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 Leff in the central direction Z = [X, Y] of Heis₃(R) carries coefficient AZ = Cₛᵤ₍₂₎ = 2, the eigenvalue of the su(2)-Casimir on the spin-$1$ module Sym²(Vρ). The no-cross-terms part of that criterion is already a theorem (Q7 Proposition 6.1). The present paper settles the isotropy condition AZ = AZ 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 Leff 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 φ Sym²(Vρ) \;~\; Wₛₚ established in Q7 (unique up to positive scalar by Schur's lemma), the image of this commutator term under φ is constrained to be proportional to the unique su(2)-invariant quadratic form on Sym²(Vρ), namely the Casimir Cₛᵤ₍₂₎ = 2 · Id. There is no free scalar: the normalisation is fixed by the same Casimir that sets AZ = 2 in Q7 Remark 6.3. The result is therefore: given bridge non-obstruction (proved in Q9, AZ = Cₛᵤ₍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 gₛₚ = AZ(kX² + kY²) + 2\,kZ² with AZ → 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 Rτ × Heis₃(R) and closes Q5b-O2 under the Q5a hypotheses.