A portal · 13 endings on one axis
The Shape of an Ending
A question closes. What is left behind is an object, and a skeptic who wants to know for themselves has to go and look at it. These objects are not remotely the same size. The smallest one here is a few dozen bytes of coordinates. The largest is almost 200 terabytes. Both can be checked all the way to the bottom. The one that cannot is in the middle.
This page teaches none of these subjects. Each one has its own full layer and every row below links straight there. What is here is a single axis and what the axis shows: how much you would personally have to check, measured as the size of the thing that ended the argument.
Then a second axis, laid over the same members: how long the question stayed open. The rows never re-sort, so switching the axis lets you watch the dots leave their order.
Drag to move the cursor along the axis. The nearest ending opens below, with its own real numbers and the concrete act a skeptic has to perform.
at this scale: 400 bytes, a line of text
Eight Cells Against a Century
what a skeptic actually has to do
The colours are the argument, and they are not sorted. Both ends of the size axis are green or amber. At the small end, Zagier's sentence is one map applied by hand to five triples, about a minute of work, and the smallest object of all, Weaire and Phelan's eight coordinate triples, is one inequality and an afternoon. At the large end the almost 200 terabyte file is checkable too, and the price is stated: its authors report about 13,000 CPU hours to regenerate the proof from the 68 gigabyte certificate and about 16,000 CPU hours to check it. Those are CPU hours, not wall-clock hours. Divided by the 800 cores they had, the check is about 20 hours; divided by one core it is about 1.8 years. The division is ours, the 16,000 is theirs, and the same division reproduces their own stated wall clock for the solving run: 35,000 CPU hours over 800 cores is 43.75 hours, which is the "about 2 days" in their abstract. So the far end of the axis is a large machine job, not a large human one, and not a short one either. The single red marker, the one thing here that no person has read end to end and no machine has checked at all, sits in the middle.
Do the two orderings agree?
ρ
| layer | size | rank by size | years open | rank by years | d |
|---|
Spearman's rank correlation,
computed in the browser from the raw figures in the rows above, and recomputed offline by
node research/the-shape-of-an-ending/verify-the-shape-of-an-ending.mjs. With this
few points a correlation of this size is not statistically significant, and this page does not
claim it is. The claim is the weaker, certain one: the two orderings
are not the same ordering.
The uncomfortable middle
Almost every intuition about proof runs on one sliding scale: short means trustworthy, enormous means suspicious. This page exists because that scale does not survive contact with the actual objects.
At one end, Zagier's proof that every prime one more than a multiple of four is a sum of two squares is a single sentence. You apply one map by hand to a handful of triples and you are finished. At the other end, the refutation saying 7825 cannot be two-coloured without a monochromatic Pythagorean triple is almost two hundred terabytes of machine derivation. No person will ever read a millionth of it, and it is completely checkable: the authors published a 68 gigabyte compressed certificate from which the whole proof can be regenerated and run through a verifier.
Between them sits the classification of the finite simple groups: somewhere between ten and fifteen thousand printed pages, spread over hundreds of papers by roughly a hundred authors. Too long for any person to have read. Too informal for any machine to check. It was announced complete in 1983, and a gap in the quasithin case went unnoticed until Aschbacher and Smith closed it in two volumes of 477 and 743 pages, published in 2004, twenty one years later.
Size is not the axis of doubt. Checkability is, and checkability is not monotonic in size.
Not the same page as Past the Last Case
This corpus already has a portal about the distance between a finite check and a claim about every case, Past the Last Case: that one is about a logical distance, whether an argument reaches past the last thing you tested, and this one is about a physical one, how large the object is that you must pick up in order to see that it does.
Three that are not on the axis at all
Three members carry no certificate object, and forcing them onto the size axis would misdescribe what they are. While the axis is size their rows carry no marker on the track at all. The hollow ring appears instead in the value column, off the axis entirely, because there is no position on this axis that would be true of them.
