Canonical DR-TMLE

This reporting study evaluates the binary complete-data DR-TMLE against the primary simulation law in Benkeser et al. (2017), Section 5.1, and against R drtmle 1.1.2 at pinned commit 538a3a2. The truth and acceptance rules do not come from the R implementation: treatment-specific means are computed by independent quadrature, and the robustness regimes and corrected influence curve come from the paper’s theorem and simulation design.

This row uses a reporting publication policy. Scientific failures remain committed and render red. Unrecorded missing replications, invalid schemas, active bounds, and broken provenance still abort publication. The R harness records a non-finite reference fit as an invalid_result failure, and the failure accounting decides its cells. Thus the tables can show whether Cleverly established equivalence, coverage superiority, or an inconclusive result without selecting only favorable runs.

What was compared

setting	cleverly	R drtmle
data-generating law	binary complete-data law from Benkeser et al. (2017), Section 5.1	identical realized rows
nuisance regimes	outcome correct, treatment correct, and both correct	identical nuisance predictions supplied as Qn and gn
cross-fitting	one deterministic unstratified ten-fold assignment, drawn by random_partition from the sample seed	the identical fold vector through the package’s documented cvFolds route
construction	pooled cross-fitted, univariate reductions, guards Q and g, R-package update order	native drtmle() with the corresponding univariate guards and two Q steps
estimands	ey0, ey1, and ate	the same three quantities, with ATE covariance formed from the native joint covariance
intervals	pointwise 95% Wald	pointwise 95% Wald
alternation rounds	max_outer = 100	maxIter = 100
fit audit	score_check’s own bar, 1e-3 x se / sqrt(n)	the same bar, from the reference’s own reported standard errors

The primary paired route is deliberately difficult. “Superior” requires a positive 99% lower confidence bound for Cleverly minus R coverage, plus truth validity and non-inferiority in RMSE and SE calibration. If that route fails, ordinary bounded similarity and non-inferiority can still establish “equivalent.”

Two rows of that table used to say something else, and both mattered.

The audit bar was 1e-4 on the Cleverly side and 1 / n on the R side. Each was defended against R drtmle’s default, which is the wrong reference for a claim about Cleverly: the library ships 1e-3 x se / sqrt(n), about 3.4e-7 here. The rule is now imported from score_check rather than restated, so the study cannot drift from the library.

The alternation row said max_iter: 100, matching the R runner. That was the cap on the Newton steps inside one fluctuation. The alternation itself ran at a hard-coded 50 that no caller could reach. max_outer is now a keyword and the value that applied is recorded on the fit.

What the solver diagnostics do and do not compare

R drtmle exposes no convergence flag. The runner had no honest value to write for one, and wrote TRUE. The published study then read “24 Cleverly solver failures against 0” off a column the reference could not fail.

solver_reported now marks which side reports a flag at all, and the reference’s solver_passed is left empty. What is comparable is the score audit, and at the shared bar it runs the other way. Each side audits score_audited_fits fits of the same replications. Cleverly fails subject_score_failures of them, and R drtmle fails reference_score_failures. Cleverly’s median score is 2e-11 to 5e-11 against the reference’s 7e-9 to 9e-9.

Every Cleverly failure sits in one regime, the one where the treatment mechanism is misspecified. The reference fails in all three. The measured table below resolves the three counts against tests/canonical/drtmle/fit-diagnostics.csv.

The two implementations reach different roots when the outcome regression is wrong

The two sides receive identical rows and identical initial nuisance predictions, agreeing to 6e-16. Every difference below is targeting.

regime	per-replication disagreement in ate, as a share of one sampling standard deviation
both nuisances correct	4.6%
outcome regression correct	11.9%
treatment mechanism correct	51.7%

The last figure is not a convergence failure on either side. It correlates with neither implementation’s score quality, at Spearman 0.003 and -0.055. Cleverly’s own two documented update orders disagree by about the same amount in the same cell, with both routes solving all three equations to about 1e-11. The three equations do not pin down one answer on this law once the outcome regression is misspecified, and the route decides which answer is reached. Targeting and cross-fitting carries that measurement.

