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Foster and Rahmstorf (2026) remove El Niño, volcanic and solar influences from five global temperature records with a regression fitted to those records. They then report high confidence that warming accelerated, using critical values that treat the adjusted series as raw data. We ran their whole procedure, adjustment and tests together, on 2000 records per dataset, simulated under their own null: a straight-line trend since 1970 plus stationary noise. With noise fitted under that null, their changepoint test at nominal level 0.05 rejected 0.059 to 0.07 of null records, and the observed statistics had full-pipeline p of 0.0005 to 0.0025 in every record. Their quadratic test fared worse, with full-pipeline p up to 0.054. Under long-memory noise the changepoint evidence weakened, to p of at most 0.04, but held. The acceleration finding rests on the changepoint test.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: A numpy port of Foster and Rahmstorf's 2026 adjustment for El Niño, volcanic and solar influences reproduces their published adjusted NASA, NOAA, HadCRU, Berkeley and ERA5 series to within 0.0001 °C, and the changepoint test of Beaulieu et al. (2024) on the adjusted annual means for 1970-2024 gives maximum slope-change statistics from 4.91 to 6.68 with the change in 2013 or 2014 and nominal AR(1) Monte Carlo p-values from 0.00014 to 0.0033, against 0.041 to 0.19 for the unadjusted series.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: When 2,000 records per dataset are simulated under Foster and Rahmstorf's null hypothesis (a linear trend since 1970, their fitted exogenous terms, and ARMA(1,1) noise fitted under the null) and passed through their whole adjustment and changepoint test, the nominal 5% test rejects 5.9% to 7.0% of null records, and the observed statistics have full-pipeline p-values from 0.0005 to 0.0025 in all five records, so the changepoint finding is robust by the pre-registered rule.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Under the same full-pipeline null, Foster and Rahmstorf's quadratic test with the ARMA(1,1) correction of Foster and Rahmstorf (2011) rejects 5.8% to 8.0% of null records at nominal 5% on monthly values and 7.2% to 9.2% on annual means, and the observed quadratic terms have full-pipeline p-values from 0.0015 to 0.054 (monthly) and 0.0005 to 0.032 (annual), below 0.02 in only three records for monthly values and two for annual means, so the reported confidence above 98% in all five records does not hold and the finding is mixed by the pre-registered rule.

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Knuth's draft of section 7.2.2.4 of The Art of Computer Programming defines the digraph $SB(m, 3)$, whose vertices are the strings $xyz$ over $\{0, \dots, m-1\}$ and whose arcs go from $xyz$ to $yzx$ ("save") and to $yz(x+1 \bmod m)$ ("bump"), and poses exercise 225, rated as a research problem: construct a Hamiltonian cycle in $SB(m, 3)$ for all $m > 1$. We give a successor rule whose choice between save and bump at $xyz$ depends only on $y$, $z$ and $m$, through comparisons with $0$, $1$, $2$, $\lfloor m/2 \rfloor + 1$ and $\lfloor m/2 \rfloor + 2$, and prove with the Lean proof assistant that for every $m \ge 2$ its walk from $000$ visits all $m^3$ vertices and returns to $000$ after exactly $m^3$ steps. The proof follows the walk's first returns to the vertices $0yz$, which have closed forms region by region, and checks the small cases $m \le 11$ by computation in the kernel. This answers the exercise for every $m > 1$.

• Published, Reviewed, Formally verified.Published, Reviewed, Formally verified: For every integer m ≥ 2, the following successor rule traces a Hamiltonian cycle of SB(m, 3), the digraph whose vertices are the strings xyz with 0 ≤ x, y, z < m and whose arcs go from xyz to yzx and to yz(x+1 mod m). With k = ⌊m/2⌋ + 1, the successor of xyz is yzx when y ≥ k and z < k, when y = 0 and z ≠ 0, or when z = 1 and y ≥ 2, except that for odd m ≥ 7 it is yz(x+1 mod m) when z = 1 and y ≥ k + 1; every other xyz has successor yz(x+1 mod m). Starting from 000, the first m³ steps of the walk visit m³ distinct vertices, which is every vertex, and the walk is back at 000 after m³ steps.

The Mandelbrot set M is the set of complex numbers c for which the orbit of $0$ under $z \mapsto z^2 + c$ stays bounded. Its area is unknown. Pixel-counting estimates agree on many digits, but their error bars are statistical, and the bounds anyone has proven are much further apart. This work proves that the area of M is greater than 1.5065108996. The proof is a computation in ball arithmetic. It certifies a hyperbolic component of M of a stated period at each of 617591 approximate centers, and bounds each one's area from below with the area theorem applied to the inverse of its multiplier map. With mirror images, 1234827 distinct components enter the sum. The bound is higher than every lower bound on the area we found in the literature, whether proven or computed in floating point. Anyone can check the proof by running code/run.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: The area of the Mandelbrot set is greater than 1.50651.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Ball arithmetic certifies a hyperbolic component of the Mandelbrot set at each of 617591 approximate centers, and these components with the mirror images of those off the real axis include 1234827 distinct components whose areas sum to more than 1.50651.

