkanis (2026), Theorem 4 | a stacked estimating equation of the mechanism score and the second-moment equation, then the chain rule and the delta method | F26 records no curve for an arbitrary learner. That case stays refused | qualifies for implementation, only when the fit's own treatment learner is an unpenalized logistic or multinomial model | [X26](../roadmap.md#x26-plug-in-omitted-variable-limits-with-a-declared-parametric-mechanism) |
| (i) stratified incremental targets, stratified nonlinear or continuous MSMs, and `DRTMLE` with strata | none; the refusals are removed | Kennedy (2019), arXiv v3, Section 3.1 and Theorem 2; the [MSM projection](msm-projections.md) contract; Benkeser et al. (2017), Theorem 1 | a finite partition into a fixed number of baseline strata | the earlier note on dose strata is a design note, not a theory gap. Kennedy (2019), arXiv v3, Section 6, page 20, lists "estimation of how mean outcomes under different interventions vary with covariates (effect modification)" as future work. That is an unaddressed conditional curve, not a recorded defect of a fixed finite partition | shipped. The fold-evaluated fit qualifies for implementation, and it stays refused until [X28](../roadmap.md#x28-fold-evaluated-cv-tmle-with-baseline-strata) delivers it | [point-treatment TMLE, baseline strata](point-treatment-tmle.md#weights-strata-and-clusters) |
| (j) a `discrete` C-TMLE fit whose one declared candidate is the full adjustment set | the `working_mechanism_plugin` status | the ordinary TMLE interval result | none. The fit selects nothing, so the estimator is TMLE | none | shipped | [collaborative TMLE](collaborative-tmle.md) |
| (k) the outcome-adaptive C-TMLE design with one scalar design for each arm, in sample and cross-fitted, with fixed weights and repeated splits | the `generated_design_plugin` status; `oat_design="per_arm"` is the default design and reports its point estimate | Benkeser, Cai and van der Laan (2020), Theorem 1, and Appendix D, "Cross-validated CTMLE", printed page 26 of the author preprint | an indicator reduction to $1\{A = a\}$ for each arm (the Remark, p. 10), a fixed-dimension stack, linearity and the delta method; cross-fitted nuisances with one pooled coefficient for each arm, as Appendix D states; the weight-tilted law; the median over splits | the base result's condition (v) fails for an estimated outcome regression, so its remainder $R_{24}$ is first order. The paper's Section 4.1 and Discussion report standard errors that underestimate the variability, and the registered study `ctmle-oat-per-arm` measured them 8 to 24 percent too small | does not qualify for an interval. The point estimate ships | [the applied revert](collaborative-tmle.md#the-applied-revert); [F19](../roadmap.md#f19-outcome-adaptive-c-tmle-generated-design-inference) |
| (l) cross-fitted longitudinal `msm=` | none; the refusal is removed | Díaz et al. (2023), Section 5.2, page 852, and Theorem 3, page 853, for each regimen and horizon cell; the shipped in-sample [longitudinal projection](msm-projections.md#the-longitudinal-projection) | a fixed-dimension stack of the cross-fitted cells, the delta method for the projection, and the chain rule for the stacked update over every follower | the earlier refusal named an evidence gap, not a theory gap. No read source records a defect | shipped | [the longitudinal projection](msm-projections.md#the-longitudinal-projection) |
| (m) `incremental=` on data that declares its treatment mechanism | none; the refusal is removed | Kennedy (2019), arXiv v3, Theorem 2 and Corollary 2, Section 3.3, and Lemma 2 of Section 8.2 for a known stochastic intervention | the chain rule with a known weight: with $g_0$ known, $q_\delta(g_0)$ is a known stochastic regime, and the efficient curve is the full-model curve projected off the mechanism tangent space, which removes the term in $(A-g)$ | Kennedy (2019) states after Corollary 2 that with known propensity scores the efficient influence function is the first weighted average term. That is this step, not an objection | shipped | [known treatment mechanism](point-treatment-tmle.md#known-treatment-mechanism) |
