The whole life of one table, one move at a time: a particle ticks, a detector meets it, the joint table turns, and two books settle it into a fact. Everything before the last step is reversible and writes nothing.
Two separate tables. Tick each one as often as you like.
1
The threads
a tick turns the pair of cells along that thread's own axis
P · the particle
P's axis
P = 0
P = 1
M · the detector
M's axis
M = 0
M = 1
Every cell holds one complex number, not a fraction. The arrow's length is how much of the wave sits in that cell, a real number between 0 and 1, drawn as the bar under it. The angle between the two arrows is the phase, and the phase is what lets two contributions cancel later. Only that angle is real: turn both arrows of a thread together and nothing anyone can measure changes, which is why the phase slider moves one cell against the other. A thread ticking from a fresh start holds the two arrows at a right angle, one cell real and one imaginary, which is why they sit on the axes. Move the phase, or turn the joint table along both axes, and they leave them.
The meeting is the only move that touches both axes: the product of the two tables, then one swap.
2
The meeting, step by step
product, then swap
3
The joint table
P turns the columns · M turns the rows
M's axis
P's axis
M = 0
M = 1
P = 0
P = 1
books that must agree:
Two books, two clocks, one line. This is the only move that writes anything, and the only one you cannot undo.
4
The settlement
each book lays the four lengths end to end and signs for the line its unitdraw falls in
Four lines, because a line names both parties. It is not P's answer and then M's answer; it is one entry, "P reads this and M reads that", the way a double entry names both sides of a transaction. Two values each gives four of them. Four values each would give sixteen, every combination of the two multiplied, not added; the ones the meeting left empty simply take no room on the ruler. A book's unitdraw is not its dial: the dial is the angle it turns the table by, while the unitdraw is what its own clock shows at that tick, a number between 0 and 1 that nobody chooses.
puts the table back as it was before the fact and runs another settlement
5
One book, two books, three books
the same table, settled thousands of times
Why the square, and whose lengths make it. A book signs for a line in proportion to that line's length. One book alone is a needle dropped on a ruler: the chance of landing under a length is the length. Two books are two needles on two rulers, and the chance that both land under the same length is that length times itself, an area. The two sides of the square are the same line's length twice, one taken from each book: different books, same line, same length, which is why the square is a square and not a rectangle. Three books would be a volume.
The number of books is the exponent. Each extra book that has to agree multiplies a line's chance by its own length once more, so with k books a line is posted in proportion to its length to the power k. That is the whole ladder, and it says where the three rules hide their difference: when two lines are equally long, every power treats them alike and all three agree, so they only pull apart when one line is longer than the other, and further the more lopsided it gets. Quantum mechanics says the exponent is two. Watch it from the other side by settling with one book and with three in step 4: one book never refuses, because there is nobody to disagree with, and three refuse constantly and hand the long line a bigger win than it has earned.
The third book is a hypothetical third signer, not a thread in this toy.