Sub-Principal Symbol of the Effective Operator and Casimir Rigidity of the Central Direction

Paper Q7 reduced the open problem Q5b-O2 of to a single computable criterion: whether the sub-principal symbol of the effective operator L₄₅₅ in the central direction Z = X, Y of Heis₃ (R) carries coefficient Aₙ = Cₒₔ (₂) = 2, the eigenvalue of the su (2) -Casimir on the spin-1 module Sym^2 (V_). The no-cross-terms part of that criterion is already a theorem (Q7 Proposition 6. 1). The present paper settles the isotropy condition Aₙ = Aₙ by a structural argument that does not require a full computation of the hypoelliptic symbol. The key observation is that the sub-principal term in L₄₅₅ along Z has a unique algebraic source: the Heisenberg commutator X, Y = Z, which reflects the nilpotency class~2 of Heis₃ (R) and cannot be mimicked by any other generator. Under the equivariant bridge Sym^2 (V_) \;\; Wₒ₏ established in Q7 (unique up to positive scalar by Schur's lemma), the image of this commutator term under is constrained to be proportional to the unique su (2) -invariant quadratic form on Sym^2 (V_), namely the Casimir Cₒₔ (₂) = 2 Id. There is no free scalar: the normalisation is fixed by the same Casimir that sets Aₙ = 2 in Q7 Remark 6. 3. The result is therefore: given bridge non-obstruction (proved in Q9, Aₙ = Cₒₔ (₂) = 2 unconditionally under the Q5a hypotheses. The lifting hypothesis H-lift, previously an additional assumption of Q5b, is now a theorem of Q9 under the Q5a hypotheses alone, so no extra condition is needed. The effective spatial co-metric is gₒ₏ = Aₙ (kX^2 + kY^2) + 2\, kZ^2 with Aₙ 2 as q, proved in Q10 and U1 under the Q5a hypotheses and the O-series spectral universality U. Together with Q5b Theorem 6. 1, this yields a non-degenerate rank-4 Lorentzian metric on R_ Heis₃ (R) and closes Q5b-O2 under the Q5a hypotheses.