| (n) a discrete `policies=` fit, or a modified treatment policy at a categorical `LTMLE` node, on a declared mechanism | none; the refusals are removed | Díaz, Williams, Hoffman and Schenck (2023), Section 4 (the discrete formula) and Theorem 3, page 853 | the degenerate estimator $g_n=g_0$: the remainder is a product of the ratio error and the outcome error, so it is zero, and the shipped curve at $(\bar Q_\infty,g_0)$ is exact | none. The ratio $g^d/g$ is the only place the fit reads $g$, and the plug-in reads the observed treatment | shipped | [known treatment mechanism](point-treatment-tmle.md#known-treatment-mechanism) |
| (o) the per-arm outcome-adaptive C-TMLE design with baseline strata, a missing outcome or a missing treatment | none; the refusals are removed for point estimates. Every such fit takes the `generated_design_plugin` status, and a cross-fitted fit with a missing treatment stays refused | Benkeser, Cai and van der Laan (2020), Theorem 1, as part (k) ships it | (g) a finite partition with a stratum-local design; (h) the composite-indicator reduction to $C_a = \Delta_A \Delta 1\{A = a\}$ for a missing outcome, with each factor regressed on $\bar Q_n(a, W)$; (i) the same reduction with a treatment observation factor, in sample; (j) the natural-course mean with $\Delta$ as the treatment and the design $(A, \bar Q_n(A, W))$ | Appendix C, preprint page 24, ends: "We leave to future work the identification of general strategies for selecting adaptive censoring mechanism target parameters." Steps (h) to (j) are not such a general strategy: each data structure has the form of Theorem 1, and the target is the same projection, refined by a finite partition. The objection of part (k) applies to the interval | shipped (point estimate). Its registered study, [`ctmle-oat-per-arm-strata-missing`](method-evidence/outcome-adaptive-per-arm-c-tmle-strata-missing.md), pairs all 31 rows with R `drtmle` and passes every `robustness_contract` cell. [F19](../roadmap.md#f19-outcome-adaptive-c-tmle-generated-design-inference) owns its red truth rows and its three red ladder rungs | [baseline strata and missing data in the per-arm design](collaborative-tmle.md#baseline-strata-and-missing-data-in-the-per-arm-design); [F19](../roadmap.md#f19-outcome-adaptive-c-tmle-generated-design-inference) |
| (p) the sequentially doubly robust estimator `SDR`, cross-fitted, with survival, competing risks, known policies, modified treatment policies, weights, clusters, contrasts, bands and `g_bounds` | none; a new estimator | Díaz, Williams, Hoffman and Schenck (2023), Equation (4), Theorem 2, and Section 5.3, page 854: Lemma 3, Lemma 4 and Theorem 4 | (1) survival and competing risks: page 850 places $Y_t$ in $L_{t+1}$, and a cause's incidence is the indicator reduction, so the node target is $Y^{(j)}_t + (1 - Y_t)\varphi_{t+1}$. (2) known policies and modified treatment policies: the randomizer integrated out by linearity. (3) weights: the chain rule with a known weight. (4) clusters: a cluster as the unit, with whole-cluster folds. (5) contrasts and bands: a fixed-dimension stack. (6) cross-fitting: the proof template of Theorem 3 (supplement Section 6) with Lemma 3, applied to a training-fold recursion that is fixed given its training rows. A continuous node's fold density is the one exception: its bin edges are quantiles of the dose over every row at risk. (7) `g_bounds`: a fixed truncation of the estimated partial path probability, a deterministic map of $\hat r$ | the printed Step 2 and the `lmtp` construction are not fold-measurable, so the shipped fit uses neither. Rotnitzky, Robins and Babino (2017), after Equation (72), call the rate of an in-sample transform's regression a conjecture. The supplement prints no proof of Theorem 4 | shipped, with an interval under the Theorem 4 rate condition. Where a bound binds, no SDR property is claimed on that path | [the sequentially doubly robust estimator](longitudinal-tmle.md#the-sequentially-doubly-robust-estimator) |
