Toward Four-Dimensional Lorentzian Geometry from BFS Shell Stratification

The companion paper Q5a shows that, with the pipeline's initial vector, the canonical filtration of the admissible fibre on the Heisenberg carrier of ${\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})}$ is a growing toric Fourier window, that the published admissibility forms converge to the zero form on it, and that no common scalar normalisation 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, on balls of radius $n \ll \sqrt q$, where the BFS balls of ${G_q} = {\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})}$ coincide with those of the integer Heisenberg group, the rescaled BFS shells converge to the sphere foliation of a sub-Finsler Carnot–Car\-a\-théo\-do\-ry metric on ${\mathrm{Heis}_3(\mathbb{R})}$; the homogeneous dimension ${D_{\mathrm{hom}}} = 4$ (Bass–Guivarc'h) gives the limiting geometry the volume growth and spectral dimension of a four-dimensional space. Second, under [H-L] and the lifting hypothesis [H-lift], which identifies ${L_\Pi}$ with $A$ times the kinetic sector of the nonnegative operator $-{\Delta_H}$ under the Schrödinger representation, the principal symbol of the effective operator on $\mathbb{R}_\tau \times {\mathrm{Heis}_3(\mathbb{R})}$ gives a leading-order co-metric $\mathrm{diag}(-A_\tau, A_H, A_H, 0)$ in the left-invariant frame. Hypothesis [H-lift] is open: Q9 gives sufficient conditions for a kinetic Mosco limit and does not discharge it. The central slot is empty and cannot be filled by lower-order terms, since the sub-Laplacian has no first-order part and a rank-two principal symbol; a full-rank extension requires a new operator (open problem Q5b-O2). No value of $A_\tau$, $A_H$ or of a central coefficient $A_z$ is established: Q10 derives no value of $A_H$, Q8 shows that invariance leaves the common value of an isotropic form free, and Q11 shows that spatial $\mathrm{SU}(2)$ invariance leaves $A_\tau$ independent of the spatial coefficients. Third, conditionally on the same two hypotheses and on the hyperbolicity hypothesis [H-hyp] of the companion signature analysis, applied to the sector of the effective operator that does not depend on the central variable, the signature of the non-degenerate block is $(-,+,+)$ with $\tau$ time-like; an extension $\mathrm{diag}(-A_\tau, A_H, A_H, A_z)$ with $A_z > 0$ in the left-invariant frame would be Lorentzian. Each result carries an explicit status: structural, or conditional on named hypotheses. Interpretive outlook: the four-dimensionality of the emergent space is carried by the homogeneous dimension of the Heisenberg carrier, while its metric completion, the length of the central direction and the values of the coefficients remain open; Q5 is open.