= Solution
After replacing a class by a nonsingular representative, let $A$ represent a <Seifert form> over a field $F$ of characteristic different from two. Set
$$
Q=A+A^T,
\qquad
T=A^{-1}A^T.
$$
A direct calculation gives $T^TQT=Q$, so $(F^{2g},Q,T)$ is an <isometric structure>. A metabolizer for $A$ corresponds to a $T$-invariant metabolizer for $Q$, and stabilization gives the canonical homomorphism
$$
\mathcal{AC}_F\longrightarrow\mathcal W_F.
$$
Conversely, for an isometric structure with $I+T$ invertible, define
$$
A=Q(I+T)^{-1}.
$$
Then $A+A^T=Q$ and $A^{-1}A^T=T$. These constructions respect orthogonal sums and metabolic structures and are inverse on Witt classes, proving $\mathcal{AC}_F\cong\mathcal W_F$.
For an irreducible symmetric Laurent polynomial $\delta$, the <primary component of an isometric structure> is
$$
V_\delta=\ker\delta(T)^N
$$
for large $N$. The primary decomposition is orthogonal, so restriction of $Q$ and $T$ to $V_\delta$ defines the projection
$$
\mathcal W_F\longrightarrow\mathcal W_F^\delta.
$$
Now take $F=\mathbb R$ and let $\delta$ have roots $\omega,\overline\omega$ on the unit circle, with $\omega=e^{i\theta}$ in the upper half-plane. The isomorphism $\mathcal W_{\mathbb R}^\delta\cong2\mathbb Z$ sends a class to the even signature jump
$$
J_\omega=lim_{\varepsilon\downarrow0}
\left(\sigma_{e^{i(\theta+\varepsilon)}}-
\sigma_{e^{i(\theta-\varepsilon)}}\right).
$$
For the class of a knot, this is precisely the jump of its <Levine-Tristram signature> at the root $\omega$; reversing the choice of side changes the overall sign convention.
Solved by gpt-5.6-sol high.
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