| (q) `SDR` with `msm=` | none; a new composition | part (p), and the shipped [longitudinal projection](msm-projections.md#the-longitudinal-projection) | the conditional form of Lemma 1 at $H_1$, whose proof is a recursion of conditional identities (supplement Section 4), with a baseline $V$. Then the projection's stacked estimating equation, with bracket $\varphi_{1,c} - m(c, V; \beta)$, and the delta method | no published result states the composition. A transform can leave $[0, 1]$, so a logit or log link can have no finite solution | shipped for the identity link. A logit or log link ships when its projection converges, and raises `CapabilityError` otherwise | [MSM projections](msm-projections.md) |
| (r) a different $\delta_t$ at each node of `IPSI` | none; a new estimator | Kennedy (2019), arXiv v3, Theorem 2, and Appendix 8.2, Lemmas 3 and 4. Kim, Kennedy and Naimi (2021), Appendix F, proves the lemmas | the chain rule for influence functions, node by node: Lemma 4 at each node's own $\delta_t$ | none. Theorem 1 already admits any $\delta$ in a compact subset of $(0, \infty)$ | shipped | [incremental interventions over time](longitudinal-tmle.md#incremental-interventions-over-time) |
| (s) tilt nodes beside label, rule and known-policy nodes in one `IPSI` plan | none; a new estimator | Kennedy (2019), Appendix 8.2, Lemma 2 for a known stochastic intervention and Lemma 3 for one that reads $P$; Kim et al. (2021), Appendix F, Lemmas F.2 to F.5 | the chain rule with a known weight: Lemma 2 covers each node whose density is known, and Lemma 3 adds the term only at the nodes that read $P$ | Kim et al. (2021), Appendix F, call the intervention distribution "completely general". Díaz et al. (2023), page 850, concerns multiply robust estimation, which `IPSI` does not claim | shipped. A label or rule node keeps its positivity condition. A modified treatment policy or a risk-ratio tilt beside a tilt stays refused | [incremental interventions over time](longitudinal-tmle.md#incremental-interventions-over-time) |
| (t) survival and competing risks under a tilt | none; a new estimator | Kennedy (2019), Remark 1, and Kim et al. (2021), Section 2, which place earlier outcomes in the history | an indicator reduction for one cause at a time. After an absorbing event both arms predict the same value, so every node term vanishes there | none found | shipped. The risk curve can be non-monotone across horizons | [incremental interventions over time](longitudinal-tmle.md#incremental-interventions-over-time) |
| (u) observation weights on `IPSI` | none; a new estimator | Kennedy (2019), Theorem 3; Kim et al. (2021), Theorem 5.1 | the chain rule with a known weight: the tilted law of the weighted population, weighted regressions, and $w\varphi$ | none found | shipped | [incremental interventions over time](longitudinal-tmle.md#incremental-interventions-over-time) |
| (v) clusters on `IPSI`, in sample and cross-fitted | none; a new estimator | Kennedy (2019), Theorem 3 and Algorithm 1 | a cluster as the unit, with whole-cluster folds and a cluster-summed curve | none found | shipped, with the few-cluster rules of `LTMLE` | [incremental interventions over time](longitudinal-tmle.md#incremental-interventions-over-time) |
| (w) contrasts, horizons and a band over a declared grid of plans | none; a new estimator | Kennedy (2019), Theorems 3 and 4; Kim et al. (2021), Theorem 5.1, the joint process over $\delta$ and $t$ | a fixed-dimension stack. A finite grid of plans and horizons is a subset of the index set of the joint process | none. A band over the continuum of $\delta$ is not offered | shipped for the declared grid | [incremental interventions over time](longitudinal-tmle.md#incremental-interventions-over-time) |
