Cochran–Davis–Ray satellite injectivity

Use the finite-input choice qualification in [F3]: encode the actual finite knots, pattern and witnessing concordance, including the finite smooth presentation and chain homotopies when localized coefficients are involved. Execute only the unconditional geometry in $L[r]$ and decode actual compact witnesses, finite group words and chain identities into $V$. Thus the finite uses of [F4] below do not apply an AC_omega theorem without its premise. No countable collection of representatives is selected. In the conditional ordinary smooth direction, the actual contractible smooth ball and capped sphere are constructed and decoded before the supplied smooth sphere-uniqueness hypothesis is applied in $V$. No transfer of that conditional hypothesis to $L[r]$ is made.

Suppose P(K)=P(J) as concordance classes. By [F5] these values are defined in the chosen category, and the group law gives −[P(K)]#[P(J)]=0.

Apply the auxiliary-pattern reduction [F3] to obtain R with R(−K#J)=U, closure R(U)=−P(K)#P(K) ribbon, and winding w(R)=w(P).

If P has strong winding ±1, [F3] preserves strong winding for R; in the localized branch R has the same nonzero winding.

The weak-injectivity part of [F3] applies to R and yields $-K\mathrm{\#}J=U$, hence $K=J$. In ordinary smooth concordance the last step uses [F6] and assumes uniqueness of the smooth structure on S⁴.