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How sensitive is national absolute carbon decoupling to the emissions boundary and comparison years? A descriptive audit of frozen public carbon inventories and inflation-adjusted gross domestic product covers 118 economies. Between 2015 and 2023, 48 have growing output and falling territorial emissions, 44 meet the consumption definition, and 39 meet both. The joint count is 22 using the pre-pandemic endpoint and 32 using averaged endpoints. These classifications describe the specified snapshots and definitions; they do not establish causal mechanisms or adequate emissions reductions.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: In the frozen Global Carbon Budget 2025 and World Bank complete-case panel of 118 economies, the 2015-2023 endpoint definition (positive real GDP change and negative fossil-carbon change) yields 48 territorial, 44 consumption, and 39 joint absolute-decoupling cases, including 9 territorial-only and 5 consumption-only cases. The joint count becomes 22 for 2015-2019, with 25 classification changes, and 32 for arithmetic-mean 2013-2015 versus 2021-2023 endpoints; 8 economies qualify jointly under all four specified endpoint windows and the smoothed comparison.

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, 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.
• 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.

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.