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Exact SymPy verification of the Cartan bridge from Ω_W. The Hitchin
bilinear is derived in-file from Ω_W (contraction = 24·b, signature
(3,4)). The infinitesimal stabilizer has dimension 14, preserves b
(AᵀB+BA=0 certified for all 14 generators); true Killing form computed
via the adjoint representation — inertia (8,6), with B_Killing = 4·tr₇
verified exactly; the Cartan involution θ(X)=−Xᵀ is certified positive
definite, so k = su(2)⊕su(2) is a maximal compact subalgebra. One su(2)
factor fixes W_diag = span{e1+e4, e2+e5, e3+e6} pointwise; the other
acts irreducibly on it. Transitivity on the projective null quadric is
certified in-file: one constant nonzero 6×6 action minor per affine
chart (six polynomial charts covering Q, explicit overlap witnesses) —
no external homogeneity theorem used. Hence D = ker(ι_xΩ_W)/⟨x⟩ is
globally a (2,3,5) Cartan distribution. In the canonical
compact-orthogonal gauge the representation-spectral amplitude ratio is
ρ = s_body/s_spatial = 3, the unique maximum along gauge fibers for
every nonzero horizontal direction (9·Q_spatial − Q_body = 2c² on the
full three-parameter gauge family). 72 exact checks, integer/rational/
algebraic arithmetic only, with an explicit COMPUTED / DEDUCED /
NOT CLAIMED separation. v7 incorporates an independent external
verifier's refactor, including the transitivity certificate.
Dependencies: Proof 22 (Ω_W canonical representative), Proof 24.
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Title CCFU Script 5 — verify_cartan_bridge.py
Exact SymPy verification of the Cartan bridge from Ω_W. The Hitchin
bilinear is derived in-file from Ω_W (contraction = 24·b, signature
(3,4)). The infinitesimal stabilizer has dimension 14, preserves b
(AᵀB+BA=0 certified for all 14 generators); true Killing form computed
via the adjoint representation — inertia (8,6), with B_Killing = 4·tr₇
verified exactly; the Cartan involution θ(X)=−Xᵀ is certified positive
definite, so k = su(2)⊕su(2) is a maximal compact subalgebra. One su(2)
factor fixes W_diag = span{e1+e4, e2+e5, e3+e6} pointwise; the other
acts irreducibly on it. Transitivity on the projective null quadric is
certified in-file: one constant nonzero 6×6 action minor per affine
chart (six polynomial charts covering Q, explicit overlap witnesses) —
no external homogeneity theorem used. Hence D = ker(ι_xΩ_W)/⟨x⟩ is
globally a (2,3,5) Cartan distribution. In the canonical
compact-orthogonal gauge the representation-spectral amplitude ratio is
ρ = s_body/s_spatial = 3, the unique maximum along gauge fibers for
every nonzero horizontal direction (9·Q_spatial − Q_body = 2c² on the
full three-parameter gauge family). 72 exact checks, integer/rational/
algebraic arithmetic only, with an explicit COMPUTED / DEDUCED /
NOT CLAIMED separation. v7 incorporates an independent external
verifier's refactor, including the transitivity certificate.
Dependencies: Proof 22 (Ω_W canonical representative), Proof 24.
Work type Script
Tags ccfu, captain cookie face, script
-------------------------
Registry info in Safe Creative
Identifier 2607166402839
Entry date Jul 16, 2026, 12:48 AM UTC
License All rights reserved
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Author 100.00 %. Holder Captain Cookie Face Universe. Date Jul 16, 2026.
Information available at https://www.safecreative.org/work/2607166402839-ccfu-script-5-verify_cartan_bridge-py
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