The proof has two parts: (I) every Vassiliev invariant determines a weight system (surjectivity of $\mathcal{V}_n/\mathcal{V}_{n-1}\to\mathcal{A}_n^*$), and (II) every weight system arises from a Vassiliev invariant (injectivity, which requires constructing $Z$).

Part I: Vassiliev invariants yield weight systems.

Step 1. Let $v$ be a type-$n$ Vassiliev invariant. For a singular knot $K$ with exactly $n$ double points, $v(K)$ depends only on the "chord diagram" of $K$: the combinatorial pattern of which pairs of points on the circle are identified.

To see this, suppose $K_1$ and $K_2$ are singular knots with the same chord diagram but different embeddings (different "topology" between the double points). We show $v(K_1) = v(K_2)$.

Step 2. The knots $K_1$ and $K_2$ can be related by a sequence of crossing changes (Reidemeister moves) in the non-singular parts. Each crossing change creates a singular knot with $n+1$ double points. Since $v$ is type $n$: $v(K_+) - v(K_-) = v(K_\times) = 0$ (the singular knot has $n+1$ points). Therefore $v(K_+) = v(K_-)$: the value is unchanged by crossing changes in non-singular parts.

Step 3. Thus $v$ descends to a function on chord diagrams of degree $n$. The 4T relation follows from specific knot isotopies that relate four singular knots with $n$ double points. Therefore $v$ defines a weight system $w_v \in \mathcal{A}_n^*$.

Part II: Every weight system is realized (via the Kontsevich integral).

Step 4. Kontsevich constructs the integral $Z(K) = \sum_{m\geq 0}Z_m(K) \in \hat{\mathcal{A}}$ using iterated integrals of logarithmic 1-forms (the formula given in the concept page).

Step 5: Well-definedness. The key is to show $Z(K)$ is invariant under horizontal deformations of $K$ (deformations preserving the Morse function). This follows from the flatness of the Knizhnik-Zamolodchikov connection: the KZ equation $\frac{\partial\Phi}{\partial t_j} = \left(\sum_{i<j}\frac{\Omega_{ij}}{z_i-z_j}\right)\Phi$ has zero curvature, so the iterated integral (which computes holonomy) is deformation-invariant.

Step 6: Normalization. To handle critical points (maxima and minima of the Morse function), divide by the Kontsevich integral of a standard unknot with the same number of critical points. This cancels the contribution of critical points and yields a genuine knot invariant $\hat{Z}(K)$.

Step 7: Universality. For any weight system $w \in \mathcal{A}_n^*$, the function $v_w(K) = w(Z_n(K))$ is a type-$n$ Vassiliev invariant (since $Z_m = 0$ for singular knots with $>m$ double points, and the chord diagram of a singular knot with $n$ points equals $Z_n$ evaluated on it). The weight system of $v_w$ is $w$ itself, establishing the isomorphism. $\square$