Endomorphism counts from primitive orbits and synchronized block actions

For a finite poset P, let R(P) = |End(P)|/|Aut(P)|, where endomorphisms preserve the reflexive order. We prove R(P) ≥ 2^{|P|/(64 log₂ |P|)} for |P| ≥ 4 when every chain has at most two elements and the action on each nonsingleton automorphism orbit is primitive. The same bound holds for a class of imprimitive actions: on each interaction component the action must be the full automorphism group of a rooted tree, acting identically on all the identified orbits. The proof combines simultaneous collapses of indices with maps into principal ideals and filters. We give counting lemmas that retain the whole kernel of a block action, and a quantitative collapse, degree, or orbit-size alternative for normal products of alternating groups with natural orbits. We also construct connected twin-free posets with group S₃ wr Sₜ, constant minimum of the two maximum principal-set sizes, and imprimitive action on every orbit, although every orbit-preserving endomorphism is an automorphism. A restricted three-level extension has lower bound 2^{|P|/(128 (log₂ |P|)²)}. These results give partial answers to the finite-poset automorphism problem of Rival and Rutkowski; the unrestricted question remains beyond the arguments here.

GitHub repository: https://github.com/eulogica/endomorphism-counts-posets