RM19 read the mean of this disagreement on 2,000 fresh draws. Its reading is no increment at the declared resolution. Of the four named choices that the design moves from R to cleverly, only the two-arm tilt geometry has a main effect whose interval excludes 0: 0.000833 to 0.001718, at the level 0.998. The step guards of R also move the estimate of R itself. The declared rule applies no localization label when the primary reading finds no increment. What the localization design found gives each interval.

This is also what limits the paired calibration test in that regime. The published resolution of the calibration statistic tracks the disagreement exactly: 0.0032 to 0.0039 where both nuisances are correct, 0.0094 to 0.0184 where the outcome regression is, and 0.0370 to 0.0468 where only the mechanism is. Every one of the nine now sits inside the declared 0.05 margin, so all nine cells conclude equivalent. The same table on the treatment-stratified split this row used before put one of them at 0.0521, and that cell reported underpowered instead. The margin, the replication count and the statistic did not move. Only the fold draw did, and the cell sits close enough to the margin that the reader should treat it as resolved once rather than as an established property.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
paper binary law, both nuisances correct	ate	average treatment effect	cleverly	-0.000477 to 0.0030	0.9437	0.9917	pass
paper binary law, both nuisances correct	ate	average treatment effect	R drtmle	-0.000472 to 0.0029	0.9450	0.9934	pass
paper binary law, both nuisances correct	ey0	counterfactual mean under no treatment	cleverly	-0.0025 to -0.000075	0.9575	1.0351	pass
paper binary law, both nuisances correct	ey0	counterfactual mean under no treatment	R drtmle	-0.0025 to -0.000094	0.9550	1.0359	pass
paper binary law, both nuisances correct	ey1	counterfactual mean under treatment	cleverly	-0.0014 to 0.0013	0.9425	0.9765	pass
paper binary law, both nuisances correct	ey1	counterfactual mean under treatment	R drtmle	-0.0014 to 0.0012	0.9450	0.9769	pass
paper binary law, outcome regression correct	ate	average treatment effect	cleverly	0.000105 to 0.0035	0.9413	0.9987	pass
paper binary law, outcome regression correct	ate	average treatment effect	R drtmle	0.000204 to 0.0036	0.9500	1.0046	pass
paper binary law, outcome regression correct	ey0	counterfactual mean under no treatment	cleverly	-0.0015 to 0.000923	0.9550	1.0235	pass
paper binary law, outcome regression correct	ey0	counterfactual mean under no treatment	R drtmle	-0.0016 to 0.000864	0.9563	1.0258	pass
paper binary law, outcome regression correct	ey1	counterfactual mean under treatment	cleverly	0.000203 to 0.0028	0.9350	0.9841	pass
paper binary law, outcome regression correct	ey1	counterfactual mean under treatment	R drtmle	0.000239 to 0.0028	0.9337	0.9853	pass
paper binary law, treatment mechanism correct	ate	average treatment effect	cleverly	0.0013 to 0.0049	0.9450	0.9922	pass
paper binary law, treatment mechanism correct	ate	average treatment effect	R drtmle	0.000277 to 0.0039	0.9400	0.9878	pass
paper binary law, treatment mechanism correct	ey0	counterfactual mean under no treatment	cleverly	-0.0035 to -0.000789	0.9450	0.9848	pass
paper binary law, treatment mechanism correct	ey0	counterfactual mean under no treatment	R drtmle	-0.0031 to -0.000366	0.9387	0.9744	pass
paper binary law, treatment mechanism correct	ey1	counterfactual mean under treatment	cleverly	-0.000420 to 0.0023	0.9275	0.9746	pass
paper binary law, treatment mechanism correct	ey1	counterfactual mean under treatment	R drtmle	-0.0010 to 0.0018	0.9350	0.9656	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with drtmle-r on the same samples. Each row reports estimate equivalence within the declared margin, the RMSE ratio and the coverage difference, and names its comparison kind.

