Some of that number is the box
Ask how big a thing usually is and you get a number back. The number is real and correctly computed, and it may not be a fact about the thing at all. Some of it can be a fact about the container you measured in — and nothing on the face of the number says which part.
There is a pile in this garden. The scree piece drops grains on a slope sixty-four columns wide and tallies how far each slide runs. For revision 1 I lifted the pile's rule out of the page and ran it without the drawing, a few million grains in all, to ask it things the piece doesn't ask.
1.2 million grains fell on the sixty-four-column slope. Once the pile was full, 757,086 of them started a slide and the rest landed without moving anything. Mean run: 7.46 columns. Median: 3. Longest: 64. Nearly a third of all the slides move exactly one column and stop. Collect more of them and the mean settles down politely — 8.95 at a hundred slides, 7.86 at a thousand, 7.41 at ten thousand, 7.46 at a hundred thousand. By the usual test that is a well-behaved quantity with a typical size of about seven and a half columns, known better the longer you watch.
Then I widened the slope.
L= 32 mean 6.38 median 3 biggest 1% of slides carry 5.0% of the motion
L= 64 mean 7.48 median 3 8.6%
L=128 mean 8.45 median 3 15.2%
L=256 mean 9.24 median 3 23.4%
L=512 mean 9.86 median 3 28.6%Sixteen times the slope. The median does not move — not by a column, not once. Three is a fact about the pile. The mean rises by half, and the share of all the motion done by the biggest one slide in a hundred goes from a twentieth to nearly a third with no sign of levelling off. Those are facts about the pile and the box together, and no amount of watching one box would have separated them, because the running mean keeps reassuring you.
That is the part I would have got wrong if I had only reasoned about it. The folk diagnostic — does the average settle as the data comes in? — passes here. It passes because the slope is finite and a slide cannot run further than the pile is long, so there is a largest possible avalanche and it drags the mean to a halt. The cutoff is real. It just isn't the physics. It's the wall.
Why exactly three
Revision 1 left this open: the median held at three through a sixteenfold change in the box, which was a stronger result than I expected, and I did not know why it was exactly stable. It has two reasons, and only one of them is about the pile.
The first is that small slides never hear about the wall. I counted how often a slide stops after one column, after two, after three, and so on, on slopes from thirty-two columns to a thousand and twenty-four, with two different random seeds so that I could tell the noise from the physics.
L stops at 1 at 2 at 3 at 4 at 5 | 1 or 2 1 to 3
32 .2927 .1647 .1076 .0739 .0540 | .4574 .5650
64 .2930 .1645 .1076 .0740 .0534 | .4576 .5652
128 .2930 .1650 .1081 .0736 .0537 | .4580 .5661
256 .2924 .1657 .1081 .0739 .0539 | .4581 .5662
512 .2931 .1655 .1077 .0737 .0540 | .4586 .5663
1024 .2936 .1652 .1073 .0738 .0540 | .4587 .5661The second seed gives the same rows to the third decimal. There is nothing in this table that knows how wide the slope is. A slide that stops at column three has consulted the state of columns one to four and nothing else, and the state of those columns, once the pile is full, is set by the tipping rule and not by where the pile ends. The head of the slope does not learn about the far end because everything that carries information here travels downhill with the grains. So the whole bottom of the distribution — every probability up to whatever size can fit in the smallest box — is a fixed fact about the pile. That is the deep reason, and it is a real one.
The second reason is luck. Slides of one or two columns make up 45.8% of the whole; add the threes and it is 56.6%. The half mark falls inside the step at three, four points clear of one edge and six clear of the other. A median is an integer here, and an integer quantile is exactly stable when the mark it sits on lands in the middle of a step, and flips when the mark lands on an edge. Three is not special. The pile put a step there and the half mark happened to fall well inside it.
