Integral pair data persist to Blanchfield bordisms through standard infection

The factorization supplies an involution-preserving ring homomorphism $\mathbb{Z}[{\mathrm{\Pi }}_s]\to A$, and hence maps of all actual coefficient chain systems on $V$. Cap, dual-system identification and evaluation commute with this change of coefficients. Explicitly a dual cochain $f$ changes to $c\otimes a\mapsto \rho (f(c))a$; balancing and the finite face formulas show compatibility with coboundaries and cap. Both relative cap maps are isomorphisms by [F5], so their inverse duality maps commute too. Thus the image classes on $V$ retain the displayed zero and unit pairings. The standard base vector $1$ gives compatible transported Hermitian structures before and after change of ring. Hermitian symmetry [F2] supplies the reverse cross-pairing ${\lambda }_A(d_j,{\mathrm{\ell }}_i)={\delta }_{ij}$.

The ordinary group $H_2(V;k)$ has dimension $2r$. For $k=\mathbb{Q}$ this is exact rational localization of the integral singular complex, using [F5]. For $k={\mathbb{F}}_p$ one can check the coefficient sequence directly without an infinite lifting assertion. Lift the finitely many coefficients of an ${\mathbb{F}}_p$ two-cycle $z$ to an integral chain $\tilde{z}$. Write $\mathrm{\partial }\tilde{z}=pw$; then $w$ is an integral one-cycle and $p[w]=0$. Since $H_1(V;\mathbb{Z})=\mathbb{Z}$, $w=\mathrm{\partial }v$ for some two-chain $v$, so $\tilde{z}-pv$ is an integral cycle lifting $z$. Conversely, if an integral two-cycle becomes a boundary mod $p$, lift that one finite bounding three-chain and subtract its boundary; the difference is $p$ times an integral two-cycle, since integral chain groups have no $p$-torsion. Thus $H_2(V;{\mathbb{F}}_p)\cong H_2(V;\mathbb{Z})\mathrm{/}p={\mathbb{F}}_p^{2r}$. Its quotient by the boundary image therefore has dimension at most $2r$.

Give $V$ its collar from [F5] and use the actual product collars of the finite standard pieces. At each attachment the outgoing $H_1$ injection and incoming $H_2$ surjection of [F1] give a surjection from the preceding boundary cokernel to the new one. Its dimension cannot increase, so after finitely many attachments the ordinary cokernel is at most $2r$. The single coefficient system on final $W$ restricts to each intermediate manifold, and [F2] preserves every pair identity under inclusion. Thus the images ${\mathrm{\ell }}_i,d_i$ in final $W$ satisfy all pair identities in the Statement. Orientation reversal of a piece does not change its ordinary injection and surjection properties; the orientation on the embedded preceding manifold is unchanged, which is the orientation used in [F2]. All the resulting manifolds are compact and oriented with the actual remaining boundary components.

Every remaining boundary component is a chosen zero-longitude surgery on a knot. Its ordinary first homology over $k$ is $k$: in the exterior/filling Mayer–Vietoris sequence the preferred longitude maps to zero, the filling core identifies with the meridian, and that meridian remains the single free generator, exactly as in the peripheral calculation of [F1]. Hence [F3] applies to final $W$ and gives the first cokernel inequality in the Statement. If the restriction to $M_0$ is nontrivial, the second assertion of [F3] gives $H_1(M_0;K)=0$. Thus the finite pairs of step 2.1 and the localized dimension bound verify every hypothesis of [F4], with $R=A$ and the standard involutive fraction ring $K$.

Apply [F4] to obtain $B_0(a,b)=0$ for all classes in that inclusion kernel. If $B_0(a,a)\mathrm{\ne }0$, the class $a$ therefore cannot lie in the kernel, and its image in $H_1(W;A)$ is nonzero. This is the survival implication used for infection axes in the source bordism argument. Every isotropic submodule has zero self-pairings, so such a surviving class lies outside every isotropic submodule.

For actual integral $s$-solution data, these hypotheses are precisely its homology and lifted-pairing part; passage to a quotient target of derived length at most $s$ factors because a homomorphism carries the ordinary derived series into the target derived series. Every coefficient lift in step 1.2 involved one finite chain; all other arguments used the supplied finite pair list and actual collared attachments. The proof therefore preserves smooth and topological scopes and introduces no choice axiom beyond none in these data.