The return-level table as a certificate.

Buoy C, daily maxima, 7,437 points. A fitted distribution is usually a number an optimiser stopped at; here it is a box the Krawczyk operator has proved to contain exactly one stationary point of the likelihood — the score and its Jacobian evaluated over the whole box in arithmetic that rounds outward at every step — and over which the Hessian is proved negative definite, so the point is the likelihood's maximum there. Every later number is computed over that box, so a statistic or a level is an enclosure whose width is stated, not a float's last digits. The widest box here is the exponentiated Weibull's, ±1.4e-10. The generalized gamma is certified in Prentice's coordinates (μ, σ, Q), the same distribution with its long (α, c, λ) ridge straightened. On buoys A and B its likelihood peaks at the family's lognormal limit: the derivative of the likelihood in Q is proved negative over a stated neighbourhood of the lognormal fit, so every member there is less likely than the lognormal's maximum, and the search — started from both ends of the family — found no maximum inside it above that. It is left out of the ranking, named, because the lognormal it tends to is ranked. Where the exponentiated Weibull's climb runs past α = 10⁴ it is carried to its Gumbel coordinates and certified there, as it is at five hindcast series; on A's and C's annual maxima it runs on even there, toward k → 0 — a Gumbel law of ln H_s, the Fréchet tail the family reaches only in a double limit — so nothing is proved and those choices are REFUSED rather than handed to the families that did converge.

The first table is the paper's selector: A² for every family, "at its limit" where the generalized gamma's likelihood peaks at the lognormal, REFUSED where no maximum is certified. The unfiltered hourly series go three ways — the exponentiated Weibull on A (15.1 against the lognormal's 161), the lognormal on B (77.9 against 90.0), the generalized gamma on C (97.6 against 110.1). As the blocks grow, the winner moves, but never to the Weibull. On the annual maxima the Gumbel — the classical law of block maxima, one of the paper's six — is the decided best on B; on A and C the choice is refused, the exponentiated Weibull's climb running on toward k → 0 in its Gumbel coordinates. The second table asks the paper's other question: do the four criteria pick the same family? Mostly yes; on A's monthly maxima they split — Anderson–Darling, MSE and χ² pick the lognormal, Kolmogorov–Smirnov the exponentiated Weibull; on C's weekly maxima they split — Anderson–Darling and Kolmogorov–Smirnov pick the exponentiated Weibull, MSE and χ² the Weibull. χ² is undefined on the annual maxima of B, where no bin keeps an expected count of five — undefined, as the paper has it, not guessed. Every DECIDED choice here is decided in the sense that the enclosures do not overlap; that ranks the criterion's arithmetic on this sample, and it is not a test that one family is the truer model.

The paper's third finding — that the temporal aggregation moves the return level — is here with numbers under it. Reading only the family the statistic picks at each of the paper's four blocks, the 100-year wave at buoy B runs from 8.10 m (weekly maxima, the Gumbel) to 12.21 m (hourly, the lognormal), a factor of 1.51 from the same years of the same buoy; at A the factor is 1.33 and at C 1.33. On six of the 13 decided buoy series the family the statistic prefers puts the 100-year wave below the largest hour the buoy has already recorded — on B from the daily, weekly and monthly maxima (9.41, 8.10, 9.32 m against 9.80 m), and on C from the daily, weekly and monthly maxima (10.51, 9.19, 9.37 m against 11.25 m): the best fit by Anderson–Darling is not a fit that respects the extreme already on the record. Each level is the quantile at 1 − b/(T·8766) for a block of b hours, over the certified box; the enclosures are narrower than a millimetre and printed at their upper end. That width is the arithmetic's: the sampling uncertainty of a 1000-year level from twenty years is statistical, and it is not in these numbers — a certified fit is a certified point estimate.

