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

Abstract

Paper Q7 reduced the open problem Q5b-O2 of to a single computable criterion: whether the sub-principal symbol of the effective operator ${L_{\mathrm{eff}}}$ in the central direction ${\widetilde{Z}} = [{\widetilde{X}}, {\widetilde{Y}}]$ of ${\mathrm{Heis}_{3}(\mathbb{R})}$ carries coefficient ${A_{Z}} = {C_{{\mathfrak{su}(2)}}} = 2$, the eigenvalue of the ${\mathfrak{su}(2)}$-Casimir on the spin-$1$ module ${\operatorname{Sym}}^{2}({V_{\rho}})$. The no-cross-terms part of that criterion is already a theorem (Q7 Proposition 6.1). The present paper settles the isotropy condition ${A_{H}} = {A_{Z}}$ 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 ${L_{\mathrm{eff}}}$ along ${\widetilde{Z}}$ has a unique algebraic source: the Heisenberg commutator $[{\widetilde{X}}, {\widetilde{Y}}] = {\widetilde{Z}}$, which reflects the nilpotency class $2$ of ${\mathrm{Heis}_{3}(\mathbb{R})}$ and cannot be mimicked by any other generator. Under the equivariant bridge $\varphi \colon {\operatorname{Sym}}^{2}({V_{\rho}}) \xrightarrow{\;\sim\;} {W_{\mathrm{sp}}}$ established in Q7 (unique up to positive scalar by Schur's lemma), the image of this commutator term under $\varphi$ is constrained to be proportional to the unique ${\mathfrak{su}(2)}$-invariant quadratic form on ${\operatorname{Sym}}^{2}({V_{\rho}})$, namely the Casimir ${C_{{\mathfrak{su}(2)}}} = 2 \cdot \mathrm{Id}$. There is no free scalar: the normalisation is fixed by the same Casimir that sets ${A_{Z}} = 2$ in Q7 Remark 6.3. The result is therefore: given bridge non-obstruction (proved in Q9 ), ${A_{Z}} = {C_{{\mathfrak{su}(2)}}} = 2$ 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 ${g_{\mathrm{sp}}} = {A_{H}}(k_X^{2} + k_Y^{2}) + 2\,k_Z^{2}$ with ${A_{H}} \to 2$ as $q \to \infty$, 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 $\mathbb{R}_{\tau} \times {\mathrm{Heis}_{3}(\mathbb{R})}$ and closes Q5b-O2 under the Q5a hypotheses.