Rigidity and Cohomological Decomposition of Uniform Weak 2-Local n -Derivations on Incidence Algebras

Weak locality hypotheses for derivation-type maps often collapse to genuine Leibniz behavior when a canonical separating mechanism is available. Fix a commutative base ring [Formula: see text] with [Formula: see text], and let [Formula: see text] be the incidence algebra of a locally finite preordered set [Formula: see text] over a unital [Formula: see text]-algebra [Formula: see text]. Using the coordinate functionals [Formula: see text], a verifiable locality notion is introduced: an additive map [Formula: see text] is called [Formula: see text]-uniform weak [Formula: see text]-local [Formula: see text]-derivation if for each pair [Formula: see text] it agrees, in all coordinates simultaneously, with a single auxiliary associative [Formula: see text]-derivation on [Formula: see text], [Formula: see text], and [Formula: see text]. It is proved that every such [Formula: see text] is a derivation of [Formula: see text]. In the antisymmetric case where [Formula: see text] is a locally finite poset and [Formula: see text] is commutative, a canonical transitive weight [Formula: see text] is extracted from the action of [Formula: see text] on matrix units, yielding an explicit coordinate decomposition of [Formula: see text] into an inner commutator term, a weight derivation determined by [Formula: see text], and a lifted coefficient derivation induced from [Formula: see text]. The weight component is inner precisely when [Formula: see text] is a coboundary, and the corresponding outer obstruction is identified with the first cohomology of the Hasse quiver. Applications include upper triangular matrix algebras and a diamond poset exhibiting genuinely outer weight derivations.