The Kontsevich integral induces the identity on the associated graded

Choose pairwise disjoint balls around the $n$ singular points so each resolved pair of branches is monotone in height inside its ball. The same height function is used on every resolution, so $r(K_{ϵ})=r$ for all $ϵ$. If necessary, move the singular knot by an isotopy supported in the balls to separate the singular heights and make the crossings monotone; each corresponding resolution is carried by the same ambient isotopy, so [F6] leaves ${\mathrm{\Delta }}^{n}I(K)$ unchanged.

Expand the degree-$m$ part of ${\mathrm{\Delta }}^{n}I(K)$ as a finite sum of pairing integrals over the resolution choices. If a pairing has no chord endpoint in the $i$th crossing ball, its integrand and diagram are independent of the sign ${ϵ}_i$, because the two resolved knots agree outside that ball. Summing over ${ϵ}_i=+1,-1$ therefore cancels that term. Hence every nonzero contribution to the $n$-fold difference must meet each of the $n$ balls.

Apply the principal-symbol theorem [F7] to the common singular Morse presentation. It gives $D_{n}Z(K{)}_m=0$ for $m<n$ and $D_{n}Z(K{)}_n=D$. The local hypotheses hold by step 1.1 and the unframed target agrees by [F2, F3].

For one singular point, let $w_{\pm }(t)=z_{1,\pm }(t)-z_{2,\pm }(t)$ be the relative complex coordinate of its two monotone branches in the positive and negative resolutions. The loop obtained by following $w_{+}$ and then the reverse of $w_{-}$ winds once positively around $0$: the two resolutions are the opposite half-turns of the relative vector. Hence $\frac{1}{2\pi i}(\int d\log w_{+}-\int d\log w_{-})=1$. For two upward branches this is the difference of angular changes $+\pi -(-\pi )=2\pi$. Reversing both branch orientations leaves the determinant sign unchanged and changes the down count by two; reversing exactly one branch reverses both the crossing sign and the factor $(-1{)}^{\downarrow }$. Thus under [F4] the coefficient is $+1$ for every oriented positive resolution. The localization theorem [F7] identifies this local winding contribution with the leading term of the full one-singular tangle. Thus the leading term is the one-chord diagram joining the two preimages.

The degree-$n$ diagram in [F7] pairs exactly the two preimages of each marked double point. Step 3.1 checks that its coefficient has sign $+1$ under the present oriented resolution convention. Local factors compose in height order, and their endpoints occur on the original circle skeleton; hence the resulting class is precisely $D$ in the unframed 4T+1T quotient. This agrees with the full principal-symbol assertion in step 2.1, including diagrams that vanish under 1T.

The common factor $Z(H{)}^{-r}$ can be pulled outside the alternating sum because all resolutions have the same $r$. Its degree-zero coefficient is $1$, so multiplication by it preserves the first nonzero homogeneous component: the degree-$n$ component remains $D$, and all lower degrees remain zero. For $n=0$, the alternating sum is $I(K)$ and its degree-zero coefficient is $1$ by [F2]. Thus ${\mathrm{\Delta }}^{n}I(K)=D+{\widehat{\mathcal{A}}}_{>n}^{\mathrm{u}\mathrm{n}}$ in every case.