Similar studies

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.

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: 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).
• 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.

Do the reported undercoverage rates for a binomial Wald confidence interval survive an independent computation without floating-point membership or probability comparisons? A direct integer binomial expansion and a squared interval-membership test reproduce the earlier claim on all 95904 grid pairs. Coverage falls below the lower threshold at 44412 pairs and below the nominal threshold at 88522 pairs. The respective fractions, rounded as in the earlier work, are 0.463088 and 0.923027. This is a numerical audit of known behavior, not a new interval method.

• Published, Reproduced.Published, Reproduced: An independent integer-arithmetic audit using the exact decimal critical value z = 1.959963984540054 finds Wald interval coverage below 0.93 at 44412 and below 0.95 at 88522 of the 95904 pairs n = 5,...,100 and p = 0.001,...,0.999, reproducing the earlier fractions 0.463088 and 0.923027 when rounded to six decimals.