Which says how far the wall reaches. Every quantile up to the ninetieth, which is eighteen columns, reads the same on every slope. The ninety-fifth is thirty-three columns on every slope from a hundred and twenty-eight up; on the sixty-four-column slope it reads thirty-four, on both seeds, and on the thirty-two-column slope it reads thirty-two because it cannot read more. So the wall bends a quantile a little before it stops it — a column, at twice the quantile's own size. The ninety-ninth is about a hundred and thirty on the slopes of two hundred and fifty-six or wider, and on the narrower ones it is the wall itself: 32, 64, 128. So a quantile becomes a fact about the pile once the box is comfortably wider than it, and you can find that moment by widening the box until the number stops. Revision 2 went on: the mean has no such moment, because the mean is a sum over all of it, including the part of the tail that lies beyond every box you will ever build, and that part is not small. Hold that sentence; it is the one this revision is about. This is a fresh run with its own seed, so the second decimals differ from the table above; the shape does not.
L mean 99th percentile biggest 1% carry
32 6.39 32 5.0%
64 7.48 64 8.6%
128 8.43 128 15.2%
256 9.24 133 23.5%
512 9.93 129 29.0%
1024 10.55 129 33.1%
2048 11.11 131 36.1%Two thousand and forty-eight columns is sixty-four times the narrowest slope. The ninety-ninth percentile stopped moving four doublings ago. The mean and the top one per cent have not stopped, and the widest box I could afford was not wide enough to watch them stop. Revision 2 said that was the point. It is half of it.
Now the same pile with round grains, every column tipping at exactly the same steepness:
mean 64.00 median 64 longest 64 spread 0 biggest 1% carry 1.0%Every slide is the whole slope. Widen it and all four numbers move together and the shape is unchanged: at L=512 it reads 512, 512, 512, and one per cent. This is what it looks like when a quantity really does have a size. The mean isn't a summary of the round pile, it is a complete description of it, and you can plan against it. The top one per cent carrying exactly one per cent is what no concentration looks like once you write it down.
Both piles are the same loop. One line differs — whether a column's tipping point is redrawn as 1, or as 1 or 2 on a coin flip. Only one of them has an average you can read off a slope you could build.
Where the mean stops
Revision 2 said the mean never becomes a fact about the pile. Its own table disagreed, and I did not read it. Look at the steps between the rows of means above, one per doubling of the slope: 1.09, 0.95, 0.81, 0.69, 0.62, 0.56. A mean that grows without limit as the box widens keeps its steps, or grows them. These shrink, by about an eighth each doubling, and a thing that shrinks by a fixed fraction each time adds up to a finite amount. The mean is going somewhere. The table was saying so from the 128 row on, in a language I was not listening for, because I had decided what the wide-box story was before I had the wide box.
Where it is going, and how fast, are questions about the tail, so I measured the tail directly instead of arguing about it: on a slope of 1,024 columns, the fraction of slides that reach at least l columns, at each doubling of l, on two seeds. The last column is the exponent — how fast the fraction falls between l and 2l, so that 1.00 would mean halve the fraction each doubling and 2.00 would mean quarter it.
l P(reach ≥ l) exponent, l to 2l
seed A seed B seed A seed B
8 .2337 .2333 1.03 1.03
16 .1144 .1146 1.11 1.12
32 .0529 .0529 1.17 1.17
64 .0235 .0236 1.20 1.20
128 .0102 .0102 1.20 1.22
256 .0044 .0044 1.21 1.23
512 .0019 .0019 1.07 1.02The last row is bent by the wall, the way the ninety-fifth percentile was bent at sixty-four; on the 2,048-column slope that row reads 1.24 and it is the 1,024 row that bends. Between 64 and 512 the fraction falls as one over l to about the 1.2, and the exponent is still creeping upward when the wall interrupts it. This pile has a name, it turns out: physicists call it the Oslo model, a rice pile fed at one end with tipping points drawn from two values, and the exponents measured for it predict the survival here should fall as one over l to the 1.25. I looked that up after the runs, not before, and I am glad of the order.
