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How sensitive is a focused meta-analysis of walking and jogging for depressive symptoms? A registered audit of a fixed public extraction pools 21 direct-comparison trials and 994 participants. Post-treatment Hedges g is -0.762, with a confidence interval from -1.203 to -0.321. Every leave-one-trial-out interval excludes zero, but the usual-care-only interval and the between-study prediction interval include zero. No included trial has an overall low-risk-of-bias rating. These conditional computations support neither a uniform treatment effect nor a treatment recommendation.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: In the fixed Noetel et al. public extraction, the preregistered post-treatment direct comparison of walking/jogging with its specified active controls pools 21 trials and 994 participants at Hedges g=-0.762 (95% modified Hartung-Knapp confidence interval -1.203 to -0.321), all 21 single-trial-omission intervals remain below zero, but the overall prediction interval (-2.613 to 1.088) and usual-care-only confidence interval (-1.650 to 0.237) include zero and the overall-low-risk-of-bias subset has no trials, conditional on unchanged source outcomes and ratings.

Does adding a previously unextracted public registry outcome change the selected-trial evidence for ivermectin in acute outpatient COVID-19? This revised aggregate-data audit incorporates COVID-OUT's reported hospitalization/death endpoint at the common follow-up window, expanding the primary analysis to 7714 records and 98 events. The exploratory pooled risk ratio is 1.107 (95% confidence interval 0.577 to 2.123). All 550 extensions compatible with TOGETHER's separate hospitalization and death margins also include no difference. The newly retrieved appendix does not identify their overlap. A separate denominator sensitivity shows why sparse observed events cannot settle the treatment effect when outcome coverage differs. This is a correction of a targeted audit, not a comprehensive meta-analysis, proof of no effect, or treatment recommendation.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Conditional on the frozen public trial tables, registry extraction and exploratory estimators, adding COVID-OUT's reported day-28 hospitalization/death endpoint yields 7,714 analysis records and pooled risk ratio 1.107 (95% modified Hartung-Knapp interval 0.577 to 2.123), all 550 compatible TOGETHER hospitalization/death union extensions include no difference, and assignments conditional on nesting COVID-OUT's endpoint denominators within its published modified-intention-to-treat cohort span trial risk ratios 0.083 to 7.766, establishing neither benefit nor equivalence.

How much false-positive error can arise when a nominally exact test treats clustered subsamples as independent? An exact finite benchmark enumerates null distributions for 75 balanced designs. In the selected beta-binomial design, the cell-level Fisher test rejects with probability 0.21567065919813758, compared with 0.0436374106894531 for a test using the known clustered null. With a single observation per donor, the Fisher rejection probability is 0.021484375. These are exact probabilities conditional on the specified toy model. They demonstrate calibration failure from ignoring dependence, without evaluating real biological pipelines or ranking practical alternatives.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: For the specified null of five independent donors per arm and twenty conditionally independent Bernoulli observations per donor with latent Beta(5,5) success probabilities (within-donor correlation 1/11), the cell-level two-sided probability-ordering Fisher test at nominal alpha=0.05 rejects with exact-model probability 0.215670659198, versus 0.043637410689 for the known-clustered-null absolute-difference-tail oracle. The finite 75-design grid has 31 cell-level Fisher probabilities above nominal, while all 75 oracle probabilities and all 15 independent-observation controls are at or below nominal. These are conditional toy-model calibration results, not evaluations of real biological analysis methods.

Can repeated rounding hide escape even when a Mandelbrot parameter is exactly representable? We construct the first binary floating-point successor of the real cusp and give an elementary invariant argument: both separate rounding and single-round evaluation keep its computed critical orbit below an absorbing boundary, although its exact orbit is unbounded. An exact rational simulator and directed integer enclosures check 22 precisions. At single precision, the stationary transitions occur at 8999 and 9532, while the exact orbit first escapes at 18196. This supplies an adversarial numerical test family; it does not claim discovery of rounding artifacts or slow escape.

• Published, Reviewed.Published, Reviewed: For every binary significand precision p >= 3, the exactly representable parameter c_p = 1/4 + 2^(-(p+1)) has an unbounded exact critical orbit, while both nearest-even recurrences RN_p(RN_p(x_n^2)+c_p) and RN_p(x_n^2+c_p), starting at zero, stay in [0,1/2] for every iterate, with 1/2 absorbing.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: For all 22 precisions p from 3 through 24, the supplied exact-dyadic benchmark reaches the stationary value 1/2 under both rounding models and certifies finite exact first escapes, with first stationary transitions 8999 and 9532 and exact first escape 18196 at p=24.

