The preceding papers on continuous and discrete hybrid fractional calculus constructed boundary-augmented ambient Banach spaces carrying extended tuples of partial Caputo operators. In those models the only source of noncommutativity is the ordered creation of boundary-trace words, while the spectral multipliers remain diagonal and commute with the entire one-sided block. In the present paper we isolate this mechanism abstractly and develop an ordered boundary trace calculus that applies simultaneously to the continuous and discrete ambient models. We show that both ambient theories are realizations of one common coefficient-space model. On this common model we define ordered trace operators \ (Trₖ: X, ^p Tw^p\; (wᵣ), \) prove their recursive composition law, and decompose the defect part of each extended Caputo operator into a residual backward shift plus a boundary-trace creation term. We then define the ordering-defect space \ (N: = span\tₖ, ₊, -tₖ', ₊,: supp (w) =supp (w') \, \) and prove that it is exactly the smallest closed invariant subspace containing the ranges of all commutators. Equivalently, if \ (J₂₎₌ A: = Alg\C₁, , Cᵣ, M_\\) denotes the closed two-sided ideal generated by the commutators of the partial Caputo tuple, then the associated range space satisfies \ (R (J₂₎₌) =N. \) Thus the commutator ideal is precisely the operator-theoretic shadow of ordered boundary ambiguity. Finally, we pass to the quotient \ (X: =X, , ^p/N, \) where ordered trace words collapse to unordered boundary supports. On this quotient the induced partial Caputo tuple is commuting, and the quotient enjoys a universal property: every bounded intertwining of the ambient calculus with a commuting target tuple factors uniquely through \ (X\). Hence the quotient by the commutator ideal provides a canonical commuting reduction complementary to the maximal commuting sectors obtained earlier by restriction.
Ariel Daley (Wed,) studied this question.