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title:mandelbrot - SummaryWords in its Summary, the paper’s abstract
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published:2026-10-01..2026-10-15 - ImportanceIts importance score, such as 70 or more
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cites:doi:10.1214/ss/1009213286 - IsRetracted, a correction, or scored
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Searching for Published Oct 1, 2026 to Oct 15, 2026 Clear the words
Studies
- Its most important claim’s importance 60 out of 100: meaningful importance, under edition 3 of the rubric, being rated again
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: 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: 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: 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.
- Its most important claim’s importance 49 out of 100: limited importance, under edition 3 of the rubric, being rated again
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 $m > 1$.
- Its most important claim’s importance 48 out of 100: limited importance, under edition 3 of the rubric, being rated again
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: The area of the Mandelbrot set is greater than 1.50651.
- 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.
- Its most important claim’s importance 46 out of 100: limited importance, under edition 3 of the rubric, being rated again
Neurons spend most of their ATP on ion pumping (Attwell & Laughlin 2001) and receive oxygen by diffusion from capillaries, so energy-hungry neuron types might sit closer to capillaries. In whole-brain MERFISH atlases (Yao et al. 2023; Zhang et al. 2023) we measured each neuron's distance to the nearest endothelial cell relative to neighbouring neurons in the same local tile, and correlated each subclass's excess with an OXPHOS plus Na/K-pump expression score (preregistration). Across 239 subclasses there was no relation: rho = -0.018960304 (CI -0.14959145 to 0.11132168; p = 0.7770223), with no consistent sign across the replication brains. Every subclass sits within a few percent of its neighbours' capillary distance. Decision: refute.
- Its most important claim’s importance 45 out of 100: limited importance, under edition 3 of the rubric, being rated again
How sensitive is evidence for recent global warming acceleration to the final years included and the model of natural variability? A registered audit uses a frozen NASA global annual temperature series. A fixed-breakpoint model estimates a warming-rate increase of 0.232 °C per decade, with a confidence interval from 0.106 to 0.358. A search over possible breakpoints gives p = 0.0271 with fitted autocorrelation, but p = 0.4078 when the recent endpoint years are excluded. Stronger assumed autocorrelation also weakens the test. These are conditional statistical results about an unadjusted series, with limited simulated power for smaller slope changes.
- Its most important claim’s importance 40 out of 100: limited importance, under edition 3 of the rubric, being rated again
Evidence that small fMRI studies overstate effects comes mostly from reported peaks, which are inflated by selection whatever the sample size. We asked the question with a whole-brain measure that has no peak selection. For each published (DOI-linked), group-level, unthresholded task-fMRI T or Z map on NeuroVault (Gorgolewski et al. 2015), we estimated the root-mean-square standardized effect over a fixed grey-matter mask from the mean squared statistic. We then regressed its log on log N with map-type, smoothing, upload-year and paradigm fixed effects plus a random intercept per collection. Everything was preregistered (prereg:f7ecdaf6…) on a frozen map list. After QC and de-duplication, 1245 maps from 200 collections were analysed. The slope was β = -0.101387 (CI -0.272087 to 0.069313), so the locked decision is inconclusive. There is no significant small-study inflation, and steep slopes are excluded. The data cannot rule out a modest slope at the preregistered boundary.
- Its most important claim’s importance 39 out of 100: limited importance, under edition 3 of the rubric, being rated again
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.
- Its most important claim’s importance 39 out of 100: limited importance, under edition 3 of the rubric, being rated again
Why did United States heart-disease deaths increase between 2018 and 2024? An updated analysis of final national death counts and published population estimates averages all six orders of a demographic decomposition. Known-age deaths increase by 28135. Population size contributes 25981.4 deaths, age composition 26512.2, and age-specific rates -24358.5. Demographic contributions outweigh the negative rate contribution. These are arithmetic contributions conditional on the source denominators, which change estimation methods during the period, rather than causal effects or individual mortality risks.
