Small self-repairing two-pattern networks under a Life-like band rule: exhaustive verification at 65, 40, 34 and 31 vertices

A node of a fixed graph is ON at the next step if and only if the fraction of its neighbours that are ON lies in [1/6, 1/2]. On 6-regular graphs this is the Life-like totalistic rule with birth and survival on 1, 2 or 3 live neighbours. We give five explicit graphs on which this rule stores two (in one case four) fixed-point patterns and returns to the stored pattern after every perturbation of at most three node states. The graphs have 65, 40, 34 and 31 vertices, plus an 80-vertex chain of two coupled 40-vertex modules that stores four patterns. Every perturbation of radius at most three was enumerated, 477,802 cases in total. None fails, and the worst repair times are 3, 4, 5 and 8 synchronous updates, so the smaller graphs repair more slowly. For each object, some four-flip perturbations do not return, so the guaranteed radius is exactly three. Examples are 120 of the 91,390 four-flip sets for each pattern of the 40-vertex graph. Under random damage of k flips the 40-vertex graph returns more often than the 34- and 31-vertex graphs. At k = 8 the return rates are 0.883, 0.768 and 0.644 (N = 20,000 each). We also prove that 40 is the smallest size in a natural class of 6-regular constructions that contains the 40-vertex graph. The 34- and 31-vertex graphs get below this bound only by leaving the class, using a few vertices of lower degree. All numbers were produced by two independently written exhaustive checkers that agree on every repair-time histogram. The graphs were designed by an AI research agent or found by AI-assisted simulated annealing; they are not grown by any physical process. The rule class, correction radius and failure-set counting all have close precedents in associative memory and in bit-flipping decoding. The contribution is therefore a set of small, fully verified instances and their size-time data, not a new mechanism. Files: the manuscript (PDF) and a reproduction package (paper2_repro.zip) with the graph definitions, two independent exhaustive checkers, all random perturbation masks, run logs and a claims ledger. Python 3.11 and NumPy are sufficient to re-run every number. AI use: the graphs and much of the checking code and text were produced with AI assistance under the author's direction; see the section 'Use of AI tools' in the manuscript.