Geometric intersection numbers are isotopy invariants

The exceptional cases. Suppose the pair has no common boundary endpoint and $c_0\simeq c_1$. For closed curves the prescribed value is $2$. For arcs, isotopy preserves their endpoint set, so both endpoints must lie in $\mathrm{\Delta }$. Each relative minimal model in [L4] has just those two common marked endpoints, giving $I=1$. These values depend only on the isotopy classes.

Other minimal representatives give the same count. Suppose $c_0\simeq ̸c_1$ and let $c_1^{\mathrm{\prime }},c_1^{\mathrm{\prime }\mathrm{\prime }}$ be minimal representatives of $c_1$ relative to $c_0$. The relative comparison [L4] gives an ambient isotopy preserving $c_0$ setwise and carrying $c_1^{\mathrm{\prime }}$ to $c_1^{\mathrm{\prime }\mathrm{\prime }}$. Its endpoint map bijects intersections with $c_0$ and preserves $\mathrm{\Delta }$, so the ordinary and marked intersection counts agree. With step 1.1 this proves representative independence away from boundary intersections. The AC-dependent arc inputs [L2] and [L3] retain their hypotheses; the stronger relative comparison is the source lemma [L4].

Isotopy invariance away from boundary intersections. An endpoint map $F$ of an identity-component ambient isotopy carrying $c_0$ to $c_0^{\mathrm{\prime }}$ carries a minimal representative $c_1^{\mathrm{\prime }}$ to a minimal representative of the same class of $c_1$. Simultaneous transport preserves both counts. If the pair is isotopic and closed, both values instead equal the prescribed $2$; otherwise the weighted formula applies. Hence $I(c_0^{\mathrm{\prime }},c_1)=I(c_0,c_1)$ by step 2.1. Representative independence gives invariance in the second argument as well.

The boundary push is independent of its small positive choice. Interpolate between the two positive boundary fields and their extensions by convex combination, and between sufficiently small positive flow times; write $g_s$ for the resulting endpoint diffeomorphisms. Compactness of the parameter interval allows a common small-time bound, so each boundary endpoint of $g_s(c_0)$ stays in one complementary interval of $\mathrm{\partial }D\setminus c_1$. Larger allowed times can first be decreased within those intervals. Choose a boundary isotopy $h_s$, starting at the identity and fixing the endpoints of $c_1$, which carries these moving endpoints back to those of $g_0(c_0)$. Extend $h_s$ to $H_s$ in a thin collar preserving $c_1$ setwise and missing $\mathrm{\Delta }$: in collar coordinates straightening the endpoint germs of $c_1$ to radial segments, extend the boundary velocity tangentially along these segments, with a cutoff. Then $a_s:=H_s{g}_s(c_0)$ is a smooth isotopy of embedded arcs with fixed endpoints. It extends to a boundary-fixed ambient isotopy fixing $\mathrm{\Delta }$: extend the velocity along the moving arc over tubular charts with cutoffs; it vanishes at fixed endpoints, and transversality at boundary endpoints allows the extension to vanish on the boundary. Thus step 3.1 gives $I(a_0,c_1)=I(a_1,c_1)$. Simultaneous transport by $H_1$, which preserves $c_1$, bijects intersections and marked subsets and preserves the Jordan-disk condition; therefore $I(g_1(c_0),c_1)=I(a_1,c_1)=I(g_0(c_0),c_1)$. The comparison is between isotopy classes before minimization; no isotopy preserving $c_1$ is asserted between the arbitrary pushed arcs themselves.

Invariance with the boundary convention. A boundary-fixed endpoint map $F$ transports a positive field $Z$ to $F_{\ast }Z$ and conjugates their flows, so simultaneous transport identifies $I(f_t(c_0),c_1)$ with $I(F{f}_t(c_0),F(c_1))$. Step 4.1 permits the transported push for $F(c_0)$. Since the pushed pair has disjoint boundary endpoints and $F(c_1)\simeq c_1$, step 3.1 identifies the latter count with the pushed count for $(F(c_0),c_1)$. This proves invariance in the first argument. For an isotopy in the second argument, use the same fixed push of $c_0$; its boundary endpoints are disjoint from the fixed endpoints of every curve in that isotopy, so step 3.1 applies directly.

Conclusion. The ordinary weighted formula, the exceptional closed value, and the positive-boundary extension all define numbers independent of minimal representatives and invariant under the stated isotopies. The AC-dependent arc inputs retain the hypothesis in the Statement; the finite counts and explicit collar comparison require no additional choice.