- Its most important claim’s importance 38 out of 100: limited importance, under edition 3 of the rubric, being rated again
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.
- Its most important claim’s importance 30 out of 100: limited importance, under edition 3 of the rubric, being rated again
Preregistered TWFE DiD test of whether pharmacist direct-authority naloxone laws cut CDC WONDER age-adjusted opioid overdose mortality before synthetic opioids dominate a state's mix, then attenuate to null once T40.4 accounts for at least half of opioid overdose deaths. On the pinned panel (661 state-years after dropping 53 suppressed or unreliable cells, 2008–2021), ATT before dominance is -0.285902 (CI -3.772196 to 3.200392); the Law×SyntheticDominant interaction is -0.926561; ATT when dominant is -1.212463. Locked success flag false is false (decision refute). Re-run from code/analyze.py.
- Its most important claim’s importance 29 out of 100: limited importance, under edition 3 of the rubric, being rated again
Can simple daily sleep and positive-experience reports improve forecasts of emotional recovery beyond recent mood and stress? We fitted a fixed model in an earlier MIDUS daily-diary wave and evaluated it in held-out families there and in a later wave. The later-wave evaluation included 529 participants and 790 stressful days with elevated negative affect. Adding sleep duration and four common positive-event categories changed prediction error by -0.5602920767206551 percent in the improvement direction, with a conditional confidence interval from -1.4264876842069119 to 0.26668493108151836 percent. Thus the additional reports did not establish better predictions in this model. This does not show that sleep or pleasant experiences cannot help people recover. The plan was amended after a data-quality problem in an initial run; the amendment and original results are retained.
- Its most important claim’s importance 27 out of 100: limited importance, under edition 3 of the rubric, being rated again
Preregistered TWFE DiD test of whether pharmacist direct-authority naloxone laws cut crude opioid overdose mortality before synthetic opioids dominate a state's mix, then attenuate to null once T40.4 accounts for at least half of opioid overdose deaths. On the pinned panel (601 state-years, 2008–2021), ATT before dominance is -0.632217 (CI -4.144376 to 2.879942); the Law×SyntheticDominant interaction is 0.231578; ATT when dominant is -0.400639. Locked success flag false is false (decision refute). Re-run from code/analyze.py.
- Its most important claim’s importance 26 out of 100: limited importance, under edition 3 of the rubric, being rated again
Could unobserved cognitive-test scores overturn the low sensitivity of a question about worsening memory or confusion? We preregistered an analysis of public National Health and Nutrition Examination Survey data and assigned missing low-performance labels in the most and least favorable possible ways. Among 3469 examined older adults with a recorded answer, 457 lacked a Digit Symbol Substitution Test score. Observed-score sensitivity was 24.15%. Allowing arbitrary missing-score labels gives a sensitivity range of 19.27% to 33.78%; the upper endpoint has a confidence interval from 30.75% to 36.82%. Missing scores alone therefore cannot make the question identify most low performers under the prespecified definition. This extends an established low-sensitivity result with a missingness robustness analysis; it is not a dementia diagnostic study.
- Its most important claim’s importance 25 out of 100: limited importance, under edition 3 of the rubric, being rated again
Implementations of this protocol must agree byte for byte on claim IDs, bundle hashes, Merkle log proofs, signatures, and the IDs operators and volunteers go by. This fifth version adds vectors for IDs no log assigns: an operator's ID is op: and the SHA-256 of its first public key, a volunteer's is obs: and that of their first passkey, and an entry's digest as signed lets two logs compare entries. The log's example leaves now name their operator by ID. An independent checker recomputes every vector with 0 claim-ID, 0 bundle, 0 log, 0 signature, and 0 ID mismatches, and accepts none of the invalid cases. Hashing Python's json.dumps output still gets 3 of 20 claim IDs wrong.
- Published, Reproduced, Reviewed: The claim-ID test vectors in data/claim-id-vectors.json are correct: an implementation written independently of the reference node recomputes every listed claim ID from the claims.json text, verification inputs, and declared results given with it, rejects every claims file listed as invalid, and computes no ID for any case listed as unresolved, where a claim names a result the bundle doesn't declare.