Ifremer's WAVEWATCH III hindcast GLOBMULTI_ERA5_GLOBCUR_01 — the paper's data, 3-hourly on a 0.5° grid, 1993–2024 — read at 13 of its nodes: the Campos and Santos basins and four more on the Brazilian margin, and points in the regions the paper describes by name. Every hs chunk of every monthly file was fetched by byte range and hashed as read (corpus/ww3-points). On these points: the exponentiated Weibull is the decided best of the unfiltered series on 6 of 13; the Weibull is the decided best of none of the 52 paper-block series; the four criteria split on 20; the generalized gamma is certified on 15 and the decided best on 12, and on the other 37 its likelihood peaks at the lognormal limit.

Three things a fitting routine can do silently are decided here. On six series — the monthly maxima at Campos (α 1,287), the weekly maxima at Foz do Amazonas (α 951), the daily maxima in the Drake Passage (α 1,670), the unfiltered series in the central Antarctic Circumpolar Current (α 1,370), the daily maxima in the central Antarctic Circumpolar Current (α 2,323) and the weekly maxima in the North Atlantic (α 2,315) — the generalized gamma's maximum lies beside its lognormal limit, far out on the ridge where α, c and λ run together; certified there, it wins 17 of their 24 choices — Anderson–Darling's on five of the six — and a climb that stops at α = 500 and takes the limit for the answer hands every one of them to another family. Its certificate there is in Prentice's coordinates, with the likelihood written as power series in Q so that nothing is divided by the small number the ridge is made of. On five series the exponentiated Weibull's climb runs to α of 10⁴, 10⁹, 10¹⁰, 10¹⁷, 10²⁷: in its Gumbel coordinates the same family is a Gumbel law of H_s^k, its maximum an ordinary point, and it is certified there. And the Arabian Sea shows what the selector does with that: the exponentiated Weibull is the decided choice at every block, and its 100-year wave is 360.9 m, 128.5 m, 46.3 m and 23.6 m from the 3-hourly, daily, weekly and monthly maxima, against a largest value in thirty-two years of 6.87 m. Its A² on the 3-hourly series is 965 — the best of six families none of which captures what the paper itself calls, there, "effectively multimodal or strongly skewed distributions that are poorly captured by a single parametric model" (§3.2) — and the "marked reduction in projected extremes as block size increases" the paper reports there comes out of a tail no sea has. Where the unfiltered choice is decided, the preferred family's 100-year wave runs from the record itself (at the central Antarctic Circumpolar Current) to 53 times it (at the Arabian Sea), and stays within 10 % of it at Campos and Santos. A handful of nodes is not the paper's global map — the return-level atlas certifies the same method at every cell of its lattices — but it is the paper's method, decided, on the paper's own numbers, at places anyone can name.

The environmental-contour benchmark (Haselsteiner et al., Ocean Engineering 2021) gave nine teams the same ten years of buoy data; its appendix prints the Hs marginal each fitted. Every printed row is decided here against those hours. Contribution 8's two-parameter Weibull for A and B is exactly the maximum-likelihood fit, to its last digit; its row for C is also a certified maximum-likelihood Weibull — of the zero-up-crossing period, not of the wave height, so the table's Hs marginal for C describes the wrong variable. Contribution 3's three-parameter lognormal, fitted with scipy, is a certified local maximum on all three buoys, of a likelihood that has no global maximum (it grows without bound as the location approaches the smallest hour; Hill 1963) — which is what a free location buys. The baseline's location sits on the smallest hour of each buoy to every printed digit; whether that hour is inside the support is below the printed precision, and the page leaves it undecided. Contribution 9's locations leave 10,426, 10,904, 9,410 hours of A, B and C below the support — hours its model says cannot happen, among the low sea states where the benchmark paper found that contribution's thousands of exceedances. Contribution 4 fitted by weighted least squares, deliberately not by likelihood, and sits 877 or more log-likelihood units below the exponentiated Weibull's maximum on A — a different estimator, not an error. Together: from the same ten years of one buoy, a 100-year wave anywhere from 5.25 to 14.53 m.

