The CDR concordance categories are abelian groups

Specialize the TOP/exotic external assertion [F5] at each finite list of knot representatives and witnessing annuli actually used. For TOP these lie in the standard product; for exotic concordance the supplied boundary-marked homeomorphism to that product is part of the input code. The finite chart and pair-chart extraction in [F6] uses compactness, not a choice of an atlas for a countable family. Inside $L[r]$ use the ordinary CDR group operation, retain the actual output annulus and marked product homeomorphism, and export their compact forward/inverse and collar certificates. This proves the required finite compatibility and transitivity instances in $V$. Since each instance starts with its own given finite data, no simultaneous choice of representatives or witnesses for all classes is made. The quotient group operations and laws follow by universal quantification over these instances; the original ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis is therefore respected, rather than inferred from the unqualified source assertion. The localized category is handled directly by the subsequent homology calculations, with no TOP normal theorem needed.

Let $D\subset B^4$ be the smooth ribbon disk for $K\mathrm{\#}{K}^{†}$ supplied by [F2]. By local smooth coordinates and straightening of the embedded disk, choose an ambiently standard interior 4-ball $B$ such that $B\cap D$ is a standard 2-disk. An ambient isotopy supported in the interior carries $B$ to a concentric standard ball, so its complement in $B^4$ is diffeomorphic to $S^3\times I$. Removing $\mathrm{int}B$ from $B^4$ and the interior of $B\cap D$ from $D$ gives a smooth annulus. The outer and inner boundary collars give product collars for it. Order the ends with $K\mathrm{\#}{K}^{†}$ at the lower end and the inner standard unknot at the upper end, and orient the annulus so its boundary is $-(K\mathrm{\#}{K}^{†})$ at the lower end and $U$ at the upper end. It is therefore an admissible inverse concordance in the smooth category, and the same product annulus satisfies the TOP, exotic and localized ambient conditions.

The supplied connected-sum theorem gives associativity, commutativity and the unknot identity at the level of oriented knot types. The smooth conclusion of [F4] gives concordance compatibility in the standard smooth product; the TOP/exotic compatibility is the external input [F5], and the localized ambient verification is given by the subsequent homology calculations. The knot-type isotopies themselves are admissible annuli in every category.

The external input [F5] supplies the TOP and exotic ambient gluing/compatibility assertions at exactly the definitions in [F1]. For the localized smooth category, transitivity is stacking along the marked $S^3$: Mayer–Vietoris and the two boundary homology isomorphisms give another $R_n$-homology $S^3\times I$. Reflection gives symmetry and a product gives reflexivity, all with their smooth annulus collars.

For localized connected sum choose a smooth product neighborhood $D^3\times I$ of a proper spanning arc inside each witnessing annulus, with the annulus strip in its standard local position, and remove its interior. If $Q$ is the remaining ambient piece, excision gives $H_j(W,Q;R_n)=R_n$ only for $j=3$. The map $H_3(W;R_n)\to H_3(W,Q;R_n)$ is an isomorphism: the boundary fundamental class meets the spanning arc once. The pair sequence therefore makes $Q$ an $R_n$-homology ball. Glue the two such complements on $S^2\times I$, joining the annulus strips. Mayer–Vietoris gives $R_n$ in degrees0 and3 and zero in other positive degrees; the endpoint connected-sum spheres give the degree3 generator, so each end inclusion is a homology isomorphism. The resulting smooth annulus witnesses connected-sum compatibility in the original localized category. This calculation uses the product strip neighborhood along an arc in the smooth annulus; it does not apply the standard-product conclusion of [F4] to a homology product. These calculations use no homeomorphism-to-product assertion.

Thus connected sum is well defined in every category: the retained standard smooth construction handles ${\mathcal{C}}_{\mathrm{s}\mathrm{m}}$, [F5] handles TOP/exotic, and steps 2.2–3.1 handle the localized case. The oriented knot connected-sum monoid [F3] descends to each quotient, giving associativity, commutativity and the unknot identity. The ribbon inverse in [F2], admissible in all four ambients, supplies the stated reverse-mirror inverse. All ambient and coefficient conditions are those of [F1].