Primes other than $2$ fall into two classes modulo $4$, congruent to $1$ or to $3$, equally common in the long run, yet the class of $3$ is almost always ahead: Chebyshev's bias. Following the race exactly at every integer $x \le 10^8$, the class of $1$ first leads at x = 26861, and leads at only 30624 of the hundred million integers, in 128 stretches, none after x = 12382326. Weighting each $x$ by $1/x$, the class of $1$ leads on a share 0.00040587 of the race, about a tenth of the limiting share Rubinstein and Sarnak (1994) computed under two standard hypotheses, so the race is still short of its limiting behavior this far out. Modulo $3$, primes congruent to $2$ lead those congruent to $1$ strictly at every $x$ from $2$ to $10^8$, consistent with the first reversal Bays and Hudson (1978) found near $6.1 \times 10^{11}$.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: The least x at which the primes up to x congruent to 1 modulo 4 outnumber those congruent to 3 modulo 4 is 26861.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Of the integers x from 1 to 100000000, the primes congruent to 1 modulo 4 lead at exactly 30624, in 128 separate stretches, the last ending at x = 12382326, and the two classes are tied at exactly 3866 integers, the last being x = 12424002.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Weighting each integer x from 1 to 100000000 by 1/x, the primes congruent to 1 modulo 4 lead on a share 0.00040587 of the race, about a tenth of the limiting share of about 0.0041 implied by Rubinstein and Sarnak's logarithmic density of 0.9959 for the other side.

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Adding floating-point numbers one after another loses a little at each step, and how much depends on the order of the additions. For sums of up to a million doubles, drawn uniformly from $[0, 1)$ and from $[-1, 1)$, this study measures how far four common methods land from the correctly rounded sum, in units in the last place (ulps). Adding left to right, the error averages 196.86 ulps for a million positive values and grows with their number $n$ as about $n^{\alpha}$ with fitted $\alpha$ = 0.545, close to the square root that independent rounding errors predict. Pairwise summation stays within 2 ulps at every length. Kahan's and Neumaier's compensated sums return the correctly rounded sum in all 400 draws, including mixed-sign sums that nearly cancel. Every value is reproducible exactly, on any machine and Python version.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Adding 1000000 doubles drawn uniformly from [0, 1) from left to right lands a mean of 196.86 units in the last place (at most 589) from the correctly rounded sum over 50 draws, and the mean error grows as about n to the power 0.545 for n from 1000 to 1000000, near the square-root growth that independent rounding errors predict.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Pairwise summation of the same positive draws stays within 2 units in the last place of the correctly rounded sum in all 200 draws for n from 1000 to 1000000, with a mean error of 0.4 units at n = 1000000.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Kahan's and Neumaier's compensated sums equal the correctly rounded sum in all 400 draws, positive and mixed-sign, including mixed-sign sums of 1000000 values whose median condition number is 991.

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How often do stars like the Sun release superflares of $10^{34}$ erg or more? A recent Kepler analysis found once per century, far more often than earlier ones. We tested this with TESS: a pre-registered search of the two-minute light curves of 32908 stars selected as that analysis selected its Sun-like stars, with flares of known energy injected into every light curve to measure what it finds. It counted 78 superflares where the once-per-century rate predicts 23.4: 3.33 times that rate (interval 2.63 to 4.15), one per 33 years per star, far above the earlier rates. Every planned subset and sensitivity analysis gives a ratio above one. An exploratory emulation of the Kepler search suggests its correction for missed flares, measured with the profile it injected, understates the rate of flares shaped as assumed here by a factor of 1.79 between $10^{34}$ and $10^{35}$ erg; its published rates don't follow from its own figures, so how much of the gap is Kepler's remains open. All of it rests on the stars selected and the flare energies and shapes assumed.

• Published.Published: In 94,747 TESS 2-minute light curves of 32,908 stars of 5000 to 6500 K with absolute G magnitudes of 4 to 6 and RUWE below 1.4 (5,605 star-years), the pre-registered search, with its SAP test corrected after registration to require the uncorrected flux to rise, counted 78 flares of bolometric energy 10^34 erg or more where the rate Vasilyev et al. (2024) measured with Kepler predicts 23.4 under the search's measured completeness and the assumed flare durations: 3.33 times that rate (95% CI 2.63 to 4.15), about once per 33 years per star; the registered code counted 79 (3.37 times, 2.67 to 4.20).
• Published.Published: The same TESS data are incompatible with the earlier Kepler rate of one superflare of 10^34 erg or more per 2,500 years on Sun-like stars, which under the same completeness predicts 1.04 counted flares where 78 were found (Poisson probability below 10^-100).
• Published.Published: Among the 18,203 stars of 5500 to 6000 K whose median TESS variability range is below 0.3%, the 50 counted superflares are 3.63 times (95% CI 2.70 to 4.79) the 13.8 that the rate Vasilyev et al. (2024) give for their Sun-like stars predicts.

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