C.40:5.19 - Reaching a target does not establish persistence
A stipulated three-cell process has state (left, centre, right), with each cell either 0 or 1. All cells update simultaneously from the previous state; positions outside the three-cell row are fixed at 0. Each cell reads itself and its immediate neighbours. The developer must obtain 111 from the seed 010, preserve it during continued execution, and recover after the centre is removed. In this exercise the three bits are the complete state and can be copied.
Compare two available rules. G outputs 1 when its neighbourhood contains at least one 1. P outputs 1 when the neighbourhood contains one or two 1s, but outputs 0 when it contains zero or three. Both reach 111 from 010 in one update.
| Trial start | G after one update | G after two updates | P after one update | P after two updates |
|---|---|---|---|---|
Original 010 | 111 | 111 | 111 | 101 |
Reached target 111 | 111 | 111 | 101 | 111 |
Damaged target 101 | 111 | 111 | 111 | 101 |
A first-attainment test cannot choose between the rules. Return the reached 111 as another trial start while retaining the seed 010; continuing from those starts exposes P’s alternating 111 and 101. Running longer from the seed with repeated checks would expose the same failure without a pool. Under G, 111 is unchanged by another update, which also supplies a direct persistence argument for this deterministic case.
Now remove the centre of 111, obtaining 101. Both rules restore the target in one update. Continue: P destroys it again, whereas G preserves it. A one-step damage test would therefore miss the consequential difference. A complete comparison of these two rules is enough to select G for the stipulated use; no neural training is needed. The example explains the trial construction and selection, not a claim that learning will find such a rule in any search space.
Finally remove all three cells. Both rules leave 000 unchanged. More generally, if an empty neighbourhood is forbidden to create a 1, simultaneous application cannot leave this all-zero state. Recovery now needs a new seed or another permitted operation. Repeating the same trials cannot supply either.