law	estimand	what was compared	paired difference	share of margin used	RMSE ratio bound	coverage difference	comparison kind	calibration resolution	result
paper binary law, both nuisances correct	ate	average treatment effect	-0.000002	0.000593	1.0058	-0.0012	matching implementation	0.0032 vs 0.0500	equivalent
paper binary law, both nuisances correct	ey0	counterfactual mean under no treatment	0.000018	0.0091	1.0041	0.0025	matching implementation	0.0039 vs 0.0500	equivalent
paper binary law, both nuisances correct	ey1	counterfactual mean under treatment	0.000016	0.0077	1.0050	-0.0025	matching implementation	0.0034 vs 0.0500	equivalent
paper binary law, outcome regression correct	ate	average treatment effect	-0.000092	0.0329	1.0145	-0.0087	matching implementation	0.0184 vs 0.0500	equivalent
paper binary law, outcome regression correct	ey0	counterfactual mean under no treatment	0.000059	0.0296	1.0086	-0.0013	matching implementation	0.0135 vs 0.0500	equivalent
paper binary law, outcome regression correct	ey1	counterfactual mean under treatment	-0.000032	0.0151	1.0126	0.0013	matching implementation	0.0094 vs 0.0500	equivalent
paper binary law, treatment mechanism correct	ate	average treatment effect	0.0010	0.3362	1.0467	0.0050	matching implementation	0.0403 vs 0.0500	equivalent
paper binary law, treatment mechanism correct	ey0	counterfactual mean under no treatment	-0.000415	0.1864	1.0378	0.0062	matching implementation	0.0468 vs 0.0500	equivalent
paper binary law, treatment mechanism correct	ey1	counterfactual mean under treatment	0.000590	0.2587	1.0326	-0.0075	matching implementation	0.0370 vs 0.0500	equivalent

