CCFU Proof 29 — C₂ Constructs G₂(split)

2605265789974

CCFU Proof 29 — C₂ Constructs G₂(split)

05/26/2026

Theorem.
C₂: x(n+1) = x(n) + x(n-1) determines a stable 3-form Ω_W
on R⁷, unique up to GL(7)-equivalence, whose stabilizer
is G₂(split). One volume choice on W, absorbed by GL(7).

Proof.

(i) C₂ forces φ = (1+√5)/2 [Proof 1].

(ii) spec(A₂) = {φ, −1/φ} gives two completions:
     factor → sig(3,1), symmetry → sig(2,2) [Proof 17].

(iii) sig(2,2) confirmed universal at fixed point for all
      degree-2 memory maps [Proof 26].

(iv) Minimal parent: sig(3,2), dim 5 = Δ(C₂) [Proofs 5, 17].

(v) Parent decomposes: W = S₁ ∩ S₂, dim W = 3.
    W nondegenerate, sig(2,1) [Proof 20].

(vi) Higher alternating closure [Proof 21]:
     V_W = R ⊕ W ⊕ W*, dim = 1+3+3 = 7.
     Ω_W = τ∧ev + vol_W + vol_{W*}.
     Five terms. Volume choice on W absorbed by GL(7).
     dim 7 is minimal.

(vii) dim Stab(Ω_W) = 14, exact [Proof 23].

(viii) sig(b_{Ω_W}) = (3,4), algebraic [Proof 24].

(ix) Ω_W stable: dim orbit = 35 = dim Λ³ [Proof 25].

(x) GL(7)-orbit unique: all Ω_{a,b,c} with abc≠0 equivalent
    [Proof 27].

(xi) Stab(Ω_W) ≅ g₂(split) [Proof 28].  ∎

Uniqueness.
    C₂ is unique among ±1 recurrences (Proof 18).
    W is forced by signature constraints (Proof 20).
    V_W = R ⊕ W ⊕ W* is the canonical higher closure (Proof 21).
    Ω_W is the unique natural 3-form up to GL(7) (Proof 21).
    One volume choice on W. It does not affect the GL(7)-orbit.

[Dependencies: Proofs 1, 5, 17, 18, 20-28.
External: Killing 1887, Cartan 1914, Cartan criterion, Schur.]