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

Abstract

Earlier versions of the companion paper Q5a claimed, conditionally, that the admissibility forms ${\mathcal{E}_q}$ on the Heisenberg carrier 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 $L^2(\mathbb{R})$. Version 3.0 of Q5a withdraws that derivation: the canonical filtration of the admissible fibre is exactly a growing toric Fourier window, the published admissibility form converges to the zero form on it, and no common scalar normalisation of the form produces a non-trivial toric differential operator. The spatial input of the present paper is therefore not established; we formalise it as an explicit hypothesis [H-L] (existence of a spatial second-order limit operator ${L_\Pi} = -A\partial_x^2$ on $L^2(\mathbb{R})$) and state every result that consumes it conditionally on [H-L]. We present 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, under [H-L] 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] has been proved in Q9, so the conditionality of Theorem thm:metric reduces to [H-L] itself. The metric coefficients are determined by companion papers: $A_H = 2$ by Q10, $A_z = 2$ by Q8, and $A_\tau = 2$ by Q11, giving, still under [H-L], the isotropic Lorentzian co-metric $g^{\mu\nu} = 2\,\eta^{\mu\nu}$ with no remaining free parameter. Third, the Born–Infeld admissibility constraint conditionally selects the Lorentzian signature $(-,+,+,+)$ for this metric (Theorem thm:lorentz), the import being conditional on the hyperbolicity hypothesis of ([H-hyp]). Each result is given with an explicit status: structural, or conditional on [H-L] (and, where indicated, on [H-hyp]). The only potential source of [H-L] currently identified in the corpus is the conditional dual-window regime of Q5a version 3.0, which requires an asymptotically non-vanishing critical coverage — a scenario the published data contradict; Q5 remains open.