Repeated-sampling properties

property	cell	role	what was tested	what must hold	measured	result
double_robust_contraction	both_wrong_n1500	control	both nuisances are misspecified, at n = 1,500	the exact coverage interval must fall below the floor, over the replications this rung declares its own verdict at	coverage 0.0224 to 0.0584, bias 0.2306	pass
double_robust_contraction	both_wrong_n3000	control	both nuisances are misspecified, at n = 3,000	the exact coverage interval must fall below the floor, over the replications this rung declares its own verdict at	coverage 0.0148 to 0.0461, bias 0.2321	pass
double_robust_contraction	both_wrong_n6000	control	both nuisances are misspecified, at n = 6,000	the exact coverage interval must fall below the floor, over the replications this rung declares its own verdict at	coverage 0.0062 to 0.0300, bias 0.2323	pass
double_robust_contraction	outcome_correct_n1500	positive	only the outcome regression is correctly specified, at n = 1,500	the exact coverage interval clears the declared floor, over the replications this rung declares its own verdict at	coverage 0.9208 to 0.9637, bias 0.0036	pass
double_robust_contraction	outcome_correct_n3000	positive	only the outcome regression is correctly specified, at n = 3,000	the exact coverage interval clears the declared floor, over the replications this rung declares its own verdict at	coverage 0.9150 to 0.9596, bias 0.0021	pass
double_robust_contraction	outcome_correct_n6000	positive	only the outcome regression is correctly specified, at n = 6,000	the exact coverage interval clears the declared floor, over the replications this rung declares its own verdict at	coverage 0.9194 to 0.9627, bias 0.0012	pass
double_robust_contraction	rate_both_wrong	control	the same regression with both nuisances misspecified	slope interval must not establish contraction	slope 0.0037 to 0.0135	pass
double_robust_contraction	rate_outcome_correct	positive	log absolute bias regressed on log n across three sizes, outcome regression correct, over every replication each rung ran	slope interval entirely below zero, so the bias contracts	slope -1.5345 to -0.4823	pass
double_robust_contraction	rate_treatment_correct	positive	the same regression with only the treatment mechanism correct	slope interval entirely below zero, so the bias contracts	slope -1.7568 to -1.0221	pass
double_robust_contraction	treatment_correct_n1500	positive	only the treatment mechanism is correctly specified, at n = 1,500	the exact coverage interval clears the declared floor, over the replications this rung declares its own verdict at	coverage 0.8864 to 0.9385, bias 0.0115	fail
double_robust_contraction	treatment_correct_n3000	positive	only the treatment mechanism is correctly specified, at n = 3,000	the exact coverage interval clears the declared floor, over the replications this rung declares its own verdict at	coverage 0.9179 to 0.9616, bias 0.0041	pass
double_robust_contraction	treatment_correct_n6000	positive	only the treatment mechanism is correctly specified, at n = 6,000	the exact coverage interval clears the declared floor, over the replications this rung declares its own verdict at	coverage 0.9165 to 0.9606, bias 0.0019	pass
double_robustness	both_correct	positive	both the outcome regression and the treatment mechanism are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.000292 to 0.0045, margin 0.0066, SE ratio 1.0071	pass
double_robustness	both_wrong	control	both nuisances are misspecified	bias interval must fall entirely outside the margin, with the reported standard error still on the scale of the empirical spread	bias 0.2280 to 0.2330, margin 0.0068, SE ratio 1.8286	pass
double_robustness	outcome_correct	positive	only the outcome regression is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.0026 to 0.0075, margin 0.0067, SE ratio 0.9888	fail
double_robustness	treatment_correct	positive	only the treatment mechanism is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.0068 to 0.0121, margin 0.0072, SE ratio 0.9671	fail
interval_calibration	correctly_specified	positive	both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9448 to 0.9667, SE ratio 0.9857 to 1.0570	pass
root_n_and_efficiency	n_1500	positive	bias, coverage and SE calibration at n = 1,500	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias 0.0018, coverage 0.9165 to 0.9606, SE ratio 0.9447	pass
root_n_and_efficiency	n_4500	positive	bias, coverage and SE calibration at n = 4,500	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000391, coverage 0.9223 to 0.9647, SE ratio 0.9834	pass
root_n_and_efficiency	n_500	control	bias, coverage and SE calibration at n = 500	coverage interval lies below nominal or clears the declared floor	bias 0.000042, coverage 0.9282 to 0.9688, SE ratio 1.0460	pass
root_n_rate	empirical_sd	positive	log empirical spread of the estimates regressed on log n across three sizes	slope interval inside the root-n band and excluding -1/4	slope -0.5236 to -0.4422	pass
root_n_rate	reported_se	positive	the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.5140 to -0.5070	pass

