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

The companion paper Q5a establishes, conditionally, that the admissibility forms Eq on the Weil representation of Heis₃ (Z/qZ) converge in the Mosco sense to a second-order operator L_ = -Aₓ² on L² (R), which is described as the ``one-dimensional operator-theoretic shadow'' of the homogeneous four-dimensional structure of Heis₃ (R). The present paper completes the geometric reading deferred by Q5a. We establish three results. First, we show that the BFS shell stratification of Gq = Heis₃ (Z/qZ) converges, in the pre-saturation regime, to the Carnot–Car\-a\-th\'eo\-do\-ry sphere foliation of Heis₃ (R) ; the homogeneous dimension D₇₎₌ = 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_ as the image, under the Schr\"odinger representation, of the kinetic sector of the sub-Riemannian Laplacian H on Heis₃ (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: AH = 2 by Q10, Aᵦ = 2 by Q8, and A_ = 2 by Q11, giving the isotropic Lorentzian co-metric g^ = 2\, ^ 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.