Sixteen thousand breaking waves, decided against the laboratory — and against the paper.

The paper's central numbers survive exact re-derivation. The three quadratic laws of Figure 3 are least squares through the origin on cb²/g, and each printed coefficient is the rounding of the exact rational — 0.8268, 0.6602, 0.2032. The standard deviation and the coefficient of variation of the aspect ratio (0.26, 46 %) are the roundings of enclosures 0.261030 and 46.150 %; the mean, 0.5656, is printed as 0.56, its truncation. The self-similar fraction — every event at or beyond mean + 2 sd on a ratio, sample sd as pandas computes it — is 4.429 % · 4.141 % · 4.013 % across the three ratios, inside the printed 4 ± 2 %, and on the 725 tail events of cb²/(g·Lb) the fit is 0.2955 with r(cb², Lb) = 0.858 — the paper's 0.30 and 0.9.

Three are not the table's. "The range of θ observed here (3° to 71°)" is not the range of the column named theta, which the histogram script plots and which runs 1.38° to 42.97° with no event beyond 45°; it is the integer parts of a different angle, tan⁻¹(Δz_max / eB), whose tangent runs from 0.0614 (3.51°) to 3.0331 (71.75°), decided through enclosed tangents. The Hs column of Table 1 is, to two decimals, the pickle's column sv_fp2 in 19 of 20 records and the pickle's column named Hs in none: the record carries a column called Hs (0.20–0.75, every value a multiple of 1/256) that is not the significant wave height the paper reports, and the earlier version of this page, which read that column, said "Hs 0.20–0.75 m" — the stats row above now reads the printed Hs. And "Ab/L²_D81 = 0.22" is the slope through the origin of the area at the frame of maximum length on the squared Duncan length (0.2243), whereas the histogram, the mean and the tail use the maximum area, whose slope is 0.2696; "Ab/Lb² = 0.27" matches no subset under either definition.

Two printed coefficients cannot be decided at all: "eA/eB = 2.37" and "Ae = 2.5 Ab" are read off RANSAC fits — a randomised robust regressor whose coefficient depends on the seed — and the scripts print them beside a Pearson r from linregress. The r beside the first (0.8) is the rounding of 0.760; the r beside the second (0.9) is not the rounding of 0.793 on the columns the script sets, though the summed columns give 0.862. "r(Ab, Δt) = 0.6" is 0.444 on the maximum area and 0.613 on the summed area. "Pearson r < 0.45 in all cases" holds for 31 of the 32 speed–geometry pairs; speed against vertical extent is 0.4536. In Table 1, 7 cells are truncations rather than roundings (Tp 4.066 printed 4.0, Hs 1.7254 printed 1.72), 3 are ties (U10 9.95 printed 9.9), and two are neither: Hs 0.6759 printed 0.65 and Tp 6.585 printed 6.7.

Duncan (1981) towed a hydrofoil through a tank and measured the breaking region it made: a wedge inclined at 10–14.7° whose cross-sectional area was 0.11 of the square of its length. The authors' scripts draw both numbers over their histograms, and this page decides both event by event with no tolerance: an inclination is INSIDE, ON or OUTSIDE the band by an exact comparison of the literal with 10 and 14.7; an aspect ratio is above or below 0.11 by an exact sign. One caveat belongs to the comparison itself and is the authors' as much as ours: Duncan's 0.11 relates a vertical cross-section to a length, whereas the area measured from above a real sea is the whitecap's plan view. The script draws the line on that histogram; the page decides it as drawn, and says so. The paper prints the same comparison as 0.56 ± 0.26 against 0.11 ± 0.01.

If breaking waves were geometrically similar to one another, a length made from the speed (cb²/g) would be a fixed multiple of the measured length, and the area a fixed multiple of the length squared: each ratio would be a constant up to measurement noise. Here each quantile is an order statistic of the exact ratios — the k-th smallest value, so a number that some event actually has — and the spread factors are exact ratios of two of them. The middle half of the events spans a factor of two in every ratio (q75/q25 = 2.062, 1.874, 1.875); the central ninety per cent, a factor of four to six (q95/q05 = 5.771, 4.263, 5.435). The paper's test marks the tail beyond the mean plus two standard deviations and calls the events in it self-similar: that threshold is enclosed here (the standard deviation needs a square root) and the count at or beyond it is exact — 725, 678 and 657 events, 4.429 % · 4.141 % · 4.013 %, with no event close enough to the threshold to be undecided. The spread factors are the same finding in a form that does not depend on a distribution.

"No predictable relationship" is a sentence about correlation, and a rank correlation is a rational number: average ranks, doubled, are integers, and Pearson's formula on integers is a ratio of two integers. The speed of a breaker ranks with its length at ρ = 0.360, with its area at 0.375, with its vertical extent at 0.410, with its duration at 0.150 and with its inclination at 0.097. Length ranks with area at 0.902. The paper prints Pearson's r for the same pairs — r(Ab, Lb) = 0.9, r(Ab, L_D81) = 0.8, r(Δz, eB) = 0.7 are the roundings of 0.898, 0.808 and 0.679 — and a breaker's speed explains at most ρ² = 0.168 of the rank variation in any property measured.

The paper's abstract says the scaling "varies with the sea state"; its conclusions say that showing it "would require a dedicated stratified analysis by Hs, Tp, or wave age, which is beyond the scope of this study". The table above is that analysis, on the paper's own data with the paper's own fit. Each record is one stereo-video run from the Katsiveli platform, its Hs, Tp and U10 constant within it (verified: no record carries two values). The fitted coefficient runs from 0.665 to 1.106 — a factor of 1.66 — around the pooled 0.827, and the four records with Hs above 1.5 m (1 October 2013, winds 15.7–18.0 m/s) carry the four lowest values, 0.66, 0.70, 0.74, 0.74: in the roughest seas a breaker of a given speed is shorter. Across the twenty records the coefficient ranks with Hs at ρ = -0.493 and with Tp at -0.492, with U10 at -0.365, and with wave age cp/U10 at -0.026 — every ρ² an exact rational on twenty ranks (the wave age needs no π: its rank is the rank of 1/(fp·U10)). Twenty points do not make a law, and the records are neither independent nor equal in size (3,320 events in the largest, 40 in the smallest); what they do make is the sentence the abstract asserts, with a number under it and the variable named. The share of events in Duncan's band, by contrast, barely moves with anything (ρ with Hs -0.29, with U10 0.02), and the self-similar tail is concentrated in the rough records: 10%, 9%, 12%, 11% of their events against 4.4% overall.