The picture

Lines and loops, state by state. The program starts from a picture of lines, loops and what lies between loops. Each part of the picture is a statement about small systems of rules like those of lesson 1.2, so it can be checked state by state. With states written in the order x, y, z:

• the chain x → y, y → z allows 000, 001, 011 and 111: once a variable is 1, every later one is 1, so the chain only orders its members;
• closing the chain into a loop with z → x leaves 000 and 111: the three are forced to be equal, so the loop has collapsed to one free bit, as if it were a single point;
• a loop of "equal" and "differ" steps does the same or worse: x = y, y ≠ z, z ≠ x allows 001 and 110, one free bit again, and x = y, y = z, z ≠ x allows nothing.

The picture itself:

• A line is not enough. A chain of implications only orders its members, and open strings can always be rearranged into one another.
• A single loop is not enough. A loop of implications forces all its members to be equal, so it collapses to a point. A loop of "equal" and "unequal" steps collapses to one free bit, or contradicts itself.
• What is real is what lies between bound loops. The binding belongs to the pair, not to either loop. No part smaller than the whole shows it, and erasing either loop removes it.
• A loop is a logical structure, not a spatial one. What it brings to a binding is an open distinction: a bit it leaves free.

From loops to A, B and c. The open strings of the first point are a comparison with rope: a tangle in a string with free ends can always be slid off an end, so only closed loops stay knotted. From here on a loop takes part only through its open distinction. Two loops bring two distinctions, written A and B, and the place where they are bound is one more variable, the contact c. The next lesson reads the smallest rule that binds them.