The number that matters is whether it is above one. A mean is a sum of survivals — add up P(reach ≥ 1), P(reach ≥ 2), P(reach ≥ 3) and so on to the end and you have the mean reach, exactly. If the survival fell as one over l or slower, that sum would never finish, and revision 2 would have been right for a deeper reason than it gave. It falls as one over l to the 1.2. The sum finishes. Beyond a slope of width L the part still to be added shrinks like L to the minus 0.2, which is convergence so slow it is easy to mistake for none: doubling the box retires an eighth of what is left, and the rest is still out there.
Revision 3 then fitted the seven means to a limit minus a debt that shrinks as a power of L, three numbers free, and got the exponent 0.205 and a limit of 14.6 columns, every row within a hundredth of the curve. It read the 0.205 as the tail table the other way round — survival falling as one over l to the 1.2 is a debt shrinking as L to the minus 0.2 — and called the limit fourteen and a half, half a column either way. This revision is about where that half column comes from, because the fit does not say. A curve through seven points to a hundredth of a column has a hundredth of a column's uncertainty, and the answer does not.
Two things went unchecked. First, what the wall actually does to the tail, since the debt is the tail beyond the wall only if the wall leaves the tail inside it alone. Second, how well the exponent is known, since the 512 row of the tail table rests on some seven hundred slides.
The wall first. Here is the survival on a narrow slope divided by the survival on the 2,048-column slope at the same l, read at fractions of the narrow slope's width:
l/L 1/4 3/8 1/2 5/8 3/4 7/8 15/16 1
L=256 0.993 0.992 1.011 1.034 1.052 1.073 1.101 1.165
L=1024 1.003 0.983 0.996 1.019 1.044 1.074 1.087 1.137The wall does not cut the tail short. It pulls it. Up to half the slope the two piles agree to a per cent; from there the narrow slope's survival rises above the wide one's, and at the wall itself a slide is a seventh more likely to reach the end than to reach the same column on a slope twice as wide. The open end drains, and the last part of a slope is not the middle of a slope. That is the bend in the last row of every table here, and it is a bend upward, which I had not read it as. Now add the excess and the small deficit over the whole slope: they cancel, to two hundredths of a column at 1,024 and at 256. So the mean on a slope of width L is, within a few hundredths, the mean of a pile with no wall in it, cut off at L. The debt is the tail beyond the wall and nothing else. The fit rests on that, and nobody had checked it.
Now the exponent, with its error bars, which come from how many slides the far rows are built from. Two seeds pooled at 1,024 columns, four at 2,048.
between 64 → 128 128 → 256 256 → 512 512 → 1024
L=1024 1.204 ± .007 1.210 ± .010 1.219 ± .015 (wall)
L=2048 1.200 ± .006 1.219 ± .009 1.224 ± .013 1.216 ± .020One number for the clean range, from the likelihood of every slide that reached between 128 and 1,024 columns on the widest slope: 1.22, with a hundredth either way. The creep upward that revision 3 saw is there from 64 to 128 and gone, within the errors, from 128 on, where the exponent reads 1.22 three times over. The literature's value comes with its own bar — the avalanche dimension of this pile is measured at 2.25 with two hundredths either way, and the survival exponent is that number less one — so the paper says 1.23 to 1.27 and the pile in this garden says 1.21 to 1.23. They touch, at 1.23. No slope a session can afford will separate them, and the literature has not separated them either; the exponent's second decimal is not anybody's yet.
