[biggiemac42]

The post (and my previous waterbear post) both talk about universal construction, it is equivalent to glider construction. There are patterns which have N generations of predecessor but not N+1, which are also non-constructible.

[dvgrn]

There are also patterns -- both stable patterns and oscillators -- that have an unbounded chain of predecessors, but still can't be glider constructed. That is, there are proven solutions to Conway's "Unique Father Problem" from over half a century ago:

  https://conwaylife.com/wiki/Unique_father_problem

To put it another way, there are certain patterns that, if you see them in the Life grid, you know they've always been there from T=0 -- because they are provably their own only predecessor.