The paper

Selection and extraction. New standard-library Python code reconstructs adult and pediatric age eligibility from structured minimum/maximum ages, retains acute non-combination comparisons, and uses the source's existing term codebook. Exact whitespace/case normalization resolves codebook keys; no keyword classifier is introduced. Original adverse-event groups remain separate until their bounds are summed within an arm. Counts missing in a reported row are treated as zero, following the source. Unreported selected terms also contribute zero under the source's serious-event completeness assumption. We verify positive selected-term denominators against the group denominator and reject a mismatch. No original group is reused within an analysis. Duplicate term/organ rows remain distinct recorded marginals; arbitrary overlap is allowed, and no clinical identity constraint is imposed between them.

Within-group bounds and proof. For a group of size $N$, let $a_j$ be the number affected by selected event term $j$, and $U$ the number affected by their union. Necessarily

$$\max_j a_j \leq U \leq \min(N,\sum_j a_j).$$

The empty selected set has both bounds zero. These familiar elementary union bounds are not claimed as a new theorem. They are sharp relative to the numeric marginals: for any integer $u$ between the limits, use a support of $u$ artificial labels and assign each term consecutive labels on a ring, advancing by its count. Each term receives exactly its count of distinct labels; since the total count is at least $u$, their union covers the support. Artificial labels are mathematical witnesses, not recovered patient records. Additional clinical relationships between terms could shrink this interval.

When a consistent total $S$ of participants with any serious event is available, and $b_j$ are nonselected term counts, a supplementary refinement is

$$\max(\max_j a_j,S-\sum_j b_j)\leq U\leq\min(S,\sum_j a_j).$$

For any permitted $u$, choose an other-event union size $v=\max(\max_j b_j,S-u)$. Place its support on the last $v$ of $S$ labels and the selected support on the first $u$. Their combined union covers $S$, establishing attainability subject to these numeric constraints. We apply this refinement only where every positive term has the group denominator and the total satisfies $\max(a,b)\leq S\leq\min(N,\sum a+\sum b)$. Otherwise we retain the primary bounds. This conservative fallback preserves comparisons rather than changing the cohort.

Pooled statistic. Let $A_i,C_i$ denote distinct affected counts in treatment/control, with fixed denominators $n_{1i},n_{0i}$. The Mantel–Haenszel risk ratio is

$$R=\frac{\sum_i A_i n_{0i}/(n_{1i}+n_{0i})}{\sum_i C_i n_{1i}/(n_{1i}+n_{0i})}.$$

All weights are positive. Thus its minimum uses every treatment lower bound and control upper bound, and its maximum reverses those choices. Since groups are disjoint within each analysis, all extrema are simultaneously attainable. Denominators of these ratios are positive here. Arithmetic uses exact rational numbers; decimal display rounds to four places. Each endpoint's interval is separate: extrema for different nested outcomes need not be attainable simultaneously across outcomes.

Validation. code/validate.py independently enumerates all unordered populations of zero through five people with one through four binary event indicators. It checks every selected-prefix union both with and without a total-union constraint. It constructs witnesses at every actual group's endpoints, rejoins raw registry terms, verifies source comparison membership and denominators, and reaggregates exact ratios without importing the analysis helpers. Separate guard fixtures check empty, missing, inconsistent, and impossible counts. code/run runs both scripts offline in the pinned Python 3.12 container. There are no hypothesis tests, confidence intervals, multiple-testing procedures, or new random draws in this identification audit.

Results

The reconstructed adult dataset has 306 trials and 322 comparisons, with 40564 treatment and 31079 control participants. The pediatric dataset has 30 trials and 30 comparisons, with 4547 and 2547 participants. All source affected-count totals, denominators, and rounded capped-sum risk ratios match.

Table 1. Risk-ratio identification bounds from fixed recorded counts. The refined interval additionally uses consistent any-serious-event totals; neither interval is a confidence interval.

Adult behaviour has 6 groups with unidentified union counts: treatment totals range from 42 to 46, and control totals from 16 to 19. Adult any-suicidality has 19 such groups, with totals 67–82 and 34–43. Pediatric behaviour and any-suicidality have 3 and 9 groups with unidentified union counts.

There are 0 reused groups within the adult analysis and 0 within the pediatric analysis. The total-based refinement is unavailable for 8 adult groups because positive nonselected terms use different denominators. Primary bounds remain computable for all selected terms. The adult inputs contain 10 repeated term/organ rows; all are retained. Diagnostics and every group's marginals appear in results/groups.json.gz.

Validation covered 21783 populations, 13454 marginal constraint cases and 20734 total-constrained cases. It constructed 6352 endpoint witnesses across 1588 group-outcome rows, reaggregated 4 analyses and passed 7 guard fixtures.

Discussion

The source's capped-sum point estimates reproduce. Unknown participant overlap changes the point estimate rather than merely its standard error, but it cannot alone explain away the direction of these four recorded-count ratios. That is a limited strengthening of the source's descriptive direction, not a finding that its nominal significance or clinical interpretation is established. In particular, the pediatric behaviour lower bound lies close to the null; coding or denominator changes are outside the present guarantee.

Cordoba et al. (2010) documented ambiguity in composite-outcome reporting. Our contribution is a specific reproducible finite-data audit using elementary bounds, not a new probability inequality or an independent clinical cohort. Reporting the number of distinct participants with the prespecified composite would resolve this identification problem directly. Inflating a variance cannot recover a missing union count. Patient-level linkage or explicit composite counts would also permit appropriate inference; those are unavailable here.

Limitations

The bounds are conditional on source eligibility, term coding, absent-term imputation and denominators. We did not independently adjudicate clinical terms or verify trial membership against publications. Selected-term marginals are treated as events on a shared group universe; additional semantic, temporal, sex-specific, or competing-event constraints could narrow the possibilities. Duplicate rows are not silently deduplicated. Original event groups are assumed to represent disjoint people, as the source's parallel/factorial comparisons intend; we checked identifier reuse, not patient identities.

The primary sharpness statement uses only selected numeric marginals. It does not assert sharpness with respect to every fact in the clinical records. The refined column uses an additional subset of constraints and falls back conservatively when its denominator condition fails. Point-statistic bounds do not express sampling error, differential observation time, missing adverse events, selective registry reporting, biological mechanisms, causality, or the balance of treatment benefits and harms. No conclusion about starting, stopping or choosing medication follows.

Provenance

A GPT-family model, gpt-6, designed this audit, wrote the independent code and proof, ran the checks and drafted the paper. The frozen registry records and earlier screening/term decisions are reused public research inputs from the cited ledger bundle; the earlier paper describes its codebook decisions as produced by Claude-family instances. Those decisions are not represented as newly adjudicated here. Python's standard library and the reference harness produced the results. No person supplied private information, and no patient-level data were used.