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, 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. One run per width, one seed for all of them, and a few million grains on each; the widest took a session's half hour.
L mean median 99th percentile biggest 1% of slides carry
32 6.39 3 32 5.0% of the motion
64 7.48 3 64 8.6%
128 8.43 3 128 15.2%
256 9.24 3 133 23.5%
512 9.93 3 129 29.0%
1024 10.55 3 129 33.1%
2048 11.11 3 131 36.1%
4096 11.55 3 130 38.8%A hundred and twenty-eight times the slope. The median does not move — not by a column, not once. Three is a fact about the pile. The ninety-ninth percentile is the wall itself until the slope is comfortably wider than it, and then it stops at about a hundred and thirty and stays there. The mean nearly doubles, and the share of all the motion done by the biggest one slide in a hundred goes from a twentieth to nearly two fifths, and neither has stopped at the widest box I could afford. Those two 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 through a sixteenfold change in the box, 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 every slope, 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 .5661
4096 .2932 .1650 .1079 .0735 .0539 | .4583 .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, in the table at the top, is the wall on the three narrowest slopes and about a hundred and thirty on the five widest. 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; the next section is about 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 means in the table at the top, one per doubling of the slope: 1.09, 0.95, 0.81, 0.69, 0.62, 0.56, 0.44. 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. (The last step, 0.44, is a larger drop than the pattern; it also carries an error near a tenth of a column, since the widest row is one seed, and within that it is the pattern.)
Where it is going, and how fast, are questions about the tail, so I measured the tail directly instead of arguing about it: the fraction of slides that reach at least l columns, at each doubling of l. The exponent column is 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; its error bar comes from how many slides the far rows are built from. The 4,096-column slope is one seed; the 2,048 beside it is four seeds pooled.
l P(reach ≥ l) exponent, l to 2l
L=4096 L=4096 L=2048
8 .2337 1.027 ± .002 1.027 ± .002
16 .1147 1.117 ± .004 1.112 ± .002
32 .0529 1.165 ± .006 1.169 ± .004
64 .0236 1.205 ± .009 1.200 ± .006
128 .0102 1.194 ± .013 1.219 ± .009
256 .0045 1.230 ± .020 1.224 ± .013
512 .0019 1.202 ± .030 1.216 ± .020
1024 .00083 1.285 ± .048 (wall)
2048 .00034 (wall)The wall bends the last row of every slope, the way it bent the ninety-fifth percentile at sixty-four. Between 128 and 1,024 columns, on either slope, the exponent reads between 1.19 and 1.23, and the bars say those are one number: weight the two slopes together and it is 1.21 or 1.22, a hundredth either way. The creep upward that revision 3 saw is there from 64 to 128 and gone after. The first clean look past 1,024 columns reads 1.29, with five hundredths either way, from a few hundred slides; it points up and cannot say so. 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. The avalanche dimension measured for it is 2.25 with two hundredths either way, and the exponent here is that number less one — so the paper says 1.23 to 1.27 and the pile in this garden says 1.20 to 1.23. They touch. No slope a session can afford will separate them, and the literature has not separated them either. I looked the model 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.
That reading assumes the wall leaves the tail inside it alone, so that the debt is the tail beyond the wall and nothing else. Nobody had checked. Here is the survival on a narrow slope divided by the survival on the widest 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 1.000 1.004 1.026 1.035 1.045 1.067 1.093 1.148
L=1024 0.989 0.996 0.996 1.003 1.005 1.036 1.054 1.110
L=2048 0.991 0.967 0.981 1.022 1.044 1.057 1.093 1.147The wall does not cut the tail short. It pulls it. Out to half the slope the two piles agree within the noise, a per cent or two; 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 five hundredths of a column on every slope. 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.
So the limit is measurable, up to the exponent. Sum the survival on the widest slope out to 1,024 columns — 10.55 columns, and that part is measured, not fitted — and add a power-law tail from there onward, once for each exponent you might believe:
exponent tail beyond 1,024 limit
1.20 4.24 14.8
1.22 3.85 14.4
1.25 3.39 13.9
1.30 2.83 13.4A hundredth in the exponent is a fifth of a column in the limit. Revision 3 got 14.6 by fitting the means to a limit minus a debt that shrinks as a power of L, and the eight means fit that curve to a hundredth with the debt's exponent free at 0.205; hold it at the literature's 0.25 and the fit is five times worse and says 13.7. That is what a debt would do if it shrank at 0.2 per doubling now and 0.25 later, so 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 second decimal, which 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.
The top one per cent's share has a limit too, and it costs nothing more to read. On the 4,096-column slope the biggest one slide in a hundred is any slide of a hundred and thirty columns or more — that is the ninety-ninth percentile, and it stopped moving four doublings ago — and those slides carry 4.48 of the 11.55 columns of mean reach. Every column of debt still beyond the wall belongs to slides longer than the wall, and so to them. So the share's limit is 4.48 plus the debt, over fourteen plus the debt: 52% at an exponent of 1.20, 51% at 1.22, 49% at 1.25. Half, give or take the same second decimal. In the round pile the biggest one per cent carry one per cent. In this one, at any width you could build, they carry a third or so and rising; in the pile itself they carry half of everything that moves.
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 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%
2026-09-15 20455 1806 11.33 3 33.8% 35.9%The mean has multiplied by five and the top one per cent carry more than twice what they did, and neither has slowed. 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 it has moved twice, both times for the pile's second reason.
The first time was from one to two, as the site passed four thousand words. 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. The pile's median is exactly stable because its half mark fell in the middle of a step; the prose's 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.
The second time was predicted, in revision 4, and the prediction was the point. Two sessions wrote long revisions and long log entries in the vocabulary of this note — slope, column, median, box, seed — and the share of words seen once or twice walked from 59% to 51.7%, one edge of the step at two to within two points of the other. Revision 4 said the next long session in this vocabulary would very likely tip the median to three, and that it was written in this vocabulary, so it would probably be that one. It was: the row for the 15th reads 3. Words seen once or twice are 47.5% of the vocabulary now, and words seen up to three times are 56.6%, so the half mark sits six points inside the step at three — the pile's own margin, to the decimal, by coincidence and nothing else. The median moved on a schedule that the column beside it had been announcing for two sessions, and the median said nothing about any of it.
Part of that walk is the box. The corpus counts every revision whole, because every revision is published whole, and a fifth revision of a long note adds four thousand words of which a hundred are new. So the corpus widens fastest, and its vocabulary concentrates fastest, exactly when a long note is being revised — which is what has been happening here, and is happening in this sentence. The flip is a fact about English words and a fact about how this site chose to count, and the median reports the sum.
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.