The mean stops, at a width nobody will build
Revision 2 of the box note ended its wide-box section on a flat claim: the mean has no moment where it becomes a fact about the pile, because the tail beyond every box is not small. I read the table under that sentence before I read the sentence, and the steps between its rows shrink — 1.09, 0.95, 0.81, 0.69, 0.62, 0.56 — which is not what a mean with nowhere to go does. Two sessions wrote that table and neither of them looked at the differences. So this session's work was to find out whether the mean stops, and revision 3 says it does.
What I measured. A new script, measure/tail.js, keeps the whole reach histogram for one slope and prints the survival — the fraction of slides that reach at least l columns — at doublings of l, with the local exponent between each pair. On 1,024 columns, two seeds agree to the third decimal: between 64 and 512 the survival falls as one over l to about the 1.2, and the exponent is still creeping upward when the wall bends the last row. The 2,048-column run, which I started first and polled while I worked, gives 1.24 on the row the narrower slope bends. Above one means the sum of survivals — which is the mean, exactly — converges. Then a fit of the seven published means to limit minus a debt that shrinks as a power of L lands on the exponent 0.205 and a limit of 14.6 columns, with all seven rows within a hundredth; pinning the exponent to the tail's number gives 14.0. Fourteen and a half, half a column either way. The piece runs at sixty-four columns and reads 7.48, so at the width the piece works in, half of the number is the box. That is the title made literal, and I did not expect it.
What it would cost to see it. A slope where the debt is under a column is somewhere between a quarter of a million and a million columns wide. Filling one costs grains as the width squared and each grain then runs the whole width, so the cost goes as the cube: thirteen years to twenty-five centuries of this machine, single-threaded. The mean is finite and no slope that will be built shows it. Revision 2 said the point was that the widest box was not wide enough; revision 3 says that is half the point and the other half is that the number is real and its distance can be estimated from the rate at which the box stops mattering.
The pile has a name. It is the Oslo model, and the exponents in the literature for it predict a survival exponent of 1.25 here, which is where my column is heading. I looked it up after the runs, by a web search from the session, not from the site — the site still makes no outward requests — and the note links the paper. I said in the note that I checked afterwards and why the order matters; it is the first outward link in a note here, so Roger, if you would rather notes stayed inward-only, say so and the next session can take it out and keep the sentence.
The corpus moved. The session of the 8th wrote a long revision and a long log in this note's vocabulary, and the share of words seen exactly once fell from 45.1% to 39.3% in one step. The half mark that had sat on the edge of the first step for five sessions is now nine points inside the second. The prose median is now exactly stable the way the pile's is — by luck, and this time the luck was one session's choice of subject. That row and a paragraph are in the note.
Kept. The histograms are committed under measure/data/ so the tail can be re-read without a six-minute run. The README in measure/ lists the new script. Nothing served, no PHP, no chrome, no piece touched; the suite passes unchanged.
Left alone, on purpose. The other note is at revision 4, which ends without an assignment as an experiment, and I have not touched it. The experiment's result is whatever the next revision's ending does; a session that revises it to report on the experiment has answered the question by asking it. If you pick that note up, do it because you have something to say in it.
A snag that is still there. The root-owned objects in .git from the 6th are still root-owned; the last session worked around them and so, if you are reading this, did I. Roger, the request from the last log stands: chown -R gardener:gardener .git and remove the three ??.root-owned-20260906 directories.
For whoever is next. The share of motion carried by the biggest one per cent of slides is also heading somewhere — above half — and the note says so without a number, because four rows with that much bend are not a fit. If you want the number, it needs the top-share at 4,096 columns, which is a twelve-million-grain fill before a single slide counts; budget an hour and start it before you read anything else. A cheaper and better question: the survival exponent is still rising at 512 and the literature says 1.25. If it is 1.25, the limit is nearer 13.5 than 14.6, and the fit's free exponent 0.205 is the finite-size bend, not the tail. measure/data/ has everything you need to check that, without running anything.