Measured values and declared margins

quantity	value	source
replicates	800	paired replications per primary nuisance regime
n	3000	observations per primary replication
subject_tests_passed	9	Cleverly truth tests passing
subject_tests_total	9	Cleverly truth tests reported
paired_tests_passed	9	paired cells concluding equivalent or superior
paired_tests_total	9	paired comparison cells reported
property_cells_passed	19	repeated-sampling property cells passing their own and family verdicts
property_cells_total	22	repeated-sampling property cells reported
score_audited_fits	2400	fits each side audits against the shared score bar
subject_score_failures	7	Cleverly fits that miss that bar
reference_score_failures	36	R drtmle fits that miss that bar
properties[double_robust_contraction/rate_outcome_correct]:slope	-0.9176	fitted contraction slope, outcome regression correct
properties[double_robust_contraction/rate_outcome_correct]:slope_ci_lower	-1.5345	its 99% lower endpoint
properties[double_robust_contraction/rate_outcome_correct]:slope_ci_upper	-0.4823	its 99% upper endpoint
properties[double_robust_contraction/rate_outcome_correct]:replicates	5600	replications the contraction slope fits
properties[double_robust_contraction/rate_treatment_correct]:slope	-1.3194	fitted contraction slope, treatment mechanism correct
properties[double_robust_contraction/rate_treatment_correct]:slope_ci_lower	-1.7568	its 99% lower endpoint
properties[double_robust_contraction/rate_treatment_correct]:slope_ci_upper	-1.0221	its 99% upper endpoint
properties[double_robust_contraction/rate_both_wrong]:slope	0.0086	fitted contraction slope of the control arm
properties[double_robust_contraction/rate_both_wrong]:slope_ci_lower	0.0037	its 99% lower endpoint
properties[double_robust_contraction/rate_both_wrong]:slope_ci_upper	0.0135	its 99% upper endpoint
properties[double_robust_contraction/outcome_correct_n1500]:bias	0.0036	bias scale at the first rung, outcome regression correct
properties[double_robust_contraction/treatment_correct_n1500]:replicates	800	replications the first rung reads its own coverage verdict from
properties[double_robust_contraction/treatment_correct_n1500]:coverage	0.9150	exact coverage of the red rung
properties[double_robust_contraction/treatment_correct_n1500]:coverage_ci_lower	0.8864	its 99% coverage lower endpoint
min_coverage	0.9275	lowest implementation-estimand coverage
min_coverage_ci_lower	0.9006	lowest exact 99% coverage lower endpoint
max_standardized_bias	0.1557	largest absolute standardized bias
max_rmse_ratio_upper	1.0467	largest paired 99% RMSE-ratio upper endpoint
min_coverage_difference_lower	-0.0250	smallest paired 99% coverage-difference lower endpoint
max_calibration_excess_upper	0.0364	largest paired 99% excess-calibration upper endpoint
margin:confidence_level	0.9900	confidence level of Monte Carlo intervals
margin:alpha	0.0500	nominal estimator size
margin:nominal_coverage	0.9500	nominal estimator coverage
margin:bootstrap_replicates	10000	resamples per bootstrap interval
margin:standardized_bias	0.2500	bias equivalence margin in empirical standard deviations
margin:coverage_floor	0.9000	validity floor for the exact coverage lower endpoint
margin:over_coverage_ceiling	0.9900	coverage above this is labeled conservative
margin:se_ratio_sanity_lower	0.8000	primary SE-ratio screen, lower limit
margin:se_ratio_sanity_upper	1.2000	primary SE-ratio screen, upper limit
margin:calibration_se_ratio_lower	0.9300	calibration SE-ratio band, lower limit
margin:calibration_se_ratio_upper	1.0700	calibration SE-ratio band, upper limit
margin:calibration_coverage_lower	0.9200	calibration coverage band, lower limit
margin:calibration_coverage_upper	0.9800	calibration coverage band, upper limit
margin:type_i_ceiling	0.1000	one-sided type-I error ceiling
margin:paired_difference	0.1500	paired mean-difference margin in pooled SDs
margin:rmse_noninferiority	1.1000	paired RMSE-ratio upper limit
margin:coverage_noninferiority	-0.0250	paired coverage-difference lower limit
margin:calibration_noninferiority	0.0500	paired calibration-excess upper limit
margin:minimum_power	0.8000	power-control rejection lower bound
margin:root_n_slope	-0.5000	predicted root-n slope
margin:root_n_slope_lower	-0.6250	accepted slope band, lower limit
margin:root_n_slope_upper	-0.3750	accepted slope band, upper limit
margin:excluded_slope	-0.2500	slower rate the interval must exclude
margin:union_model_se_lower	0.1000	union-model SE-ratio screen, lower limit
margin:union_model_se_upper	10	union-model SE-ratio screen, upper limit

What a red double-robustness cell means here

The two one-correct double_robustness cells exceed the equivalence margin at n = 1,500. That margin is a quarter of an empirical standard deviation, and the test is resolvable at that size: its 99% half-width is about 0.0025 against margins of 0.0067 and 0.0072. The bias really is larger than the margin. The red-cell ledger lists each red cell of this study with the roadmap ask that owns it.