| (x) `IPSI` with `msm=` over the plan set | none; a new composition | part (w), and the shipped [longitudinal projection](msm-projections.md#the-longitudinal-projection) | a stacked estimating equation for the projection, and the delta method | as part (q): a one-step mean can leave $[0, 1]$, so a logit or log link can have no finite solution | shipped for the identity link. A logit or log link ships when its projection converges, and raises `CapabilityError` otherwise | [incremental interventions over time](longitudinal-tmle.md#incremental-interventions-over-time) |
| (y) a tilt with a declared treatment mechanism, on `IPSI` and on `LTMLE` | `LTMLE` refuses `incremental=` and a tilt node with an estimated mechanism | Kennedy (2019), Section 3.3, the known-propensity statement after Corollary 2, and Appendix 8.2, Lemma 2 | the chain rule with a known weight, as in part (m): with $\pi_t$ known, $q_t$ is a known stochastic policy and the mechanism term is absent | none | shipped. `LTMLE` translates the tilt to a known policy and refuses a fit where `g_bounds` moves the declared mechanism | [incremental interventions over time](longitudinal-tmle.md#incremental-interventions-over-time) |
| (z) the mean of a repeated outcome at each horizon, `LTMLE` and `IPSI` | none; `outcome=Repeated([...])` declares it | Bang and Robins (2005); van der Laan and Gruber (2012), Theorem 2, page 13, and Section 4, page 19; for `IPSI`, Kim, Kennedy and Naimi (2021, arXiv v3), Theorems 3.1, 4.1 and 5.1 | none for one horizon: horizon $h$ is the end-of-study parameter of the panel truncated after $h$, and $Y_s$ for $s < h$ is a covariate of node $s + 1$ in time order. A fixed-dimension stack over the horizons and plans, linearity, the delta method, and a stacked estimating equation for `msm=` | none found. Kim, Kennedy and Naimi (2021, arXiv v3) treat the joint law over horizons and tilts: Theorem 5.1 gives a Gaussian process over $(\delta, t)$. The package reports the finite grid of declared tilts and horizons | shipped for `LTMLE` and `IPSI`. `SDR` refuses until X32b | [repeated outcomes](longitudinal-tmle.md#repeated-outcomes) |
| (aa) sequential g-computation and IPW, with survival, competing risks, known policies, weights, clusters and contrasts | none; new estimators | Bang and Robins (2005), Section 4, p. 969, for the recursion; Stefanski and Boos (2002), Section 2, for the stacked estimating equation | (1) the stacked score equations of the `GLM` fits and the estimate equation, with a block-triangular Jacobian. (2) the chain rule through the cumulative ratio for IPW. (3) the delta method for the Hajek forms, the hazard product and the Aalen-Johansen sum. (4) the chain rule with a known weight for a policy node. (5) the branch derivative of a modified treatment policy, for g-computation only. (6) a cluster as the unit | no regression reads an estimated index, so the linear-remainder pattern does not apply. With a data-adaptive learner no result gives asymptotic linearity over several time points. Ertefaie, Hejazi and van der Laan (2023), Theorem 1, covers one time point only | shipped, with an interval when every regression that the estimate reads is a `GLM`. Otherwise `adaptive_baseline_plugin`, which [F35](../roadmap.md#f35-inference-for-sequential-baselines-with-data-adaptive-nuisances) owns | [sequential g-computation and IPW](sequential-baselines.md) |