- Published, Reproduced, Reviewed: The bundle test vectors in data/bundle-vectors.json are correct: an independent implementation recomputes every listed file digest, bundle hash, and verification-inputs digest over the files under code/, env/, data/, and proofs/, and rejects every path set listed as invalid.
- Published, Reproduced, Reviewed: The log test vectors in data/log-vectors.json are correct: an independent implementation of RFC 9162 recomputes the empty-tree root and every listed root hash, inclusion proof, consistency proof, and example leaf hash; every listed proof verifies, and every proof listed as invalid fails; and each example leaf holds its entry with every signature replaced by the SHA-256 of the signature's canonical JSON, over a signed entry that verifies against its operator's key; and each key entry's leaf names its operator by the ID that key makes.
- Its most important claim’s importance 23 out of 100: trivial or highly circumscribed, under edition 4 of the rubric
An account of depression as accumulated physical load distributed across body systems predicts that each system's state tracks the symptoms it produces, that bodily measures relate to somatic symptoms beyond mood, and that socially distributed load appears somatically. We tested these predictions in a registered analysis of 14722 adults in the US National Health and Nutrition Examination Survey (NHANES), relating limitation in load-bearing tasks, slow bowel transit, body mass index (BMI), an allostatic load index and income to PHQ-9 depressive symptom items. Load-bearing limitation, BMI and allostatic load related to somatic symptoms beyond cognitive-affective ones, and more than to cognitive-affective symptoms beyond somatic ones. But the predicted system-specific pattern did not appear: load-bearing limitation tracked sleep, energy and appetite (0.2946) far more than slow transit did (0.0404). Low income related more to cognitive-affective than to somatic symptoms (difference -0.0996). Bodily load is linked to somatic depression generally, not system by system.
- Published, Reproduced, Reviewed: In 14722 NHANES 2005-2010 adults, limitation in load-bearing tasks is associated with the PHQ-9 psychomotor item more than slow bowel transit is (age-, sex- and cycle-adjusted partial Spearman 0.2038 vs 0.0296), but also with the sleep, energy and appetite items more than slow transit is (0.2946 vs 0.0404; difference -0.2542, 95% CI -0.2762 to -0.2312), so the predicted double dissociation between a structural and a transport system does not appear.
- Published, Reproduced, Reviewed: In the same adults, load-bearing limitation, body mass index and a nine-marker allostatic load index are each associated with somatic depressive symptoms after adjusting for cognitive-affective symptoms (partial Spearman 0.1916, 0.0558, 0.08) and more strongly than with cognitive-affective symptoms after adjusting for somatic ones (differences 0.0711, 0.0399, 0.0646; Holm-adjusted p 0.002, 0.016, 0.002), whereas slow bowel transit shows no such specificity (difference 0.0074, 95% CI -0.0209 to 0.0347).
- Published, Reproduced, Reviewed: Lower family income relates more to cognitive-affective than to somatic depressive symptoms in these adults (partial Spearman 0.1275 vs 0.0279; difference -0.0996, 95% CI -0.1299 to -0.0695), opposite to the prediction that socially distributed load expresses itself somatically.
- Its most important claim’s importance 23 out of 100: trivial or highly circumscribed, under edition 3 of the rubric, being rated again
This work tests a preregistered question on open New York City data: whether a fixed lag of CDC National Wastewater Surveillance System (NWSS) SARS-CoV-2 percentile predicts rises in confirmed COVID-19 cases better than the same-day percentile. The locked primary contrast is the difference in area under the receiver operating characteristic curve (AUROC) between a seven-day lag and a zero-day lag. On the pinned public snapshots, the lag-seven AUROC is 0.449326 and the lag-zero AUROC is 0.522145, so the delta is -0.07282 and the preregistered success flag false is false. The negative result is deterministic and re-run from code/analyze.py.
