The unique monodromy-invariant alternating form of type (1,6) on a (3,4,∞) lattice representation, and the restriction of a functorial symplectic structure

Let V = ℤ4 carry the representation of the triangle group Δ(3,4,∞) given by the matrices T1, T2 ∈ SL(4,ℤ) in Alpöge's self-hosted note on a family of complex 2-tori. The ℚ-vector space of alternating bilinear forms on Vℚ invariant under T1 and T2 is one-dimensional. A primitive integral generator is the matrix Ω0 of determinant 36 and elementary-divisor type (1,6). The unipotent T0 = (T1T2)−1 preserves the same form, and N = T0 − I lies in 𝔰𝔭(Vℚ, ω0). Consequently any monodromy-invariant alternating form on this local system—including the restriction of a functorial symplectic 2-form on families of complex 2-tori—is a unique rational multiple of ω0. This organises the fibrewise pairing used in vanishing-cycle and unipotent-cusp calculations. It does not construct a compactification of the family, and it does not prove X ≃ S6. The accompanying Lean 4 library OmegaZero34 is the kernel-checked form of Theorems A and B and of the implication in Theorem C. Alpöge is cited for the matrices T1, T2 and the compactification claim, not as a co-author.