Toward Four-Dimensional Lorentzian Geometry from BFS Shell Stratification

Current Zenodo release: version 2.3 (2026-10-27). Official title: Toward Four-Dimensional Lorentzian Geometry from BFS Shell Stratification.

Overview

Q5b asks which geometric conclusions follow from the breadth-first (BFS) shell structure of a supplied finite Heisenberg group, and which require further hypotheses. Its structural result stands on its own: the BFS shells converge to a Carnot geometry of homogeneous dimension four. Its metric results are conditional.

Q5a does not derive a spatial limit operator: the published admissibility forms converge to the zero form on the canonical filtration. The spatial input of Q5b is therefore the explicit, unestablished hypothesis [H-L]. The identification of that operator with a multiple of the kinetic sector of the nonnegative operator $-\Delta_H$ is the lifting hypothesis [H-lift], which is open.

Central result. On balls of radius $n \ll \sqrt q$, BFS stratification of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ has a Carnot limit with $D_{\mathrm{hom}}=4$. Under [H-L] and [H-lift], the effective co-metric is $\mathrm{diag}(-A_\tau, A_H, A_H, 0)$ in the left-invariant frame: rank three, with an empty central slot. No coefficient value is established.

Quick facts

Author

Jérôme Beau

Work type

Preprint

Focus

BFS shell stratification, Carnot geometry, sub-Laplacian, principal symbol, Lorentzian signature

Key result

Carnot limit with $D_{\mathrm{hom}}=4$; under [H-L] and [H-lift], a rank-three co-metric whose non-degenerate block has signature $(-,+,+)$ under [H-hyp] applied to the reduced sector

Status

Structural for the geometry; conditional on [H-L], [H-lift] and [H-hyp] for the co-metric and its signature; Q5b-O1, Q5b-O2 and Q5b-O3 open

Depends on

Q5a, Q9

Feeds into

Q7, Q8, Q10, Q11, Q12

Core contributions

Carnot convergence (structural): on balls of radius $n \ll \sqrt q$, where they coincide with those of the integer Heisenberg group, the BFS shells of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ converge in the Gromov–Hausdorff sense to the sphere foliation of a sub-Finsler Carnot–Carathéodory metric on $\mathrm{Heis}_3(\mathbb{R})$ (Pansu, Bass–Guivarc'h). The homogeneous dimension four gives the limit the volume growth and heat-kernel decay of a four-dimensional space.
Metric extraction (conditional on [H-L] and [H-lift]): the effective operator $-A_\tau\partial_\tau^2 + A_H\Delta_H$ on $\mathbb{R}_\tau \times \mathrm{Heis}_3(\mathbb{R})$ has the co-metric $\mathrm{diag}(-A_\tau, A_H, A_H, 0)$ in the left-invariant frame. Its rank is three.
Signature (conditional also on [H-hyp]): the hyperbolicity hypothesis, applied to the sector independent of the central variable, gives the signature $(-,+,+)$ on the non-degenerate block with $\tau$ time-like; an extension $\mathrm{diag}(-A_\tau, A_H, A_H, A_z)$ with $A_z > 0$ would be Lorentzian.
Consistent conventions: the group law, the left-invariant fields $\tilde X = \partial_x + \tfrac{y}{2}\partial_z$, $\tilde Y = \partial_y - \tfrac{x}{2}\partial_z$, the commutator $[\tilde X, \tilde Y] = -\tilde Z$, the Schrödinger representation and the symbol convention are stated so that they agree with each other.

BFS stratification and spacetime

The emergent space is four-dimensional in two distinct senses that Q5b keeps apart: as the manifold $\mathbb{R}_\tau \times \mathrm{Heis}_3(\mathbb{R})$, with $\tau$ the projective ordering index, and through the homogeneous dimension of its spatial factor, which comes from the weights of the Carnot stratification.

What is not yet emergent is the metric in the central direction $\tilde Z$. The sub-Laplacian has no first-order part and a rank-two principal symbol, so no correction of lower order fills the central slot (Q8). A full-rank extension needs a new operator with a term $\tilde Z^2$ of homogeneous degree four, which carries a length scale.

Interpretation. The non-commutativity of the horizontal directions makes the central direction reachable but does not give it a length; the shape of the effective spacetime is structurally supported, while its full metric and its scale remain to be constructed.

Relation to the Cosmochrony programme

Q9: gives sufficient conditions [K] and [R] for a kinetic Mosco limit; it does not discharge [H-lift].
Q7: states the requirements an equivariant bridge to a supplied positive rank-three target would meet; Q5b's rank-two symbol cannot be that target.
Q8: the commutator supplies no central coefficient; isotropy between horizontal and central directions selects a length scale; invariance leaves the common value free.
Q10: bounds the ratio $A_H/A_Z$ under named inputs and derives no value.
Q11: shows that spatial $\mathrm{SU}(2)$ invariance leaves $A_\tau$ independent of the spatial coefficients; no value of $A_\tau$ is established.
Q12: its fixed-base Yang–Mills theorem does not depend on Q5b; only its interpretation on the emergent base uses the Q5b geometry under [H-L], [H-lift] and a supplied rank-four extension.

Open directions

Q5b-O1: derive [H-lift], for instance through the Plancherel decomposition.
Q5b-O2: construct a full-rank extension, which needs a new operator and a length scale.
Q5b-O3: identify $A_\tau$ and $A_H$ from admissibility data; no value is established.
Q5: establish [H-L], or replace it.

References

Jérôme Beau. Toward Four-Dimensional Lorentzian Geometry from BFS Shell Stratification, 2026. doi:10.5281/zenodo.19686700