scipy 1.18.1, the routine a Python pipeline reaches for, called on the same block maxima two ways. With the location fixed at zero — the families as the paper writes them — it agrees with the certificate to half a centimetre on 45 fits of 48; on 3 it stops short — the generalized gamma on C's hourly series 60.8 log-likelihood units below the maximum, its 100-year wave 1.25 m low; the exponentiated Weibull on A's weekly maxima just 0.00082 units below, and 7.4 cm low, because that likelihood is nearly flat along a ridge; for the ten generalized-gamma series whose likelihood peaks at the lognormal limit it prints a point on the way — 0.69 to 2,529 log-likelihood units below the lognormal's maximum — and a return level with it; and where the certificate refuses the exponentiated Weibull (A's and C's annual maxima) it prints one too, which nothing here can decide. Called the default way, location free, it fits a different model whose likelihood is unbounded (Smith 1985), and prints where its optimiser stopped: the 100-year wave moves by −8.12 to +4.01 m against the certified fixed-location fit; one Weibull puts its location above one of A's monthly maxima (1.265 m against 0.7390 m), so an observed month has zero density; and 7 default fits lie below a member of their own family — on A's hourly series by 19,850 log-likelihood units, with a 100-year wave of 7.08 m against the certified 15.20. None of this is a bug in scipy. It is what "fit the family by maximum likelihood" leaves unsaid: where the location sits, what to do when the likelihood runs to the edge of the family, and when to stop. That packages disagree on the three-parameter Weibull is old news (Harper, Eschenbach & James, The American Statistician, 2011); what is new here is that each disagreement is decided.

Blocks in the order unfiltered, daily, weekly, monthly (and annual for the buoys). ξ: the GEV's, certified (each the upper end of a box narrower than its last digit). Among seven: the family Anderson–Darling decides once the GEV joins the paper's six — N normal, LN lognormal, W Weibull, EW exponentiated Weibull, GG generalized gamma, Gu Gumbel; — refused. The daily 100-year Hs of the family the six decide, in metres, certified; in brackets its 95% interval from the delta method, which is STATISTICAL, not certified.

The paper's six hold one extreme-value law, the Gumbel — the generalized extreme value law (GEV) at ξ = 0, the limit law of block maxima. Here the GEV joins the six on every series, written so that ξ = 0 is an ordinary point (as Prentice's Q = 0 is for the generalized gamma) and refused where its box reaches ξ = −0.5, where the maximum is no longer regular (Smith 1985) — and never ranked among them: the tables above are the paper's. It is certified on 67 of 67 series; its ξ is certified above zero — a Fréchet-type tail, heavier than any exponential — on 35 and below zero — a finite upper end — on 32. Anderson–Darling's choice among the seven names the GEV on 16 of the 65 series where it is decided, displacing the family the six decided on 16. On three unfiltered series the statistic has weighed the bulk of every value and names it — buoy A's hourly series (a 100-year wave of 43.22 m against a record of 11.80 m), the 3-hourly series at Pelotas (a 100-year wave of 12.80 m against a record of 10.11 m) and the 3-hourly series in the Arabian Sea (a 100-year wave of 1,278 m against a record of 6.87 m): a fit to the bulk, read in the tail. On the block maxima of the Brazilian margin it wins with its ξ certified below zero — Campos monthly, Santos weekly, Espírito Santo daily, weekly and monthly and Potiguar daily and weekly — that is, with a finite upper end, which none of the paper's six can have. On A's and C's annual maxima the choice stays refused: the exponentiated Weibull runs to its Fréchet corner there — a Gumbel law of ln Hs is a Fréchet law — and the GEV, certified with ξ = 0.257 and 0.311, finds that tail; deciding the choice needs the proof that the exponentiated Weibull's supremum is the Fréchet limit itself, the way the generalized gamma's lognormal limit is decided.

What no certificate here says is how far another record of the same length could move a number — the sampling width. The ledger carries it apart, STATISTICAL and labelled so: the delta method's 95% intervals, from the observed information at the fitted point in the certificate's own coordinates, asymptotic, with the blocks taken as independent and the family as right. Buoy A's daily maxima: the exponentiated Weibull's certified 100-year wave, 15.05 m, carries 13.71–16.51 m. The interval is what a design basis would argue about; the certificate is what it would not have to.