CCFU Proof 24 — sig(b_{Ω_W}) = (3,4)

2605245776109

CCFU Proof 24 — sig(b_{Ω_W}) = (3,4)

05/24/2026

Given.
Ω_W from Proof 22.

The Hitchin bilinear form [Proof 21, Step 4]:

    b_{Ω_W} = [[-6, 0, 0, 0, 0, 0, 0],
               [ 0, 0, 0, 0, 3, 0, 0],
               [ 0, 0, 0, 0, 0, 3, 0],
               [ 0, 0, 0, 0, 0, 0, 3],
               [ 0, 3, 0, 0, 0, 0, 0],
               [ 0, 0, 3, 0, 0, 0, 0],
               [ 0, 0, 0, 3, 0, 0, 0]]

Eigenvalues: −6 (×1), −3 (×3), +3 (×3).
sig(b_{Ω_W}) = (3,4).  ∎

This can be computed by hand: the matrix is block-diagonal
with a 1×1 block (−6) and three 2×2 blocks [[0,3],[3,0]]
each with eigenvalues ±3.

[Dependency: Proof 22. Algebraic — hand computation.]