| (ab) C-TMLE0 derivative columns, `CTMLE(strategy="derivative")` | none; the strategy is new | van der Laan and Rubin (2006), Sections 1, 3 and 4.2; Zheng and van der Laan (2011), Theorems 1 and 2, for the cross-fitted fit | none beyond the base. The extra derivative columns enlarge the submodel, and the clever covariate of each arm stays in its span, so the update solves the efficient influence curve equation. The indicator reduction for each arm and the chain rule are the steps the shipped `TMLE` uses | Ju, Wyss et al. (2019), Section 4.2, assert asymptotic linearity by citation only. Algorithm 2 of the same paper (reading D1), display (33) of Ju, Chambaz and van der Laan (2018) (reading D2), and the sum form of Algorithm 2 (reading D5) are printing defects. The goal equation $P_n d_h D^* = 0$ resolves each one. The collaborative claim, an interval under an inconsistent outcome regression, stays with [F18](../roadmap.md#f18-selector-path-c-tmle-inference) | qualifies, and ships. The record restates the base result and adds no step | [C-TMLE0](collaborative-tmle.md#c-tmle0-the-derivative-covariate) |
| (ac) the density route of a modified treatment policy on a vector with a dose, at a point treatment and at a longitudinal node | none; `ratio="density"` ships with this part. `ratio="classifier"` rests on published support: Díaz et al. (2023), Section 2 for a vector $A_t$, Theorem 1, Section 5.4 and Theorem 3 | Díaz et al. (2023), Equation (3) and the discrete formula of Section 4, with Theorem 3, page 853 | (1) the chain rule for the joint law: the categorical components first, then the doses in a declared order. (2) a finite partition over the categorical preimages of a cell. (3) Equation (3) applied to each conditional factor of a dose, one dose at a time. Each step reuses the base argument for a scalar $A_t$ | Section 2 lets $A_t$ be a vector. Assumption 4 and Equation (3) are written for a scalar map, and Hoffman et al. (2024), Introduction, cover several exposures and require a piecewise smooth invertible map for a continuous one. No source read records the vector step as open or as a defect. A joint map that is not triangular is outside this part, because a determinant checked on a grid does not establish that the map is one-to-one | shipped, for dose maps that read the covariates, the natural categorical components and the doses before them in a declared order | [vector treatments](point-treatment-tmle.md#vector-treatments) and [vector nodes with a continuous component](longitudinal-tmle.md#vector-nodes-with-a-continuous-component) |
| (ad) the parametric g-formula by forward simulation, with survival, competing risks, known policies, tilts, modified treatment policies, a fixed or rule-set continuous dose, weights, clusters and contrasts | none; a new estimator | Robins (1986), equation (4.7), Section 4E, p. 1423, for the formula, and Section 4I, pp. 1426 to 1427, for the Monte Carlo evaluation; Stefanski and Boos (2002), Section 2, for the stacked estimating equation | (1) the stacked score equations of the `GLM` fits, with a block-diagonal Jacobian. (2) the observation augmented with its uniforms, which makes the pairs independent and identically distributed. (3) the score-function identity for a drawn variable and the pathwise derivative for the outcome mean and the hazards. (4) stochastic equicontinuity of the simulated estimating equation: Pakes and Pollard (1989), Lemma 2.17, p. 1037, and Theorem 3.3, p. 1040, and Newey and McFadden (1994), Theorem 7.2, p. 2186. (5) the delta method for a fixed or rule-set dose inside the parametric model. (6) a cluster as the unit | no published analytic variance or Monte Carlo accounting exists for the simulation form. McGrath et al. (2020) and `gfoRmula` 1.1.1 report the bootstrap, and Zivich et al. (arXiv 2306.10976v3) give a sandwich variance for the iterated conditional expectation form only. Theorem 7.3, p. 2188, of Newey and McFadden does not cover a discontinuous simulator. Pages 2111 and 2112 of that chapter and the proofs of its Section 7 are not read | shipped, with an interval when every model that the estimate reads is a `GLM`. Otherwise `adaptive_baseline_plugin`. The repeated-sampling evidence is the registered study `parametric-gformula` | [parametric g-formula](parametric-gformula.md#why-the-argument-holds) |
