An Alexander-one Seifert form is S-equivalent to the empty form

By the actual determinant supplier, the supplied Seifert matrix has Laurent-unit determinant. Its skew part is the integral unimodular surface intersection form. The following finite matrix reduction applies to this very matrix and preserves the positive-push convention.

Let V be an integral 2g-by-2g matrix and J=V−V^T unimodular alternating. Suppose P(t)=det(V−tV^T) is a Laurent unit. Since P is a polynomial and P(t)=t^(2g)P(t^−1), it is t^g: a unit polynomial is ±t^m, reciprocity gives m=g, and P(1)=det J=1 fixes the sign. (The determinant of an integral unimodular alternating matrix is +1: choose any primitive vector, use the primitive row of J and Bezout to split off a symplectic pair, and repeat on its integral orthogonal direct summand. This determinant argument does not require a kernel vector of V, so it precedes the following deduction det V=0 without circularity.) For g>0 its constant coefficient det V is zero. Choose a nonzero rational kernel vector of V, clear denominators and divide by its coordinate gcd. This gives a primitive integral x with Vx=0.

The row x^T V equals x^T J, since x^T V^T=(Vx)^T=0. It is primitive because multiplication by the unimodular J preserves primitive vectors. Bezout therefore gives an integral y with x^T V y=1. Then omega(x,y)=x^T J y=1. Put L={z:omega(x,z)=omega(y,z)=0}. Every z decomposes uniquely as [F1, F2, F3, algebra] z = omega(z,y)x + omega(x,z)y + z_0, z_0 in L.

Thus Z^(2g)=Zx direct-sum Zy direct-sum L, and omega restricted to L is unimodular alternating. L is a free integral direct summand; its basis is obtained by applying the displayed integral projection to the coordinate generators and elementary Euclidean column operations. Repeating the primitive-vector/Bezout splitting in L gives an integral symplectic basis there. In that basis, followed by x,y, the Seifert matrix is [F1, F2, F3, algebra] [[B,0,u],[0,0,1],[u^T,0,w]].

Indeed Vx=0 gives its zero x-column, and x^T V=x^T J gives the x-row. Orthogonality to y gives equality of the two old-to-y coupling vectors. Replace y by y−wx. This is a symplectic integral basis change and makes w zero, without changing B,u. We have exhibited exactly the algebraic tube enlargement in the ACTUAL S-equivalence Definition, not a rational enlargement.

Expanding det(tV−V^T) along the x-row and then x-column gives [F1, F2, F3, algebra] det(tV−V^T)=t det(tB−B^T).

For even matrix size det(tV−V^T)=P(t) by transposition and multiplication by −1. Hence the reduced matrix B again has unit polynomial t^(g−1) and unimodular alternating skew part. Induct on g. This supplies a finite sequence of integral congruences and exact inverse tube additions down to the empty matrix. No stabilization theorem is assumed. It proves the full matrix assertion needed by the original S-equivalence carrier once its actual Seifert determinant/skew-pairing interfaces are applied.