- Its most important claim’s importance 22 out of 100: trivial or highly circumscribed, under edition 3 of the rubric, being rated again
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.
- Its most important claim’s importance 21 out of 100: trivial or highly circumscribed, under edition 4 of the rubric
Embodied accounts of affect propose that position, weight, tension and shape are bodily templates for mood and evaluation: down and heavy for sadness, up and light for happiness, tension for anger, crookedness for badness. If such mappings are deep, many unrelated languages should use one word for both senses. In two registered analyses of the CLICS4 colexification database, "light" and "easy" share a word in {{R2.pairs.LIGHT (WEIGHT)|EASY.families_colexifying}} of {{R2.pairs.LIGHT (WEIGHT)|EASY.families_with_both}} language families, mirroring "heavy" and "difficult", and "straight" shares a word with "correct" in {{R1.pairs.STRAIGHT|CORRECT (RIGHT).families_colexifying}} families. By contrast, "up" or "high" and "happy", "down" or "low" and "sad", and "light" and "happy" share a word in at most one family each, and "hard" never shares one with "angry". Across languages, weight is lexicalised as effort and straightness as rightness, but emotion is kept lexically apart from posture and weight.
- Published, Reproduced, Reviewed: In CLICS4 v1.0, one word means both 'light (in weight)' and 'easy' in 11 of 50 language families with words for both (rate 0.22), more often than 'light' pairs with any of its reference concepts, and 'heavy' and 'difficult' share a word in 7 of 57 families, so weight words double as words for ease and difficulty at both poles.
- Published, Reproduced: Words for vertical position or weight almost never double as words for happiness or sadness: up-happy 1 of 66 families, high-happy 0 of 145, down-sad 0 of 21, low-sad 0 of 108, light-happy 0 of 146.
- Published, Reproduced, Reviewed: 'Straight' shares a word with 'correct' in 38 of 155 families and with 'good' in 7 of 200, whereas 'crooked' shares a word with 'wrong' in 4 of 113 and with 'bad' in 1 of 171, so the straightness-evaluation link is lexicalised mainly at its positive pole.
- Its most important claim’s importance 21 out of 100: trivial or highly circumscribed, under edition 3 of the rubric, being rated again
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: 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: 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.
- Its most important claim’s importance 20 out of 100: trivial or highly circumscribed, under edition 4 of the rubric
Accounts that treat the felt weight of depression as more than metaphor rest on a premise: that distress is described in gravitational terms across unrelated languages. We tested a lexical form of that premise in a registered analysis of the CLICS4 database of cross-linguistic colexifications, asking whether languages use one word for "heavy" and for difficulty, grief, sadness or tiredness more often than for other concepts, and whether gravity-related words share words with sadness more than other physical-property words do. "Heavy" and "difficult" share a word in 7 of 57 families, more often than "heavy" pairs with any of its 1374 reference concepts, and "heavy" and "grief" in 2 families. "Heavy" never shares a word with "sad" (zero of 156 families) or "tired", and gravity-related words show no detectable excess with sadness (permutation p = 0.2232). Lexically, weight is widely the word for burden, not for sadness.
- Published, Reproduced, Reviewed: In CLICS4 v1.0 (3447 varieties, 247 families), a single word means both 'heavy' and 'difficult' in at least one language of 7 of the 57 families that have words for both (family-level rate 0.1228), higher than the rate for every one of 'heavy's 1374 reference partner concepts, whose 95th percentile is 0.0051; 'heavy' and 'grief' share a word in 2 of 69 families (rate 0.029, percentile 0.9964).
- Published, Reproduced, Reviewed: In the same data, 'heavy' never shares a word with 'sad' (0 of 156 families) or 'tired' (0 of 73), and 'sad' or 'grief' shares a word with any of 15 physical-property concepts in at most 2 families each, so gravity-related words (heavy, low, down) do not colexify with sadness detectably more than other physical-property words (difference in family-level rates 0.0031, one-sided permutation p = 0.2232).