Two are questions that have not ended: R(5,5) is known only to lie between 43 and 46, and the chromatic number of the plane is known only to be 5, 6 or 7. The third is stranger. Bipartite Ramanujan graphs of every degree were proved to exist by Marcus, Spielman and Srivastava in 2015, and the proof produces no example. There is nothing to inspect. That is a different epistemic kind, not a very large or a very small number.
The check · every figure, its unit, and whether it was measured
The page is the numbers, so one unsourced figure would undermine all of it. Every row below carries the raw figure its axis position is computed from, how that figure was obtained, and where it came from. There are three ways a size got here and the rail draws each one differently. Sizes marked measured are byte counts of an object this document is physically holding, printed in full underneath, and drawn as discs. Sizes marked estimated are conversions through a stated convention, and drawn as diamonds. Sizes marked published are file sizes quoted from the authors, measured by nobody here, and drawn as squares. Those last two are the two largest cards on the axis, which is exactly why they must not look measured.
Sizes first, then the dates. On a narrow screen each table scrolls sideways.
| layer | the object | raw figure | bytes | how obtained |
|---|
And the two dates on each card
Years open is the difference of these two, recomputed in the browser and again by the verifier. The two still-open questions are measured to 2026, and one card has no dates at all, so nine of the thirteen enter the correlation.
| layer | asked | what was asked | ended | what ended it | years |
|---|
The measured objects, in full
Four sizes here are byte counts of text present in this document. The browser counts them
with TextEncoder; the offline verifier counts the same bytes out of the shipped
HTML, so neither can agree with a number the page merely printed.
Weaire and Phelan: the eight generating sites ()
(0,0,0) (2,2,2) (0,1,2) (0,3,2) (2,0,1) (2,0,3) (1,2,0) (3,2,0)
Bhargava and Hanke: the 29 critical integers ()
1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290
Zagier: the whole proof (, transcribed into plain ASCII)
The involution on the finite set S = {(x,y,z) in N^3 : x^2 + 4yz = p} defined by (x,y,z) |-> (x+2z, z, y-x-z) if x < y-z, (2y-x, y, x-y+z) if y-z < x < 2y, (x-2y, x-y+z, y) if x > 2y has exactly one fixed point, so |S| is odd and the involution defined by (x,y,z) |-> (x,z,y) also has a fixed point.
EQP: the two input equations and the fifteen it derived (, semicolon separated)
n(x + y) = n(x); n(n(n(x) + y) + n(x + y)) = y; n(n(n(x + y) + n(x) + y) + y) = n(x + y); n(n(n(n(x) + y) + x + y) + y) = n(n(x) + y); n(n(n(n(x) + y) + x + y + y) + n(n(x) + y)) = y; n(n(n(n(n(x) + y) + x + y + y) + n(n(x) + y) + z) + n(y + z)) = z; n(n(n(n(x) + y) + n(n(x) + y) + x + y + y) + y) = n(n(x) + y); n(n(n(n(x) + y) + n(n(x) + y) + x + y + y + y) + n(n(x) + y)) = y; n(n(n(n(n(x) + y) + x + y + y) + n(n(x) + y) + n(y + z) + z) + z) = n(y + z); n(n(n(n(n(n(x) + y) + x + y + y) + n(n(x) + y) + n(y + z) + z) + z + u) + n(n(y + z) + u)) = u; n(n(n(n(x) + x) + x + x + x) + x) = n(n(x) + x); n(n(n(n(n(x) + x) + x + x + x) + x + y) + n(n(n(x) + x) + y)) = y; n(n(n(n(x) + x) + x + x + x + x) + n(n(x) + x)) = x; n(n(n(n(x) + x) + n(n(x) + x) + x + x + x + x) + x) = n(n(x) + x); n(n(n(n(n(x) + x) + n(n(x) + x) + x + x + x + x) + x + y) + n(n(n(x) + x) + y)) = y; n(n(n(n(n(x) + x) + n(n(x) + x) + x + x + x + x) + n(n(n(x) + x) + x + x + x) + x) + x) = n(n(n(x) + x) + n(n(x) + x) + x + x + x + x); n(n(n(x) + x) + n(n(x) + x) + x + x + x + x) = n(n(n(x) + x) + x + x + x)
Every free choice, named
- The unit is crude on purpose. Four cards count bytes of a text or a data object, three convert printed pages through a convention, two quote a file size the authors published, one derives an edge list from a published edge count. These are not the same kind of byte. A finer unit would not change the finding, because the finding is a gap of twelve orders of magnitude and the disagreement between the units is a factor of two.
