CCFU Proof 21 — Higher Alternating Closure

2605245766544

CCFU Proof 21 — Higher Alternating Closure

05/24/2026

Given.
W = S₁ ∩ S₂, dim W = 3 [Proof 20].

Step 1 — The higher closure of W.
    Define:
        V_W = R·τ ⊕ W ⊕ W*
        dim V_W = 1 + 3 + 3 = 7.
    W* is the linear dual of W. τ is a formal line.  ∎

Step 2 — The natural 3-forms on V_W.
    Λ³(V_W*) decomposes under SL(W) as:
        Λ³W* ⊕ Λ³W ⊕ (R⊗Λ²W*) ⊕ (R⊗Λ²W)
        ⊕ (R⊗W*⊗W) ⊕ (Λ²W*⊗W) ⊕ (W*⊗Λ²W)

    SL(W)-invariant 3-forms in each component:
        Λ³W*:      dim 1 → vol_W (volume form of W)
        Λ³W:       dim 1 → vol_{W*} (volume form of W*)
        R⊗W*⊗W:   dim 1 → τ ∧ ev (evaluation pairing)
        All others: dim 0.

    Total SL(W)-invariant 3-forms: exactly 3.
    Basis: {τ ∧ ev, vol_W, vol_{W*}}.

    Note: τ∧ev is GL(W)-invariant (canonical).
    vol_W and vol_{W*} are SL(W)-invariant but not GL(W)-invariant:
    under g ∈ GL(W) with det(g) = δ,
        vol_W → δ · vol_W,  vol_{W*} → δ⁻¹ · vol_{W*}.
    Therefore a volume choice on W is needed to write Ω_W.
    The GL(7)-orbit absorbs this choice (Step 5).  ∎

Step 3 — The generic natural 3-form.
    Ω_{a,b,c} = a·(τ ∧ ev) + b·vol_W + c·vol_{W*}

    In coordinates (τ, x₁,x₂,x₃, y₁,y₂,y₃):
        τ ∧ ev = e₀₁₄ + e₀₂₅ + e₀₃₆
        vol_W = e₁₂₃
        vol_{W*} = e₄₅₆

    So: Ω_{a,b,c} = a·e₀₁₄ + a·e₀₂₅ + a·e₀₃₆ + b·e₁₂₃ + c·e₄₅₆.
    Five terms. Volume choice on W required;
    GL(7)-orbit independent of choice [Step 5].  ∎

Step 4 — Stability.
    The Hitchin bilinear form of Ω_{a,b,c} has block structure:
    b[0,0] = -a³ and three 2×2 blocks [[0, abc/2], [abc/2, 0]]
    pairing W with W*.

    det(b_Ω) = 4374 · a⁹ · b⁶ · c⁶.

    Therefore Ω_{a,b,c} is stable if and only if abc ≠ 0.  ∎

Step 5 — All stable forms are in one GL(7)-orbit.
    For any a,b,c with abc ≠ 0, take g = diag(α, λI₃, μI₃) with
        λ = b^{-1/3},  μ = c^{-1/3},  α = (bc)^{1/3}/a.
    All cube roots are real (over R). Under pullback (covariant):
        τ∧ev coefficient: a · α · λ · μ = 1.
        vol_W coefficient: b · λ³ = 1.
        vol_{W*} coefficient: c · μ³ = 1.
    Therefore (g⁻¹)*Ω_{a,b,c} = Ω_{1,1,1}.
    All Ω_{a,b,c} with abc ≠ 0 are GL(7)-equivalent.  ∎

    Note: the volume choice on W gives a one-parameter family
    Ω_t = τ∧ev + t·vol_W + t⁻¹·vol_{W*} (since ν⁻¹ scales
    inversely). This is a subfamily with a=1, bc=1.
    Step 5 covers the general case.

Step 6 — Minimality of dim 7.
    On W ⊕ W* (dim 6), the natural 3-forms are vol_W + vol_{W*}.
    These have stabilizer SL(3) × SL(3), dim 16. Classical.
    The evaluation pairing ev ∈ W* ⊗ W is degree 2.
    To include ev in a 3-form requires one additional covector τ:
        τ ∧ ev ∈ Λ³(R ⊕ W ⊕ W*).
    This cannot live in Λ³(W ⊕ W*).
    Therefore dim 7 = 1 + 3 + 3 is minimal.  ∎

Conclusion.
    C₂ determines W, dim 3 [Proof 20].
    W determines V_W = R ⊕ W ⊕ W*, dim 7 (minimal).
    The natural 3-form Ω_W = τ∧ev + vol_W + vol_{W*}
    requires a volume choice on W. But:

    The volume choice is absorbed already by GL(W):
    replacing ν by t·ν gives Ω_t = τ∧ev + t·vol_W + t⁻¹·vol_{W*},
    and g = diag(1, t^{-1/3}I₃, t^{1/3}I₃) ∈ GL(W) ⊂ GL(7) sends
    Ω_t back to Ω₁. No full GL(7) needed.

    Step 5 proves a stronger statement: all Ω_{a,b,c} with abc≠0
    (including a≠1, bc≠1) are GL(7)-equivalent. This goes beyond
    volume choice — it covers deformations that break W/W* duality.

    C₂ determines the GL(7)-orbit. The representative requires
    a volume choice. The choice is absorbed by GL(W).

[Dependencies: Proof 20. Algebraic — no computation needed.]