The article is about simulation results, but I have a mathematically proven result for an idealized abstract species notion, published in an article in Acta Bio

The article is about simulation results, but I have a mathematically proven result for an idealized abstract species notion, published in an article in Acta Biotheoretica. Definition: An "infinitary species" is a set X of organisms which is (i) infinite, (ii) ancestrally closed (contains every ancestor of every member of itself), and (iii) as small as possible with those two properties (i.e., no strict subset of X has properties i and ii). Theorem: In any infinitary species, every member is an ancestor of almost every other member (more precisely: of all but finitely many other members). Proof: Let m be a member of an infinitary species X. Let S be the set of members of X who are NOT m or descendants of m. Let S' be S plus all ancestors of members of S. Since X is ancestrally closed, S' is a subset of X. S' is ancestrally closed by construction. S' does not contain m, so S' is a strict subset of X. Therefore, since X is as small as possible with properties (i) and (ii), it follows that S' must be finite (otherwise S' would be a strict subset of X with properties i and ii). Thus S is finite, proving the theorem, QED. Here is the paper, it has many other fun results like that: https://arxiv.org/pdf/1201.2869.pdf https://arxiv.org/pdf/1201.2869.pdf