Topological reuse · Figure 1
A small network learned to add on a 31-hour clock. This is what it built: nine rings, each with the thirty-one numbers around it, each counting its way around at a different rate. Every point below is read from the trained network itself. Turn the rings and you have taught it a + b + 3. Mirror them and you have taught it a − b.
Take any group of hidden neurons that read the same directions of the number table, and look at how the thirty-one numbers sit in that group’s two strongest directions. For a network that learned a + b, every such group places the numbers on a circle. The network found this on its own; it was never shown a clock.
The nine rings differ in their frequency: how many times the numbers go around before they are all placed. On the ring marked f = 1 the numbers sit in counting order like hours on a dial; on f = 11 each step skips eleven hours, so consecutive numbers sit far apart and the counting trace draws a star.
Switch to the network that learned a × b. It has rings too, but the counting trace through them is a scribble. Trace the numbers in the order 1, 3, 9, 27, 19, … (each three times the last, wrapping at 31) and the stars come back.
That is the whole difference between the two circuits. Measured on the trained networks: the add rings carry 98–100% of their wave along counting order and 4% along powers of 3; the mul rings carry 96–100% along powers of 3 and 4% along counting order. A reader that sees which arrangement a probe is building knows which circuit can serve it.
Turn every ring of input a by three numbers and the frozen circuit computes a + b + 3. Mirror the rings of input b (number n goes where 31 − n was) and it computes a − b. A turn or a mirror of a flat ring is a 2 × 2 map, four numbers per ring per input; that is why a ring lens stores so little, and why it does nothing on a task whose arrangement is the other clock.
In notebook 5 the lenses trained for subtraction came out as mirrors on b (determinant near −1) and near-identity on a, in 4 of 5 seeds. Nobody told them to.
Drag the training slider back. For the first thousand steps the network has memorized its 480 training pairs and gets 0% of the unseen pairs right; the readings are a cloud. Between about 1,000 and 3,000 steps the cloud crystallizes into rings and the unseen pairs follow: 5% at 1,000, 70% at 2,000, 100% by 5,000. The shape and the generalization arrive together.
This is why a 500-step probe is enough for the reader: the arrangement is legible at the output end before the ring is finished at the input.