BFS Shell Stratification and the Emergence of Four-Dimensional Lorentzian Geometry

The companion paper Q5a establishes, conditionally, that the admissibility forms $\mathcal{E}_q$ on the Weil representation of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ converge in the Mosco sense to a second-order operator $L_\Pi = -A\partial_x^2$ on $L^2(\mathbb{R})$, which is described as the ``one-dimensional operator-theoretic shadow'' of the homogeneous four-dimensional structure of $\mathrm{Heis}_3(\mathbb{R})$. The present paper completes the geometric reading deferred by Q5a. We establish three results. First, we show that the BFS shell stratification of $G_q = \mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ converges, in the pre-saturation regime, to the Carnot–Car\-a\-th\'eo\-do\-ry sphere foliation of $\mathrm{Heis}_3(\mathbb{R})$; the homogeneous dimension $D_{\mathrm{hom}} = 4$ (Bass–Guivarc'h) endows the limiting geometry with the spectral and volume-growth properties of a four-dimensional space (Theorem thm:carnot). Second, we identify $L_\Pi$ as the image, under the Schr"odinger representation, of the kinetic sector of the sub-Riemannian Laplacian $\Delta_H$ on $\mathrm{Heis}_3(\mathbb{R})$, and extract an effective co-metric tensor from the principal symbol of the full effective operator (Theorem thm:metric). The lifting hypothesis [H-lift] on which this result was originally conditional has since been proved in Q9 , making Theorem thm:metric unconditional. The metric coefficients have since been fully determined: $A_H = 2$ by Q10 , $A_z = 2$ by Q8 , and $A_\tau = 2$ by Q11 , giving the isotropic Lorentzian co-metric $g^{\mu\nu} = 2\,\eta^{\mu\nu}$ with no remaining free parameter. Third, the Born–Infeld admissibility constraint forces this metric to carry Lorentzian signature $(-,+,+,+)$ (Theorem thm:lorentz, following ). Each result is given with an explicit status: structural, unconditional (Q9), or imported from .