Five Zero Products in a Four-Node Path Algebra: The Taxonomy's Count and the Quiver's Arrow Roles Partition

Two partitions fixed independently agree, including on which cell is empty: the five non-empty error types plus one impossible sixth of *The Five Classes of Error* (Stewart, 2026z), and the partition of the six arrows of a four-vertex quiver Q by their role in its path algebra kQ (*The Path Algebra of the Endomorphism*, Stewart, 2026f). The five types are five zero products, and the sixth cell is the one pair of vertices kQ leaves with no product to form. Each type names a pair of stages whose order was reversed, and each reversal is the attempt to concatenate two paths across a prerequisite arrow that has not been traversed, which the algebra evaluates to zero. The five signatures are given below by the missing arrow in each case: a₁ for Type 2-1, a₂ for Type 3-2, a₃ for Type 4-3, a₅ for Type 3-1, and a₁, a₂, and a₃ together for Type 4-1, the triple that nulls every interior concatenation of γ₁ = a₁ · a₂ · a₃ · a₄. The sixth cell, 4-2, has no signature at all, because the S-C non-edge leaves no product to evaluate (*On Occurrence: The Four-Gon*, Stewart, 2026g), so the cell is not a product that comes out zero but a product that is never formed. The arrow roles reach the count by a route with no stages in it. Q's six arrows divide by role. The three forward-chain prerequisites a₁, a₂, and a₃ are each the sole in-arrow of their target, so each is the unique factor whose absence kills its product. The direct admission a₅ is a fourth prerequisite, and its independence from a₂ is what separates Type 3-1 from Type 3-2. The remaining two, a₄ and a₆, occupy only the right slot of the products the taxonomy records, so no failure is indexed to their absence. Four prerequisite arrows give four types, the triple conjunction gives the fifth, and no pairwise conjunction gives a sixth, because a pairwise conjunction nulls only the products its two constituents already null. The construction hands one falsifier forward in algebraic form. *The Five Classes of Error* stakes exhaustiveness on there being no clean 4-2 instance, a contextualization-to-commitment step that does not pass through resolution (Stewart, 2026z). Such an instance would be a nonzero element of e_S · kQ · e_C at length one, and exhibiting one is what the taxonomy's falsifier becomes once it is read in kQ. **Keywords:** path algebra, quiver, quiver representations, representation theory, finite-dimensional algebra, vertex idempotent, zero product, path composition, directed graph, error taxonomy