- Printed pages become bytes at a rate you can change. The default is 2,000 characters per printed page. Move the slider below and watch the three page-counted cards slide. Nothing else moves, the ordering does not change anywhere in the slider's range, and the verifier asserts exactly that.
- The classification's page count is a range and a convention, not a measurement. Gorenstein's 1983 description says about 10,000 printed pages; the AMS description of GLS Number 7 gives about 15,000. Their scopes are not identical. The marker is a hatched band across the whole range, and its rank uses the geometric middle.
- Zagier's sentence is not the first ending of its question, it is the smallest one. Fermat stated the two-squares theorem to Mersenne on 25 December 1640 and Euler communicated a complete proof on 12 April 1749. The size axis uses Zagier's 1990 sentence; the years axis uses Euler. Those are two different events, and the card says so.
- One card has no dates at all. The 290 theorem's deciding document is a preprint with no date, no version and no journal, as that layer's own research record states. It therefore carries no years figure and is excluded from the rank correlation. The exclusion is printed, not hidden.
- One start date is a bound, not a year. Heule, Kullmann and Marek's abstract says Ronald Graham offered a prize "over two decades ago", writing in 2016, and does not name a year. That card is ranked at 20 years, which is a lower bound, and the verifier also recomputes the correlation with the card moved to the very top of the years ordering, to show the conclusion does not depend on the choice.
- Weaire and Phelan are counted as seed data only. The eight sites are the whole of what the numerical search returned. The weighted-Voronoi rule that grows cells from them, and the area comparison that makes them a counterexample, are not counted. That card also carries two different area comparisons and this page keeps them apart: the historical relaxed-face result of about 0.3 percent, whose meshes neither this page nor its member layer possesses, and the rigorous flat-face inequality of 18.57752 against 18.581656367, a margin near 0.022 percent, which is the one a reader can perform from the eight sites.
- The bunkbed size is an encoding, not a file. Nobody published that graph as a file. The figure is its 14,442 edges written in the plainest encoding, two vertex indices and a separator, derived live from the published edge count.
- Certificate sizes are quoted in the authors' own units. 850.2 MB, 68 GB and "almost 200 terabytes" are as printed. Whether each is decimal or binary was not resolved here, which is at most a ten percent effect on a log axis spanning twelve orders. The word almost is stored on the row beside the raw figure and printed with it everywhere, because the stored figure is exactly 200 TB and the source says almost. These two cards are the only sizes on the page that nobody here measured, so they are drawn as squares and their value column says quoted.
- The cap set start date is an attribution we did not check at source. 1987, for Frankl, Graham and Rödl, is the standard account of the problem's history. We did not obtain that paper and did not verify its page range.
- The two open questions are measured to 2026, the year this page was written. Those two figures grow. Every other figure on the page does not.
Only the three page-counted cards move. Watch the correlation above: it does not.
What this page does not establish
It does not establish that any of these results are correct. Each is the subject of its own layer, with its own verifier, and the correctness claims live there and are attributed there.
It does not establish a relationship between size and difficulty, or size and importance, or size and how long a question stayed open. The rank correlation above is computed from nine points. Nine points cannot support a claim about proofs in general, and none is made. What nine points can do, and what they do here, is refute the specific claim that the two orderings are the same ordering.
It does not establish that the classification of the finite simple groups is wrong, or in doubt. Its own layer is careful about this and so is this one: the classification is a cited theorem with a repaired branch and an ongoing second-generation write-up. What this page says about it concerns only its size, and what that size costs a person who wants to check it for themselves.