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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.

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.

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, 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, 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.

What are the exact finite-sample distributions of longest increasing subsequence lengths in random permutations, and how rapidly do their moments approach the limiting Tracy-Widom law? We compute the exact distribution of longest increasing subsequence lengths for every sample size up to 50 using the Robinson-Schensted-Knuth correspondence and the hook-length formula. At the largest sample size, the exact expected length is 11.309389 with variance 1.94134, corresponding to a normalized centered mean of -1.475863 and normalized variance of 0.526961. Across all 50 sample sizes, the expected length remains strictly below two times the square root of the sample size, with maximum ratio 0.799695. Exhaustive enumeration via patience sorting independently confirms every distribution count across 409113 permutations.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: The benchmark independently verifies the exact distribution of longest increasing subsequence lengths against exhaustive enumeration via patience sorting for all 409113 permutations across n from 1 through 9, and confirms the Robinson-Schensted-Knuth sum-of-squares identity across all 50 sample sizes.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: For uniformly random permutations of size n from 1 through 50, the exact distribution of longest increasing subsequence lengths computed via the Robinson-Schensted-Knuth correspondence yields an expected length of 11.309389 at n=50, with all expected lengths strictly bounded below 2*sqrt(n).

How many longitudinal observations support a species-resolved microbial compensation analysis in the public inflammatory-bowel-disease multi-omics dataset? A reproducible eligibility audit finds 277 candidate intervals after specimen matching and sequencing-depth filtering. Four broad enzyme annotations yield [6,6,4,0] eligible contributor-decline intervals, in the stated annotation order; only [1,1,0,0] also contain paired numeric calprotectin metadata. These counts describe the specified measurement design, not whether biological compensation succeeds or fails.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Among 277 consecutive retained Crohn’s specimen intervals under the specified read-filtered, species-relative-abundance-decline and paired-transcript eligibility rules, EC 2.8.3.8, 2.7.2.7, 2.8.3.9 and 2.3.1.19 yield respectively 6, 6, 4 and 0 eligible intervals, of which respectively 1, 1, 0 and 0 also have agreeing numeric calprotectin metadata at both visits.

Can a simple electricity-use schedule transfer across years? We selected seasonal three-hour windows using 2023 data and evaluated them in later Great Britain grid records. Relative to midnight use, assigned average carbon intensity was 5.78% lower in 2024 and 13.15% lower in 2025. The schedule performed worse on 137 and 111 days, respectively. A single year-round window did not improve both years. These retrospective accounting results describe the recorded grid; they do not estimate emissions avoided by changing electricity demand.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: In the frozen Great Britain national Carbon Intensity API actual series, three-hour seasonal UTC windows selected only from 2023 yield mean assigned-intensity reductions versus 00:00-03:00 UTC of 5.778479% across 364 complete days in 2024 and 13.150070% across 365 days in 2025, compared with -9.430428% and 2.353958% for the single 2023-selected year-round window, while the seasonal rule is worse on 137 and 111 days respectively; these are finite-period accounting comparisons and do not estimate causal emissions reductions.

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.