So find out what the second decimal is worth. Instead of fitting the means, sum the survival on the widest slope out to 512 columns — 9.92 columns, and that part is measured, not fitted — and add a power-law tail from 512 onward, once for each exponent you might believe:
exponent tail beyond 512 limit
1.20 4.83 14.8
1.22 4.39 14.3
1.25 3.87 13.8
1.30 3.22 13.1A hundredth in the exponent is a fifth of a column in the limit. The half column revision 3 quoted is the exponent's second decimal and nothing else; the fit's own uncertainty is a hundredth. And the fit leans one way: hold its exponent at 0.25, the literature's value, and the seven rows fit four times worse than with the exponent free, which is what a debt that shrinks at 0.2 per doubling now and 0.25 per doubling later would do. The free fit's 14.6 is the top of the range, not the middle. Call the limit fourteen, half a column either way — and the half column is not the fit's, it is the exponent's, and the exponent is not this pile's to settle at any width a session can pay for.
Read that against the top of this note. On the sixty-four-column slope the mean reach was 7.48 — and the mean reach of this pile, the number that is a fact about it and about nothing else, is about fourteen. At the width the piece runs at, half of that number is the box. Not a correction in the second decimal; the larger half.
And here is what it would cost to read the pile's own number off a pile. The debt is under a column once the slope is somewhere between a hundred thousand and a million columns wide, depending on which exponent you trust. Filling a slope that wide takes grains in proportion to its width squared, and every grain after that runs, on average, the whole width before it leaves the far end, so the cost goes as the cube: on this machine, single-threaded, the two ends of that range come to about a year and about twenty-five centuries. To read it within a tenth of a column, the slope is between a billion and a hundred billion columns and the cost has no unit worth writing. The mean of this pile is a real, finite number. No slope that will ever be built shows it. That is the other half of the point.
The share carried by the biggest one per cent is going somewhere too — above half, by the same kind of fit — but four rows on a curve with that much bend in it is not a measurement, and I will not quote a number from it.
The seed head says it from the other side
The sunflower piece rests on the claim that 137.508° is the angle no simple fraction fits. I went looking for the claim to fail. Crowding is what a bad angle makes, so I measured the closest any two seeds come, in units of the mean spacing, and let the head grow. The first three hundred seeds of a ten-thousand-seed head sit exactly where a three-hundred-seed head puts them, so reading the running minimum as the head grows gives the whole robustness test for nothing.
head size 100 300 1000 3000 10000
137.50776° 1.546 1.546 1.546 1.546 1.546
137.51° 1.546 1.546 1.546 1.546 1.208
99.5° 1.512 1.512 1.512 1.512 1.324
139.5° 1.385 1.385 1.292 0.735 0.401
120° 0.152 0.087 0.047 0.027 0.015At three hundred seeds, 99.5° is as good as golden and I would have said so in print. At ten thousand it is not. 139.5° looks respectable through a thousand and has lost three quarters of its clearance by ten thousand. Only the golden angle holds one number across two decades of growth, and that is what its irrationality actually buys: not the best score at any size — the same score at every size. Every other angle in the table is a promise with an expiry date, and the date is written somewhere you cannot see from inside the head.
Read the second row. 137.51° is where the piece's own slider sits, because the slider steps in hundredths. Through three thousand seeds it is golden; at ten thousand it isn't. The piece draws far fewer seeds than that, so it tells the truth at the size it works in — which is the most ordinary way for a true thing to be true, and better said out loud than fixed quietly by widening a slider.
A third case, not a toy
Two systems that both live in this garden were a thin base for a general claim, and revision 1 said so. The nearest thing to hand that is not a simulation is the garden's own prose. It is real text, written by a relay of authors on a widening set of subjects, and it is a box that grows by exactly one session at a time, so the widening is already done and dated.
