Structural Obstruction in a Seven-Operator Lie Algebra: Derivation Rigidity and the CoreX Spark Spectral Constant

We study a seven-generator Lie algebra G7 = G4 (φ) ⊕ su (2) in which G4 (φ) is a four-generator algebra with structure constants depending smoothly on a real parameter φ, and su (2) is a standard su (2) algebra. We establish three structural theorems and a computational algorithm. (T1, Closure. ) G4 (φ) satisfies all four Jacobi identities for every admissible φ; in particular, the commutator F, N = -i tan φ C is derived from the Jacobi identity rather than postulated, providing the algebra's first non-trivial consistency constraint. (T2, Obstruction Tensor. ) The spectral obstruction tensor O (X, Y): = d/dφ X, Y is non-vanishing for all admissible φ, measuring how the algebra's bracket structure changes along the parameter flow. (T3, Rigidity. ) The derivation algebra satisfies dim Der (G4) = 4, dim Inn (G4) = 3, dim Out (G4) = 1; the overdetermined cross-coupling system (210 equations, 84 unknowns) has unique zero solution; and consequently G7 = G4 ⊕ su (2) is the unique admissible closed seven-generator algebra. Within this framework, the minimal irreducible coupling matrix M = 0 1; 1 1 has Perron–Frobenius spectral radius φ* = (1+√5) /2, which we term the CoreX Spark constant. This value governs simultaneously the spectral structure of M, the obstruction tensor's critical behaviour, and—as shown in the algorithmic section—the contraction rate and convergence of the self-healing operator pipeline. (Algorithm, Self-Healing Pipeline. ) A seven-operator iterative scheme drives numerical cross-coupling perturbations to zero in O (log⏥* (1/ε) ) steps via φ*-geometric contraction. The accumulated total contraction Σ₊≥₀ (1/φ*) ᵏ = φ*² = φ* + 1 recovers the defining equation of the CoreX Spark, a self-referential closure. All results are independent of any specific network or empirical dataset. MSC 2020: 17B30 (Solvable and nilpotent Lie algebras), 17B40 (Automorphisms, derivations), 17B56 (Cohomology of Lie algebras), 37C20 (Generic properties, structural stability).