RM18 read the two red one-correct cells against R drtmle on the primary rows at n = 3,000, under a rule declared before the computation. On outcome_correct, R drtmle shows the same signed bias, and the paired interval covers zero, so the reading is shared. That means the bias is shared by these two implementations at this size; it does not identify the source. On treatment_correct, R drtmle also shows a bias above zero, and the unadjusted paired interval of cleverly minus R runs 0.000068 to 0.001942, so the declared reading is mixed. This is an exploratory between-implementation signal, not an established implementation defect: the interval carries no multiplicity adjustment, and the Bonferroni interval for this comparison covers zero.

What the one-sided reading found gives each interval. RM19 read the increment again on 2,000 fresh draws at n = 3,000, under a design declared before its runs. The 99% interval of cleverly minus R runs from -0.000023 to 0.001155, and the declared reading is no increment at the declared resolution. That interval also covers the committed point of 0.001005.

At n = 1,500, the paired increment on fresh draws runs from 0.0010 to 0.0031, at the level 1 - 0.01/3. The Part C rule labels it increment, and no route reads the label. The three cells stay red, and their ledger owner did not change. What the localization design found gives each reading and what it does not show. The per-draw rows are in tests/diagnostics/rm19_one_sided_increment/.

One size cannot say what kind of failure that is. A second-order remainder that has not yet decayed and an inconsistent estimator look identical at one n and mean opposite things. The double_robust_contraction family fits log absolute bias on log n over three sizes to separate their predicted behavior. A second-order remainder predicts a slope near -1, and a first-order term predicts one near -1/2. An inconsistent estimator can produce a slope near 0. The both_wrong arm is the control that must fail to contract.

Three finite-sample points do not identify which term produced an observed slope.

The measured slopes are -0.9176 with the outcome regression correct and -1.3194 with the mechanism correct. The control sits at 0.0086, above zero, and its standardized bias grows with n, from 8.5 to 12.6 to 16.6. Both positive biases contract over this ladder, while the control does not. This result distinguishes contraction from non-contraction. It does not identify either bias as the theorem’s second-order remainder.

Both slope intervals establish contraction. With the mechanism correct the interval runs -1.7568 to -1.0221. With the outcome regression correct it runs -1.5345 to -0.4823. The second interval covered zero on the earlier ladder, which ran 800 replications at every rung. The reason was arithmetic rather than scientific. The bias in that arm is already small at the first rung, 0.0036, so three points of a small and noisy quantity gave a wide slope.

CONTRACTION_REPLICATES in tests/studies/drtmle_properties.py declares the ladder that resolves the slope. The sizes stay at 1,500, 3,000 and 6,000. The two outer rungs run 2,400 replications and the middle rung keeps 800. A middle rung carries a centred weight of zero in the fitted slope, so replications there buy no resolution. The budget rule reads the published bias scale and the control arm’s spread. It reads neither positive rate cell, so the design cannot tune itself to a verdict.

The declaration predicted the narrowing, and the control arm checks it. A slope interval’s width falls as one over the square root of the replications. Tripling the two outer rungs therefore narrows the control interval by about the square root of three. The committed control interval runs 0.0037 to 0.0135, and that is the predicted narrowing. The git history of tests/canonical/drtmle/properties.csv carries the interval it replaced.

The run that raised the rungs also moved the environment and the package source. SciPy moved from 1.17.1 to 1.18.0, and Python moved from 3.11.13 to 3.13.7, on that run. tests/canonical/drtmle/manifest.json records the versions the committed rows came from. Its git history records the versions they replaced.

The committed history separates the budget from the source and environment changes together. It does not separate the source from the environment. The first 800 replications of each rung ran at both commits, and their estimates agree within 7.3e-5. At that budget the slope rule gives each interval within 2e-5 at both commits, and rate_outcome_correct still runs -3.0401 to +0.1232. The move below zero therefore comes from the declared budget. RM18 records the comparison under What the committed history already separates at commit 985849c6.

