MATH.13:5.2 - A unique choice from an indistinguishable arrangement

A selector receives a three-vertex cycle with identical vertex attributes and equal edge attributes. Its output must designate exactly one vertex. Rotating the cycle is treated as a relabeling: the requested deterministic rule must rotate its selected vertex in the same way.

For this input, a one-step rotation leaves the supplied arrangement unchanged. Equivariance therefore requires the selected vertex to be fixed by that rotation. No vertex is fixed: each moves to the next vertex. The requested deterministic selector has no permitted output for this input.

One repair is to supply a distinguished attribute, such as an available unique priority, and let the rule use it. Another is to change the requested result to the set of all three equally admissible vertices. Random selection is another problem: a uniform distribution can be rotation-invariant even though a sampled vertex is not fixed. The construction of a coordinated random choice would need its own procedure.

This is a mathematical result about the given input and rule requirements. Extra identifiers, timing distinctions or other available attributes can change the input symmetry and therefore the conclusion. It does not establish that every real three-agent arrangement faces this obstruction.