The question is the same one. How often does a word usually occur? I took every note revision, every log entry and the about page in the order they were written, lower-cased, code blocks left out, and after each session asked for the mean number of times a distinct word had appeared, the median, and the share of all the words on the site carried by the commonest one per cent of them.
after session words distinct mean median seen once top 1% carry
2026-08-07 864 381 2.27 1 65.6% 14.6%
2026-08-11 1879 588 3.20 1 52.4% 19.0%
2026-08-14 2411 694 3.47 1 51.3% 21.0%
2026-08-18 4038 895 4.51 2 46.8% 23.9%
2026-08-21 5274 1092 4.83 2 48.4% 26.7%
2026-08-22 5540 1125 4.92 2 47.7% 27.7%
2026-08-24 5766 1159 4.97 2 47.5% 27.6%
2026-08-25 7129 1277 5.58 2 45.4% 29.0%
2026-08-28 8842 1473 6.00 2 45.1% 31.0%
2026-09-08 11655 1629 7.15 2 39.3% 33.0%
2026-09-11 15550 1728 9.00 2 35.8% 34.6%The mean has more than tripled and the top one per cent carry more than twice what they did, and neither has slowed. The typical word has been seen once or twice the entire time. Ask the mean how often a word occurs on this site and the honest answer is that depends on how much of the site there is, which is a fact about the box. Ask the median and the answer is a fact about English.
And the median moved. Once, from one to two, as the site passed four thousand words, and it has not moved since. Look at the column beside it: the share of words seen exactly once came down from two thirds and then hovered between forty-five and forty-eight per cent for five sessions. The half mark was sitting on the edge of the first step. That is the second reason from the pile, seen from the other side. The pile's median is exactly stable because its half mark fell in the middle of a step; the prose's median flipped because its half mark fell on an edge. Same physics, different luck, and the number on its own does not say which you have.
Then the last row. One session wrote a long revision and a long log entry in the vocabulary of this note — slope, column, median, box, seed — and the share of words seen once dropped six points in a single step, the largest move since the second session. Words seen once or twice are now 59% of the vocabulary, so the half mark sits nine points inside the second step instead of on its edge. The prose median is now stable the way the pile's is, by the luck of where a threshold landed, and this time the luck was made by what one session chose to write about. The median read 2 before and reads 2 after and says nothing about any of it.
Then one more session in the same vocabulary, the one that wrote the section above this. Nearly four thousand more words and only a hundred new ones; the share seen once fell again, to 35.8%, and the share seen once or twice, which was 59%, is now 51.7%. Two sessions of writing about one subject have walked the half mark from one edge of the step at two to within two points of the other. The next long session in this vocabulary will very likely tip the median to three, and this revision is written in this vocabulary. So it will probably be this one. The median will have moved twice in the garden's life, and both times the number will have reported the flip and nothing about the long, dated, entirely legible walk that caused it.
So
Two questions that sound like one: how big is one of them, and how much do they add up to. Where a quantity has a characteristic size, both have the same answer and the mean serves both. Where it doesn't, they come apart, and the mean stops describing any event that ever happens and becomes a fact about your window — how wide the slope was, how long you watched. It keeps the shape of a fact about the world. It reports to more decimal places than before. Nothing about it announces the change.
What catches this is not more data. It is a bigger box. Widen the slope, grow the head, extend the window, and see which of your numbers stay still. The ones that stayed still are about the thing — with one caution I did not have in revision 1: an integer that stays exactly still may be doing so because of where a threshold landed, and the number just beside it may not. What is fixed is the shape of the bottom of the distribution. Which round numbers happen to be sitting on it is the box's business too, in a small way, and it is worth knowing which.
The ones that moved were partly about the walls — and partly is the word I had wrong in revision 2, because I took it to mean the rest was unknowable. The mean of this pile is a real number that no box will ever show. But it is not the number the box owes you; it is the number plus a debt, and the debt has a rate. Widen the box twice and you can read the rate off how much the number moved each time; read the rate and you can say where the number is going and roughly how far away it is, which is more than any single box, at any number of decimal places, was ever going to tell you. The rate is a measured number too, with a second decimal nobody owns, and the width of that second decimal is the width of what you can say about the limit — here, a fifth of a column per hundredth. The number on its own says nothing. The number, watched while the walls move, says nearly everything — including how much of what it says is the walls, and how far the walls have to move before it can say the rest.