| (ae) a categorical odds tilt: `Incremental({level: factor})` at a point treatment with three or more levels and at a tilt node of `IPSI` | a scalar `Incremental` on other than two levels (`invalid_declaration`, which replaces the earlier `implementation_gap`) | Kennedy (2019), arXiv v3, Equation (1), Theorem 1, Theorem 2 and Appendix 8.2, Lemmas 2 to 4. Kim, Kennedy and Naimi (2021), Theorem 3.1, Theorem 4.1 and Appendix F | (1) a finite partition of the treatment into $K$ levels, which Theorem 1 admits without a new proof. (2) the chain rule through $g$, which is the argument of Lemma 4. For level $b$ the mechanism score is $1\{A = b\} - g_b$ and $\partial m / \partial g_b = \delta_b\{\bar Q_b - m\}/D$. The sum is $(\delta_A/D)(\bar Q_A - m)$. (3) Lemma 3, which Kim et al. (2021), Appendix F, prove for a general $Q$, so the recursion and the dropout factors are unchanged | No source read records a step as open or as a defect. A risk-ratio tilt and a cost-based reference policy are different interventions, and the package does not substitute them | shipped. The registered study covers the support rows `T35` at a point treatment and `G43` over time. Any other composition, such as weights, clusters or `delta=`, has the support label `experimental`. The evidence is Python property evidence: exact-law checks of the estimate, the curve and the remainder, a longhand, and mutation controls. No external comparator exists | [point-treatment incremental interventions](point-treatment-tmle.md#incremental-propensity-score-interventions) and [incremental interventions over time](longitudinal-tmle.md#incremental-interventions-over-time) |

## The conditions each part inherits

| part | inherited conditions |
| --- | --- |
| (a), (b) | $Y(a) \perp A \mid W$; $Y(a) \perp \Delta_A \mid A, W$; $Y \perp \Delta_Y \mid A, \Delta_A = 1, W$; composite positivity $P(C_a = 1 \mid W) > \delta$; `W` complete on every row; the Theorem 1 rate conditions |
| (c) | the Theorem 3 rate condition; bounded density ratios; positivity of $g_t$ wherever $q_t > 0$; a policy density that is known and fixed before the fit |
| (d) | randomized treatment or known treatment probabilities; the Theorem 2 conditions for each arm |
| (e) | the conditions of both parents; a PAF denominator bounded away from zero |
| (f) | independent clusters; no interference between units; bounded cluster size; the Theorem 3 conditions in the cluster count; the rules of part (g) |
| (g) | independent clusters; bounded random cluster size; cluster size as a covariate when it confounds |
| (h) | a correctly specified mechanism model; smoothness; an invertible Jacobian; a rule for rows that `g_bounds` clips |
| (i) | the conditions of each marginal result inside each stratum; a fixed number of strata |
| (j) | the conditions of the ordinary TMLE |
| (k) | the Theorem 1 conditions (i)-(vi): the curve equation solved, $\bar Q_n$ and $Q^*_n$ at $o_P(n^{-1/4})$, a treatment learner consistent for $P(A = a \mid \bar Q_n(a, W))$ at $o_P(n^{-1/4})$, the empirical-process condition (in sample only; cross-fitting replaces it), a bounded derivative of the projection in $\bar Q_0$, and a negligible term from fitting $G$ on the initial $\bar Q_n$; the projection inside `g_bounds`; iid rows or fixed known weights; a consistent outcome regression |
| (l) | the Theorem 3 rates and bounded density ratios in every cell; a fixed number of cells; a full-rank realized design; with `id=`, the rules of parts (f) and (g) |
| (m) | a declared mechanism equal to the design mechanism; positivity of $g_0$ on the arms $q_\delta$ charges; the conditions of the known-regime TMLE |
| (n) | a declared mechanism equal to the design mechanism; positivity of $g_0$ on every level the policy sends a unit to; the Theorem 3 conditions of the outcome regression |
| (o) | for the point estimate: an outcome regression consistent in $L_2(P_0)$; the factor bounds `g_bounds` and `nuisance_bound`; the composite identification conditions of parts (a) and (b); positivity $P(C_a = 1 \mid W) > 0$; a fixed number of strata, each with positive mass. An interval would also need the conditions of part (k), which [F19](../roadmap.md#f19-outcome-adaptive-c-tmle-generated-design-inference) holds |