Does maintaining normal tissue mechanics prevent cancer initiation? A focused evidence map separates the endpoints of 4 selected papers and reanalyzes an available tumor-count table. The selected contrasts occupy 4 classes, with 0 papers testing incident cancer after a mechanical intervention in humans. The table concerns already malignant cells: blocking-antibody pretreatment gives risk ratios of 0.4667 and 0.5 against its controls. These endpoints answer different questions; neither a smaller transplantation risk nor a human biomarker change establishes primary prevention. This is a bounded, exploratory source audit, not a systematic review or a new biological experiment.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: In a purposively selected four-paper evidence map, the coded contrasts occupy four distinct endpoint classes, only one paper starts its reprogramming program with normal primary cells, and none measures incident cancer after a mechanical intervention in humans, with all counts restricted to this selected set rather than the literature.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Reanalysis of Weaver et al. (1997) Table II gives 7/16 tumor-bearing mice after blocking-antibody pretreatment of already malignant cells versus 15/16 untreated and 14/16 control-antibody mice, risk ratios 0.4667 and 0.5000, with two-sided Fisher p-values 0.005923 and 0.023333 (Holm-adjusted 0.011846 and 0.023333), while conservative simultaneous 95% risk-ratio intervals are 0.1612 to 1.1614 and 0.1625 to 1.3304 under independent-binomial sampling.

We provide an explicit rational parameter for every finite iteration cutoff. Its critical orbit remains uniformly below the usual escape radius throughout the cutoff, although an elementary growth argument proves that it eventually escapes. The contribution is a deterministic adversarial test family, a short uniform proof, and a certified finite benchmark. Integer enclosures and Arb ball arithmetic agree on 7 cases, with first-escape counts [23,61,212,815,3228,12879,51482]. Slow escape near the real parabolic cusp is established in the literature; we claim no discovery of that phenomenon or of its asymptotic law.

• Published, Reviewed.Published, Reviewed: For every integer T >= 1, the real parameter c_T = 1/4 + 1/[16(T+1)^2] has critical iterates z_0 = 0 and z_(n+1) = z_n^2 + c_T in [0,17/32] for 0 <= n <= T, but an unbounded critical orbit.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: The exact-rational benchmark parameters generated from cutoffs [1, 4, 16, 64, 256, 1024, 4096] have first critical-orbit escape iterations [23, 61, 212, 815, 3228, 12879, 51482], with all 7 counts certified and matched by integer dyadic enclosures and Arb ball arithmetic, and all 49 exact-rational control checks passing.

Published aggregate follow-up tables can support a transparent sensitivity calculation without reconstructing participant records or fitting the original trial model. We examined 6 long-term recovery rows from PRINCIPLE, keeping the primary and larger sensitivity cohorts separate. For the primary six-month row, the respondent-only treatment-minus-control recovery difference is 3.071 percentage points. Unrestricted binary completion of unavailable responses gives a range from -16.54 to 21.764 percentage points, whereas assigning both arms any common missing-response recovery rate gives a positive range from 2.329 to 2.896 percentage points. These are conditional bounds, not confidence intervals. They quantify how assumptions affect a crude contrast and do not identify a clinical treatment effect.

• Published.Published: In six published PRINCIPLE long-term recovery rows, unrestricted binary completion bounds permit either sign of the crude recovery contrast, every common missing-response recovery rate preserves a positive contrast, and at six months in the primary cohort a missing-treatment recovery rate 13.206 percentage points below the respondent-control anchor erases that contrast, conditional on the stated completion model.

Can unknown overlap between adverse-event terms change the direction of published composite-event estimates? We reconstructed the frozen registry comparisons and calculated every possible range of distinct affected participants consistent with the selected term counts. For adult serious suicidal behaviour, the resulting Mantel–Haenszel risk-ratio bounds are 1.712 to 2.1962; for adult serious suicidality, 1.2154 to 1.8476. The corresponding pediatric bounds are 1.0045 to 1.4773 and 1.8144 to 2.8091. All remain above the null. Unknown overlap therefore changes the magnitude but cannot reverse these fixed-data point estimates under the source's cohort and outcome definitions. These are identification bounds, not confidence intervals or estimates of causal treatment effects.

• Published, Reproduced.Published, Reproduced: In the frozen antidepressant registry dataset, conditional on source cohort selection, denominators, term coding, and absent-term imputation, sharp Mantel-Haenszel risk-ratio bounds relative to selected within-group event-count marginals are 1.7120261533 to 2.1961879882 for adult serious suicidal behaviour, 1.2154409158 to 1.8475901940 for adult serious suicidality, 1.0045047792 to 1.4773331886 for pediatric serious behaviour, and 1.8144436696 to 2.8090979871 for pediatric serious suicidality. Thus unknown participant overlap alone cannot reverse the direction of these fixed-data point estimates; these are not confidence intervals or causal treatment effects.

Can other microbes maintain butyrate-related transcription when a leading contributor declines? A preregistered audit of public IBDMDB profiles finds that the available pathway-level species assignments are too sparse for a reliable answer. Among 375 retained paired Crohn's samples from 52 participants, the primary pathway has positive community RNA in 205 samples but named-species RNA in only 25. The specified longitudinal analysis retains 6 sustained-decline episodes from 5 people. Every retained episode has zero reported RNA from the named alternatives throughout its observation window. These zeros do not distinguish undetected or unassigned transcription from biological failure of compensation.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: In the audited public IBDMDB HUMAnN3 pathway tables, 375 retained paired Crohn's samples from 52 participants contain named-species PWY-5676 RNA in only 25 of 205 samples with positive community pathway RNA, and the preregistered sustained-contributor-decline analysis yields six episodes from five participants with zero reported alternative-species RNA at all three visits, preventing this operational analysis from distinguishing biological compensation failure from unresolved transcription.