CCFU Proof 27 — Unique GL(7)-Orbit

2605255780899

CCFU Proof 27 — Unique GL(7)-Orbit

05/25/2026

Given.
Ω_W = τ∧ev + vol_W + vol_{W*} [Proof 21].

Step 1 — Orbit uniqueness (algebraic).
    Ω_W belongs to the 3-parameter family Ω_{a,b,c}.
    All abc ≠ 0 are GL(7)-equivalent [Proof 21, Step 5].
    Therefore the GL(7)-orbit of Ω_W is unique.  ∎

Step 2 — No compact alternative.
    sig(b_{Ω_W}) = (3,4) [Proof 24]. Split, not compact.
    The unique orbit is split.  ∎

Step 3 — Consistency with exhaustive enumeration.
    Verification (CCFU Script 3):
    1524 C₂-compatible patterns with dim Stab = 14.
    1523 split, 1 compact.
    All 1523 split: Killing (8,6), ad-commutant 1 → g₂(split).
    All consistent with the unique split orbit of Ω_W.  ∎

Step 4 — Direct GL(7) certificates.
    Verification (CCFU Script 4):
    4 different patterns tested against Ω_old (7 terms).
    Ω_old and Ω_W are in the same GL(7)-orbit [Proof 22].
    All 4: h ∈ GL(7) and c ≠ 0 found such that
        h*Ω_old ≈ c·Ω,
    with residual ~ 10⁻²⁴. Since multiplication by a nonzero
    scalar is absorbed by a scalar element of GL(7), this still
    gives a GL(7)-orbit certificate.  ∎

The orbit is unique. The proof is algebraic (Steps 1-2).
The computational verification (Steps 3-4) is additional
confirmation, not the proof.

[Dependencies: Proofs 21, 22, 24.]