| (p) | identification (Theorem 1); the Theorem 2 conditions; the Theorem 4 rate $\sum_t \lVert \hat r_t - r_t \rVert \lVert \check m_t - \tilde m_t \rVert = o_P(n^{-1/2})$, with $\tilde m_t$ the regression of the estimated transform; bounded ratios; a fixed number of folds. In sample, condition DC of Luedtke et al. (2017). Efficiency also needs every nuisance consistent |
| (q) | the conditions of part (p) in every cell; a fixed number of cells; a full-rank realized design; a baseline $V$ |
| (r), (s) | the Theorem 3 conditions at every tilt node; under dropout, Kim et al. (2021), Assumption A3 and the Theorem 5.1 conditions; positivity at each label and rule node; a policy density known and fixed before the fit at each policy node |
| (t) | the conditions of part (r) at each horizon; a fixed set of horizons and causes |
| (u) | known, bounded weights; the conditions of part (r) on the weighted law |
| (v) | independent clusters; bounded cluster size; the conditions of part (r) in the cluster count; the rules of part (g) |
| (w) | the conditions of part (r) for each plan and horizon; a fixed, finite grid |
| (x) | the conditions of part (w); a fixed number of cells; a full-rank realized design |
| (y) | a declared mechanism equal to the design mechanism; the conditions of the known-policy recursion. On `LTMLE`, a declared mechanism inside `g_bounds` |
| (z) | the end-of-study conditions at each horizon: sequential randomization and positivity through $h$; censoring at $t$ that does not depend on $Y_t$ given the history; a fixed number of horizons and plans |
| (aa) | identification as for `LTMLE`; every `GLM` that the estimate reads correctly specified; a unique root of each score equation; an invertible information; a fixed number of nodes and parameters; positivity at each node, which no check verifies except the refusal of a prediction through an aliased `GLM` column; for IPW, a cumulative ratio that `g_bounds` does not bind at the truth |
| (ab) | the conditions of the ordinary TMLE; both mechanisms inside `g_bounds`; a derivative column that is nonzero and not collinear with the clever covariate of its arm |
| (ac) | the Theorem 3 rate condition on the product of the combined ratio error and the outcome-regression error; bounded density ratios; positivity of each dose in the categorical cell that the policy sends a unit to; strictly monotone piecewise dose maps with a declared inverse and derivative; a policy known and fixed before the fit. The remainder exposes no regression on an estimated index, because the preimage is a known function of observed values. The combined ratio error holds the errors of $K + 1$ factors for $K$ doses, and the density of dose $k$ conditions on $k - 1$ doses, so its learner's rate degrades with $k$ |
| (ad) | identification as for `LTMLE`; every model that the estimate reads correctly specified, including the covariate laws and the draw order within a node; a unique root of each score equation; an invertible information; a fixed number of nodes, parameters and draws; no positivity check beyond the refusal of a prediction through an aliased `GLM` column; for a fixed or rule-set dose, the fitted laws extrapolated to the dose |
| (ae) | the conditions of parts (m) and (r) with the factors $\delta_a$ in a compact subset of $(0, \infty)$; the rate condition $\lVert \hat g - g_0 \rVert(\lVert \hat g - g_0 \rVert + \lVert \hat Q - Q_0 \rVert) = o_P(n^{-1/2})$ in the $L_2(P_0)$ norm over the levels; no treatment positivity at a tilt node; under dropout, Kim et al. (2021), Assumption A3 and the Theorem 5.1 conditions. A declared mechanism removes the mechanism term. The remainder exposes no regression on an estimated index, and no term is linear in one nuisance's error |