Partial bi-orders and positive conjugacy in the derived Brown–Thompson groups

For each integer n ≥ 2, let F_n be the group of increasing piecewise-linear homeomorphisms of [0,1] with finitely many breakpoints in Z[1/n] and slopes in nℤ = {nk : k ∈ ℤ}. We prove that every partial bi-order on F_n′ extends to a total bi-order, and that F_n′ has exactly 2^n bi-orders. They are determined by the slope at an extreme support endpoint and the class of that endpoint modulo (n−1)Z[1/n]. For elements of F_n′ moving every point of one support interval to the right, these endpoint classes also characterize when their positive conjugacy semigroups intersect. We prove an exact positive factorization theorem for two maps f,g that move every point of a common support interval compactly contained in (0,1) to the right: matching endpoint germs and orbit-jump characters are necessary and sufficient for every sufficiently large power g^m to be a product of exactly m conjugates of f by elements of F_n′ preserving that interval. The passage from the dyadic case requires cancelling local abelianization defects before applying commutator-width bounds. Failure of simultaneous positivity has a witness using at most four supplied elements, yielding a decision procedure for finite inputs. Every group between F_n′ and F_n also has the order-extension property. The strongly isolated partial bi-orders on F_n′ are classified by their permitted endpoint signatures; there are exactly 9^(n−1), including the equality order. GitHub repository