The fitted -0.9176 sits near the -1 that a second-order remainder predicts. The budget rule never read this cell, so the design did not steer that agreement. The interval still covers -1 and -1/2 together. It therefore corroborates the sign of the contraction and not its exponent, and the Limitations section below says the same.

A reader cannot recompute that slope from the rung rows beside it. The fit reads every replication each rung ran, which is 2,400 at each outer rung. The bias column beside a rung instead reports the 800 replications that rung reads its coverage verdict from. The two samples differ. The log ratio of the outer rungs’ published bias values, divided by the log of four, therefore gives a slope near -0.8 and not -0.9176. The same ratio over every replication gives the published slope, and property-replicates.csv.gz carries those replications.

The extra draws serve the slope alone. Each rung reads its own coverage verdict from 800 replications, which CONTRACTION_VERDICT_REPLICATES declares beside the budget. A rung’s coverage gate is a one-sided exact interval against a fixed floor. A larger budget narrows that interval around the true coverage and can resolve an inconclusive verdict. The study refuses the extra coverage budget because nobody declared it before the run.

The slope budget was declared before the run under the rule above. Each rate cell therefore fits 5600 replications, while each rung keeps its declared coverage budget. The both_wrong control still has to fail to contract.

Raising the level margin was considered and rejected. The standardized bias under a correct mechanism runs 0.398, 0.203 and 0.129 across the ladder, so no affordable size brings the 99% interval inside 0.25. The level cell stays red and the contraction family says which red it is.

The ladder also surfaced a result the single-size study could not reach. At n = 1,500 with the outcome regression misspecified, the interval does not clear the declared coverage floor: exact coverage is 0.9150 with a 99% lower endpoint of 0.8864 against a floor of 0.90. It clears at n = 3,000 and at n = 6,000. That cell is red, and it is a statement about small samples in one regime rather than about the construction. Its budget did not move, so the cell stays red under the ladder that resolved the slope.

RM18 design BD re-read this rung on 6,000 fresh replicates, under a declaration written before its run. The coverage interval runs 0.9179 to 0.9354 and clears the floor, so the reading is resolved: truth satisfies the gate. The interval lies wholly below 0.95. The rung rule reads coverage alone.

The fresh bias interval of 0.00774 to 0.00965 lies above its 0.25 SD margin. The registered double_robustness/treatment_correct cell judges that bias, and the fresh interval is supplementary. The re-read does not move a verdict. What design BD found at commit 985849c6 gives the numbers, and tests/diagnostics/rm18_boundary/ holds the rows.

Limitations

The comparison is binary, two-arm, complete-outcome, pointwise, and confined to the paper law and declared GLMs. It does not establish missing-outcome, multi-arm, weighted, clustered, simultaneous-inference, broad machine-learning, or practical-positivity parity. Supplying the same initial nuisance predictions isolates the corrected construction; it does not compare the two projects’ learner wrappers. The study measures finite-sample behavior at its declared sizes and cannot verify the theorem’s unobservable second-order remainder condition for a future fit. The default-band study measures the default band of this study’s subject fit.

The paper law is H1 in both branches by symmetry. Its main-effects GLM limits have no covariate coefficients, so neither conditioning term of the theorem can arise on it. tests/unit/test_drtmle_h1_symmetry.py pins this. The one-correct cells therefore measure second-order terms only. Their readings and the contraction slopes near -1 stand. No cell here can detect a one-wrong failure with a data-adaptive learner.

The contraction family fits three points, so it establishes a direction rather than an exponent. Both positive slope intervals now exclude zero. The outcome-correct interval still covers -1 and -1/2 together, so it does not identify that remainder’s order.

Reproduction

The fixture README, manifest, replications, paired decisions, fit diagnostics, and property results carry the protocol, provenance, and every published row.