Map of Math · PG-01 · where it stands, 2026-10-03 · all explainers
Ten plates, each with what is settled and what is open. No moving parts, so it reads the same on paper. The numbers are the lab's; the run that produced each one is named beside it.
Each pattern is a 9 × 9 grid with 17 cells removed. A carpet repeats its pattern inside every kept cell, level after level. T keeps a spoke from the middle of each side into the block. X keeps two whole rows and two whole columns, like corridors.
Both keep 64 cells, both have the square's full symmetry, and both have the same number of links between cells. What differs is where the holes sit.
The census has 63 such patterns. T is number 26 and X is number 6.
Settled. These are definitions.
Ask yourselfBefore reading on: which one do you expect to let a path across more easily, and why?
Open the links in a random order and note the fraction open when the left side first reaches the right. That fraction is one draw's crossing point. Eight thousand draws make a pile.
A bigger carpet gives a narrower pile. The rate at which the width shrinks, each time the side grows nine times, is the sharpening rate. It is the main number the lab reports for a pattern.
Curves drawn from the measured middle and width at each size (run 083). The lean of the real piles is not drawn.
| size | T middle | T width | X middle | X width |
|---|---|---|---|---|
| 81 | 0.7761 | 0.0605 | 0.7161 | 0.0432 |
| 729 | 0.7855 | 0.0250 | 0.7220 | 0.0124 |
| 6,561 | 0.7881 | 0.0117 | 0.7229 | 0.0036 |
Settled. One sampler on both machines, checked draw for draw (run 092).
Ask yourselfThe pile's middle keeps creeping right as the carpet grows, but it stays left of the red line. What would it mean if it reached it?
For one 9 × 9 block there is an exact formula: the chance it is crossed, at each dial setting. The shortcut pretends every link of a big carpet is itself a whole block, and feeds the formula into itself.
It predicts where the pile should settle (the red line in Plate 2) and how fast it should sharpen. The real carpet settles lower on every pattern measured, and sharpens at a different rate. The programme is the search for what the shortcut leaves out.
Left: the replacement. Right: the exact crossing chance of one block of T, from its formula. Where the curve meets the diagonal, a block is as likely to be crossed as each of its links is to be open; the shortcut says the carpet settles there.
Settled. The gap between the shortcut and the carpet is real on all 49 formulas in the census, and on all but five of them does not close with size.
Ask yourselfThe shortcut asks one yes-or-no question of a block: is it crossed left to right? List two other things a route could do with a block.
The ladder is the first repair. Keep the real blocks, but join neighbours at only three rows. Then everything one block tells the next passes through three points, and the bookkeeping is exact again.
It gets the answer backwards. On the real carpet X sharpens faster than T. On the ladder T sharpens faster than X. Moving the three contacts to different rows changes nothing.
Solid bars T, hatched bars X. Carpet rates from run 083's samples; ladder rates from runs 084 and 094.
Settled today. Which rows carry the contacts does not matter (run 094). Three per side is too few, or one point per contact is the wrong idea.
Ask yourselfWhy might X, with its whole corridors, be hurt more than T when neighbours touch at only three rows?
Contacts sit on rows 1, 4 and 7, or on rows 2, 4 and 6. On X, rows 2 and 6 are the corridors: whole rows with no holes. So the second set looked like it should favour X. It did not.
Rows are numbered 0 to 8 from the top. A black dot is a contact; a line is the one link across the seam at that row.
Settled. Same reversal with either set: X minus T is −0.131 with rows 1 4 7 and −0.134 with rows 2 4 6.
A carpet of side 729 is 64 blocks of side 81. Give each block its own crossing point. If a route could pass a block only once that block is itself crossed, the carpet would cross when the crossed blocks first link up, left to right. For T that needs 50 of the 64.
The real carpet crosses with only 38 crossed. Routes run through blocks that are not crossed side to side: they turn corners inside them, or dip in and out. The carpet is ahead of its own blocks in 85% of draws.
Dark blocks are crossed. The numbers are measured averages for T at 729. Which blocks are dark here is one illustrative order, not data.
Measured. On X the same: 38 against 47. The lead grows with size and is still moving at 6,561.
Ask yourselfIn the left picture, can you find a route across that uses only dark blocks? If not, how might the carpet be crossing anyway?
A route enters a block by one side and leaves by another. Under the square's symmetries the pairs of sides fall into three kinds: opposite sides, adjacent sides, and the same side twice. The shortcut records only one member of the first kind.
The proposal is that each block carries a small table: when each pair of its sides is first joined. Run 097 records all of them and asks whether a block graph built from that table crosses when the carpet does, draw by draw, and whether the corners are what make it work.
The recorder keeps twelve such times for each of the 64 blocks, in every draw.
Open. Run 097 is drawing tonight. Prediction sealed: within 0.3 widths with the table; the corners move it by at least 0.3.
Ask yourselfA corner can never join later than the moment both along and across have joined. Why? (Two paths across a square must meet.)
Instead of opening links in a random order, grow from the left side by always taking the easiest link on the edge of what is joined. The growth runs through bursts of easy links between the hard links it is forced to accept. The hardest forced link is the same crossing point as before, exactly.
On the square, burst sizes follow a plain power law: the line stays flat at 1. On the carpets the line ripples. T and X peak near 25 links and again near 1,400, a factor of about 55 apart, close to the 64 cells in a block. H, a carpet built on a 3 × 3 pattern with 8 cells, has its strongest ripple at a factor of 8, and a much fainter one.
Burst density divided by its fitted power law (run 095, full side source, 2,000 draws each). Red ticks mark 25 and 1,400.
A reading, not a result. The seal was written on a ratio that could not be scored. The picture is clear; a re-seal on a difference would score it.
Ask yourselfH has 8 cells per level and T has 64. If the ripple is set by cells per level, where should H's next peak after 25 fall?
The lean of the pile (its skewness) rises with size under the shortcut, on every pattern. On the real T it rises to 729 and then falls back at 6,561. The first reading put the fall at −0.19, from one file.
Eight fresh files and a full redraw later, the fall is −0.09, give or take 0.02. It is real, and half the size. The one low file was a 3-in-1,000 draw.
Each dot is one file of 8,000 draws. Bands are the pooled value and its error.
Settled in size, open in cause. No split of the draws tried so far pins the fall on one part of them.
Ask yourselfBefore run 092: would you have trusted the single low file, or the two higher ones? What would you have wanted checked first?
Every run is sealed with a prediction before it computes, and scored as written. Most seals miss. The misses are where the picture changed.
Withdrawn runs failed their own controls before any scored number was read.
Four confirmed, six falsified, eight neither, two withdrawn, two running.