What is certified, and what is not
At every cell and block, each family's fit is a box the Krawczyk operator proves holds exactly one zero of the score, with the Hessian proved negative definite over it — the likelihood's one maximum there — or it is refused with the reason: the generalized gamma is left out only where its likelihood is proved to peak at its lognormal limit beside the lognormal fit; any other refusal blocks the choice. The four criteria and the levels are enclosures over the box; a family is chosen only where its enclosure lies wholly below every other's. That no better maximum lies elsewhere in a family is the search's claim, as it is any optimiser's. Not certified: that a family is the true law of the sea, the sampling width of a 100-year level, or the model's own error against the sea it simulates. A cell certified in your tab also draws its return-level plot from the same certificates: each family's quantile over return periods from two blocks to 10,000 years, every point an enclosure narrower than the line, beside the block maxima and the record.
The data: the Ifremer WAVEWATCH III hindcast GLOBMULTI_ERA5_GLOBCUR_01 (0.5°, 3-hourly, 1993-01-01 to 2024-12-31), the paper's own, CC BY-SA 4.0. Every hs chunk of every monthly file was read by byte range and hashed as read (the same hashes corpus/ww3-points pinned for the same files), and each UTC day's largest value kept at the nodes of a 4° global lattice, a 1° lattice of the Brazilian margin and the report's thirteen nodes; a cell is the node's series, not an area mean. 679 cells that the hindcast's own sea-ice field touches at some 3-hourly step of the 32 years are shown and not certified: under ice the model damps the waves to millimetres rather than leaving a gap, so the series is partly the ice's. The daily, weekly and monthly blocks are here; the unfiltered 3-hourly block is certified at the thirteen nodes of the return-level report, where six fits of 93,504 values take minutes each, and a report node's cell shows that record beside the atlas's own (certs/hseva-ledger.json of 2026-10-09, sha256 f7b037962511). Land: Natural Earth 1:50m, public domain.
A seventh family
The paper's six hold one extreme-value law, the Gumbel, and it is the ξ = 0 member of the generalized extreme value law (GEV), the limit law of block maxima. Here the GEV joins the six at every cell and block — 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 claims and the six-family views are the paper's. It is certified at 7,765 cell-blocks, refused at ξ ≤ −0.5 at 2; its ξ is certified above 0 (a Fréchet-type tail, heavier than any exponential) at 2,840 and below 0 (a finite upper end) at 4,925. Anderson–Darling's choice among the seven (the seventh-family view) follows the paper's rank rule unchanged — a family refused other than at its proved edge blocks it, so the GEV can decide only where the six are decided or tied — and names the GEV at 1,489 cell-blocks and one of the six at 6,193; at 1,488 the GEV displaces the family the six decided, and at 1 it decides a tie among them. Where it wins it can run as far from the sea as the six do: at 12° N 52° E (daily maxima, the Arabian Sea's two-mode monsoon climate) its 100-year wave is 397.2 m against a record of 5.23 m. At the 45 cell-blocks where the exponentiated Weibull runs toward k → 0 — its Fréchet corner: a Gumbel law of ln Hs is a Fréchet law — the GEV is certified at 45, its ξ positive at 45 and its A² below every certified family's at 45: the tail the corner points to. The choice among seven stays REFUSED there, because the exponentiated Weibull's supremum is not yet proved to be that Fréchet limit itself — the generalized gamma's lognormal limit is decided that way, and this is the next proof.
Statistical, not certified
A certificate says what a fit IS on these 32 years. It does not say how far another 32 years of the same sea could move it: that is a statement of another kind, and it is drawn apart — screened cells, dashed lines and dash-underlined numbers are STATISTICAL, asserted and not decided (design/grammar.js). The uncertainty view, the cell's intervals and the band on its return-level plot are the delta method's 95% intervals, taken on the log of the level so they stay positive: the observed information at the fitted point in the certificate's own coordinates, the level's gradient by central differences — asymptotic, with the blocks taken as independent and the family as right. A cell certified in your tab can also profile the likelihood of its 100-year wave, an interval that need not be symmetric. The decided family's 100-year wave carries an interval 7% of its level wide at the median cell (daily maxima), 12% (weekly), 17% (monthly); at 144 monthly cells it is wider than 40% of the level. The GEV's ξ: its 95% interval lies above 0 at 2,048 cell-blocks, below it at 3,591, and holds 0 — the Gumbel not ruled out — at 2,126 (dashed outlines in the tail view). Daily maxima come in storms: the runs estimator of the extremal index (days above the series' 95th percentile, a new storm after 3 quiet days) runs from 0.16 (15° N 62° E) to 0.68 (32° S 52° W), median 0.44 over 2,589 cells — so the design-life level's "successive maxima independent" counts more independent chances than the sea gives.
