Smooth and topological concordance respect connected sum

First consider smooth concordances. Pick one point on each endpoint knot. By [F4], apply separate endpoint ambient isotopies carrying the marked points to a common point $p\in S^3$; extend them over disjoint product collars of each concordance and interpolate by the identity between the collars. This changes the representatives only within their ambient-isotopy classes. Choose in each annulus a smooth spanning arc with image equal to $\{p\}\times I$ near its endpoints. Apply [F8] to each arc. Its ambient smooth isotopy is fixed near both end levels and carries the spanning arc to the vertical product arc; applying its terminal diffeomorphism to the annulus preserves smoothness, orientation, properness and the endpoint knots and their product collars.

For the TOP implication, first code the actual input as compact metric data. Parametrize the two annuli by the standard compact annulus, and choose finitely many local pair charts covering each, together with their inverses and the given end-collar maps. Such a finite selection is possible by compactness and finite choice; using countably many relatively compact chart restrictions instead would require only ${\mathrm{A}\mathrm{C}}_{\omega }$. Include the endpoint knots and the chosen connected-sum representatives from [F2]. For these smooth knots and the finite smooth coordinate maps used in their sums, also code the derivatives of every finite order, their continuity moduli and rational difference-quotient remainder bounds; thus the inner model sees smooth knots, not merely continuous embeddings. Continuous maps on these standard metric domains are coded by their values on fixed countable rational dense sets and uniform continuity moduli on compact exhaustions; include inverse moduli for embeddings and chart inverses. All this countable numerical data is one real $r$.

For each straightened annulus $A$, height restricts to a submersion along the vertical core, since the tangent of that core has height derivative $1$. Choose a sufficiently small strip in $A$ parametrized by $(u,t)\mapsto (a(u,t),t)$ with $a(0,t)=p$, using the product collars at its ends. Complete its tangent line in each three-dimensional level to an oriented frame and take small normal disks within that level. The resulting smooth family of embeddings ${\mathrm{\Phi }}_t:D^3\to S^3$ satisfies $A\cap ({\mathrm{\Phi }}_t(D^3)\times \{t\})={\mathrm{\Phi }}_t(D^1)\times \{t\}$, and is independent of $t$ near each end. Uniform radii exist by compactness; separation from the compact remainder of $A$ excludes other sheets. The initial small coordinate ball is standard: in a fixed coordinate chart, rescaling a local embedding toward its derivative gives an isotopy to a linear ball after shrinking, and [F3] extends that ball isotopy in $S^3$. Its complement is consequently a three-ball. Apply [F3] again to the compact ball isotopy ${\mathrm{\Phi }}_t$ in the boundaryless target $S^3$, obtaining diffeomorphisms $G_t$ with $G_t{\mathrm{\Phi }}_0={\mathrm{\Phi }}_t$. Thus $(x,t)\mapsto (G_t(x),t)$ identifies the entire tube and its complement with the products of the initial ball and its complement. In these product coordinates, remove the two tube interiors from two copies of $S^3\times I$ and glue their boundary cylinders by the fixed orientation-reversing map, independent of $t$. The ambient result is the product of two three-balls glued along their boundary spheres with $I$, hence $S^3\times I$. Removing the open strip from each annulus leaves a rectangle; glue the rectangles along their two side intervals in the connecting cylinder, compatibly with their orientations. Their lower and upper boundary knots are $-(K_0\mathrm{\#}{J}_0)$ and $K_1\mathrm{\#}{J}_1$ by [F2]. The resulting connected orientable surface has two boundary circles and Euler characteristic $1+1-2=0$, so it is an annulus. Round the product corners in the fixed local coordinates; the endpoint product collars are retained. This gives the smooth concordance.

We use certificates, not an unqualified assertion that homeomorphism existence is absolute. For every open chart domain record its rational-ball code (relative rational half-balls at a boundary chart), compact exhaustion, and positive-distance margins putting the images of exhaustion pieces in the corresponding open image. Record both map/inverse identities on dense sets. For the pair condition record the parametrization of the linear plane section and its identities with the annulus parametrization; on smaller charts also record positive separation from the compact remainder of the annulus outside that patch. A finite parameter-patch cover and a Lebesgue-number certificate cover the entire annulus. Properness is certified on the compact subannuli $S^1\times [1\mathrm{/}j,1-1\mathrm{/}j]$ by positive separation from the ambient end levels. Equalities with the oriented endpoint parametrizations and the product collars are dense-set equalities with their continuity moduli. The smooth derivative identities are certified by rational difference quotients and their coded remainder bounds at each finite order. These certificates express input validity using only rational inequalities, integer indices and quantification over natural numbers. They therefore have the same truth in $L[r]$ and in the ambient universe; in particular the coded input is valid in $L[r]$.

The preceding construction proves the smooth implication.

Apply [F6] inside $L[r]$, which satisfies full Choice by [F7]. It gives a locally flat annulus for the two connected sums. In that model choose its finite pair-chart cover, inverse maps, compact-exhaustion moduli, open-image margins and end-collar certificates of the same form as in step 2.2; collect them into a real $s\in L[r]$. Decode $s$ in the ambient universe by completion from the rational dense domains. The coded forward and inverse moduli give continuous inverse maps on their coded domains; the dense-set identities extend to all points, and the open-image and separation certificates make these actual local pair charts with no additional surface branch. The finite cover certificate, properness margins, and endpoint/collar identities likewise persist. Thus the decoded object is an actual oriented proper locally flat annulus in the ambient $S^3\times I$, with the required endpoint connected sums and product collars. This transfer uses arithmetic certificates for the particular compact objects, not preservation of arbitrary sets or homeomorphisms between inner models.

Steps 1.2, 2.2, 3.2 earn the required compact specialization of the external TOP input under the stated ${\mathrm{A}\mathrm{C}}_{\omega }$, without assuming ambient full AC. Together with the retained smooth construction in steps 1.1–3.1, this proves both implications in the unchanged Statement. No annulus-adapted transverse tube or ambient complement is inferred from the bare-arc theorem [F5]; that former local proof gap is bypassed by the explicitly external input [F6].