A closed dual test form pairs a compact three-cycle with flux and source currents

Use the canonical coefficient inclusion $\mathbb{Z}\to \mathbb{R}$ implicitly. On any singular pair $(Y,A)$, send a finite integral chain $\sum n_{\sigma }\sigma$ to the same finite real chain. An integral relative cochain $a$, vanishing on simplices in $A$, gives the real cochain $a_{\mathbb{R}}(\sum r_{\sigma }\sigma )=\sum r_{\sigma }a(\sigma )$. Face boundaries have integer coefficients, so these maps commute with boundary and coboundary and carry boundaries and coboundaries to boundaries and coboundaries. They therefore give coefficient maps on absolute and relative (co)homology. Pullback, pushforward, and pair inclusions commute with them because they use the same simplex maps and faces; the excision isomorphisms, induced by those inclusions, consequently commute as well. The cap formula on a simplex is $a\cap \sigma =a(\sigma \mathrm{\mid }[v_0,\dots ,v_p])\sigma \mathrm{\mid }[v_p,\dots ,v_n]$. Extending its finite coefficient calculation gives $j_{\ast }(a\cap c)=a_{\mathbb{R}}\cap j_{\ast }c$; the front/back cup formula has the same coefficient compatibility. Each positive integral local orientation generator maps to the positive real generator, and a normalized integral normal-disk Thom cocycle still evaluates to $1$ on that disk. The local characterization in F8 therefore identifies $j_{\ast }[X{]}_{\mathbb{Z}}=[X{]}_{\mathbb{R}}$ and $j_{\ast }[D{]}_{\mathbb{Z}}=[D{]}_{\mathbb{R}}$. Finally, cochain evaluation on every finite integral cycle is the same numerical value regarded in $\mathbb{R}$. These explicit formulas require no assertion that tensoring commutes with cohomology.

Let ${\tau }_D$ be the real coefficient image of the integral absolute class ${\tau }_{D,\mathbb{Z}}$ in F2. Apply step 1.1 to its integral cap identity to get ${\tau }_D\cap [X{]}_{\mathbb{R}}=-(i_D{)}_{\ast }[D{]}_{\mathbb{R}}$. Hence $u_D:=-{\tau }_D$ satisfies $u_D\cap [X{]}_{\mathbb{R}}=(i_D{)}_{\ast }[D{]}_{\mathbb{R}}$. By F3 choose a closed smooth three-form ${\theta }_D$ whose real de Rham class maps to $u_D$, and set ${\eta }_D:=-{\theta }_D$. It is closed. All subsequent cap classes use real coefficients, with inclusion pushforward suppressed in the notation $[D]$.

For a closed three-form $\omega$, the cap formula [F4] gives $(u_D⌣[\omega ])\cap [X]=[\omega ]\cap [D]$. Evaluating the resulting zero-cycle by the augmentation is the Kronecker pairing of $[\omega ]$ with $[D]$. Thus [F3], [F5], and [F6] give ${\int }_X{\theta }_D\wedge \omega ={\int }_D\omega$. Since both degrees are three, $\omega \wedge {\eta }_D=-\omega \wedge {\theta }_D={\theta }_D\wedge \omega$, so ${\int }_X\omega \wedge {\eta }_D={\int }_D\omega$.

The intersection set is finite by compactness and transversality. Excision identifies the restriction of the integral normal Thom class to $C$ with the direct sum of its local classes in $H^3(B^3,B^3\setminus \{0\};\mathbb{Z})$, one for each intersection point. The normal projection $C\to {\nu }_D$ is a local diffeomorphism at each such point by the inverse function theorem. With ${\nu }_D$ oriented in tangent-first order, its local degree is $-{\sigma }_x(C,D)$: moving the positive three-frame of $T_{x}C$ past the positive three-frame of $T_{x}D$ changes the ambient orientation by $(-1{)}^9$. The fiber-normal Thom class has value one on the positive normal disk, so its pullback contributes exactly that local degree. Summing the finitely many local evaluations gives $⟨{\tau }_{D,\mathbb{Z}},[C{]}_{\mathbb{Z}}⟩=-{\sum }_x{\sigma }_x(C,D)$. The evaluation compatibility in step 1.1 gives the same equality for its real image ${\tau }_D$ and $[C{]}_{\mathbb{R}}$. Hence $u_D=-{\tau }_D$ evaluates as ${\sum }_x{\sigma }_x(C,D)$; integration of its representative ${\theta }_D$ over $C$ gives this Kronecker evaluation by [F3] and [F6]. Therefore by [F1] and [F7]. Empty intersection gives zero on both sides.