Q5b — BFS Shell Stratification and the Emergence of Four-Dimensional Lorentzian Geometry

This repository contains the source of the Q5b Cosmochrony paper BFS Shell Stratification and the Emergence of Four-Dimensional Lorentzian Geometry.

Status revision (version 2.0). Version 3.0 of the companion paper Q5a withdraws its earlier derivation of a spatial limit operator $L_\Pi = -A\partial_x^2$ on $L^2(\mathbb{R})$: the canonical filtration is exactly a growing toric Fourier window, the published admissibility form converges to the zero form on it, and no common scalar normalisation produces a non-trivial toric differential operator. The spatial input of this paper is therefore formalised as an explicit, unestablished hypothesis [H-L] (existence of the spatial limit operator), and every result consuming $L_\Pi$ is stated conditionally on it. The geometric convergence results (Carnot limit, $D_{\mathrm{hom}} = 4$) are independent of [H-L] and stand.

Core Results

1. Four-dimensional limit geometry: the BFS shell stratification of $\mathrm{Heis}_3(\mathbb{Z}/q)$ converges, in the pre-saturation regime, to the Carnot–Carathéodory sphere foliation of $\mathrm{Heis}3(\mathbb{R})$; the homogeneous dimension $D{\mathrm{hom}} = 4$ (Bass–Guivarc'h) gives the limiting geometry the spectral and volume-growth properties of a four-dimensional space.
2. Metric extraction (conditional on [H-L]): under [H-L], $L_\Pi$ is the image, under the Schrödinger representation, of the kinetic sector of the sub-Riemannian Laplacian $\Delta_H$; an effective co-metric is read off the principal symbol of the effective operator.

The lifting hypothesis [H-lift] has been proved in Q9, so the conditionality of the metric result reduces to [H-L] alone. The coefficients are determined by companion papers: $A_H = 2$ (Q10), $A_z = 2$ (Q8), $A_\tau = 2$ (Q11) — all readings conditional on [H-L]. Establishing [H-L], or replacing it, is the open content of Q5.

Keywords

BFS stratification, Carnot–Carathéodory geometry, sub-Riemannian Laplacian, homogeneous dimension, Mosco convergence, Lorentzian signature, emergent spacetime.

Repository Contents

q5b/
├── tex/         # LaTeX sources (main + cosmochrony-bibliography.bib)
├── out/         # Compiled paper PDF (q5b.pdf)
├── zenodo.json  # Zenodo deposition metadata
└── README.md

Links

• 🔗 DOI: 10.5281/zenodo.19686700
• 🌐 Website: https://cosmochrony.org/science/emergent-geometry/q5/b/

Citation

J. Beau, BFS Shell Stratification and the Emergence of Four-Dimensional Lorentzian Geometry, Zenodo, 2026. DOI: 10.5281/zenodo.19686700.

Acknowledgements

Portions of the editorial refinement benefited from iterative interactions with large language models, used as analytical assistants. All claims and final formulations remain the sole responsibility of the author.