A fit someone printed
A design table, a report or a pipeline prints a fitted distribution for a place: a family, a block, parameters to so many digits. In a cell's tab it is decided against that cell's block maxima by the rule of the return-level check — one module, playground/return-level-check/printed.js, decides on both pages. The printed family is certified on the printed block by the ledger's own code, in a worker (the annual block too, which is not one of the paper's), and the digits are read as the box they allow: REPRODUCED when they are the rounding of the certified fit; CONSISTENT when they lie inside its box; OFF THE MAXIMUM when their likelihood is proved lower than the certified maximum's; NOT THE CERTIFIED FIT when the digits exclude the maximum but cannot separate the likelihoods; OUTSIDE ITS SUPPORT when a printed location, or a printed GEV's end, leaves observed values where the density is zero; NOT A MEMBER OF THE FAMILY when the digits allow a scale or a shape at or below zero; NOT DECIDED when they are too few to say. A fit made from the paper's hindcast at a cell's node is decided exactly; one made from another record is decided against this one, and the tab says how far the node lies from the place typed — to decide it on its own record, bring that record as your own site. What is typed stays in the tab: nothing is sent, and the address does not carry it.
A printed table, decided
Bhaskaran et al. (2023), Energies 16, 6935 (doi:10.3390/en16196935, CC BY) print the annual-maximum Hs fits of offshore-wind lease areas — the Gumbel by least squares, by maximum likelihood and by moments, the GEV by maximum likelihood — from a commercial hindcast (WaveClimate, 1992–2022) that cannot be re-run outside it. Their five South Atlantic sites, decided against the nearest cells of this public hindcast (31–61 km away; certs/design-table-audit.json): of the 20 printed fits, 19 are OFF THE MAXIMUM and 1 is OUTSIDE ITS SUPPORT. The printed 100-year waves differ from the certified fits' on these annual maxima by −2.03 to +3.14 m, outside the certified fit's statistical 95% interval at 19 of 20. That is not an error in the table. It is the distance between two records: another model, another point — two of the five sit in 27 m of water, where the portal offers a wave-ray translation inshore — and at Projeto Açu the two nearest cells, one degree apart, put the Gumbel's 100-year wave at 3.21 and 5.47 m. A printed number can be certified only against a record anyone can re-run. Each site opens its nearest cell, where every row is decided again in your tab: .
decided
exact arithmetic backs it
computed
float; nothing decided it
refused
no mark — the void is drawn
What this opens
The map answers where the paper's selection is a decision and where it is not, and two of the six families needed other coordinates to be decided at all. Where the exponentiated Weibull's climb runs past α = 104, write θ = λk and β = θ ln α: the family reads F = exp(eβ/θ ln(1 − e−xk/θ)), a Gumbel law of Hsk with location β and scale θ once e−xk/θ is small over the data. The runaway α is a regime the (α, k, λ) coordinates place in the millions or beyond, not an edge: in (k, θ, β) the family's maximum is an ordinary point, certified here at 219 cell-blocks. At 53 more the family is still refused, 45 of them because the climb runs on even there, toward k → 0 — a Gumbel law of ln Hs, the Fréchet tail the family reaches only in a double limit: there the seventh family below, the GEV, finds a Fréchet tail; the extended generalized Pareto of Naveau et al. (Water Resources Research, 2016) — the exponentiated Weibull’s own construction, a power of a distribution function, on a Pareto base — is the other candidate. The exp. Weibull view maps where each coordinate system holds. The generalized gamma's maximum sits beside its lognormal limit (α above 500) at 444 cell-blocks and is certified there, in Prentice's coordinates; its likelihood is proved to peak at the limit itself at 5,751. A fitter that stops at a threshold and calls it the limit names another family, or none, at 544 cell-blocks. And the choice the statistic makes is not a sanity check: the decided family’s 100-year wave reaches 18.7× the largest day on record (the exponentiated Weibull, 128.5 m against 6.87 m, 15° N 62° E, daily). The other way round proves less: the 100-year wave lies below the record at 2,607 of the 7,663 decided cell-blocks with a level (34%), and a 32-year record exceeds even a correct 100-year level with probability 1 − e−0.32 ≈ 27%. A standard for marginal fits can be read off this map — which choices it must fix, and where a certificate, not an optimiser, has to say what the fit is.
The code in your tab: playground/return-level-atlas/app.js 8838b9318484 · playground/return-level-atlas/worker.js c2d98e753808 · playground/return-level-check/parse.js 7bf7c1eed533 · playground/return-level-check/printed.js 8f54767742da · instruments/interval/interval.js 9891280d08e5 · instruments/interval/rational.js 59e97fc40328 · instruments/interval/transcendental.js 29da890cec15 · instruments/interval/radii.js 2d89926b8ee9 · instruments/hseva/special.js cdfa6ac130b8 · instruments/hseva/ad2.js 39e88af94e66 · instruments/hseva/families.js 027a859c72d7 · instruments/hseva/fit.js b6b67a513590 · instruments/hseva/blockrule.js 375250f46ae1 · instruments/hseva/atlas.js 113d8ce95d13 · instruments/hseva/atlas-claims.js 6159360b8170. Cell files served from the public repository at commit 151262781b90, each checked against its sha256 before it is used; the whole ledger is certs/hseva-atlas.json at the same commit.
instruments/hseva/.venv/bin/python3 tools/fetch-ww3-grid.py
instruments/hseva/.venv/bin/python3 tools/fetch-ww3-grid.py --ice
instruments/hseva/.venv/bin/python3 tools/build-ww3-atlas-corpus.py
node tools/run-hseva-atlas.js
node instruments/hseva/battery.js
node tools/run-design-table-audit.js
node playground/return-level-check/battery.js
nothing here is gated · every number came out of the record beside the page