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

Overview

Q5b addresses a central question of the Cosmochrony programme: how does a four-dimensional Lorentzian spacetime emerge from the relational substrate? The answer is found in the combinatorial structure of the Heisenberg group itself.

The BFS (Breadth-First Search) shell stratification of $\mathrm{Heis}_3(\mathbb{R})$ induces a natural decomposition of the tangent structure into one privileged direction — the BFS depth, identified as the temporal direction — and three horizontal directions forming the admissible sector $H_\mathrm{eff} \simeq \mathbb{C}^3$.

From this stratification, a four-velocity emerges naturally, and the co-metric takes the Lorentzian signature $\mathrm{diag}(-2, 2, 2, 2) \propto \eta^{\mu\nu}$. The temporal co-metric coefficient $A_\tau$ is identified by Q11.

Central result. BFS stratification of $\mathrm{Heis}_3(\mathbb{R})$ yields a 4D Lorentzian geometry with metric $\mathrm{diag}(-2,2,2,2) \propto \eta^{\mu\nu}$.

Quick facts

Author

Jérôme Beau

Work type

Preprint

Focus

BFS shell stratification, Lorentzian geometry, spacetime emergence, Heisenberg group

Key result

$\mathrm{Heis}_3(\mathbb{R})$ BFS stratification $\to$ 4D Lorentzian metric $\mathrm{diag}(-2,2,2,2) \propto \eta^{\mu\nu}$

Status

Fully proved: all open problems closed (Q9 closes O1, Q10+Q11 close O3; metric $g^{\mu\nu}=2\,\eta^{\mu\nu}$ unconditional)

Depends on

Q5a, O23–O29

Feeds into

Q7, Q8, Q9, Q10, Q11, Q12

Core contributions

Temporal direction from BFS depth: the BFS depth function on $\mathrm{Heis}_3(\mathbb{R})$ defines a canonical temporal foliation, with level sets serving as spatial hypersurfaces. This provides a purely combinatorial origin for the arrow of time.
Spatial sector from horizontal admissibility: the three spatial directions are identified with the horizontal admissible sector $H_\mathrm{eff} \simeq \mathbb{C}^3$, inheriting the admissibility constraints from the O-series (O23–O29).
Four-velocity from stratification: the BFS stratification generates a natural four-velocity vector field, with the temporal component fixed by the depth gradient and the spatial components by the horizontal projection.
Lorentzian signature: the co-metric $\mathrm{diag}(-2,2,2,2)$ emerges from the asymmetry between the BFS depth direction and the three horizontal admissible directions, with the temporal co-metric coefficient $A_\tau$ identified separately in Q11.
[H-lift] closed by Q9: the lifting hypothesis [H-lift] (Q5b-O1), formerly a conditional assumption, is proved unconditionally by Q9 via modulation generator suppression, removing the main conditional status of Q5b.

BFS stratification and spacetime

The BFS shell stratification provides a new mechanism for the emergence of Lorentzian geometry that is genuinely different from existing approaches. Rather than postulating a spacetime manifold and imposing a metric, the geometric structure arises from the combinatorial properties of the Heisenberg group under BFS exploration.

The key insight is that the Heisenberg group $\mathrm{Heis}_3(\mathbb{R})$ has an intrinsic asymmetry: the central direction (corresponding to BFS depth) behaves differently from the horizontal directions under the admissibility filter. This asymmetry is precisely the origin of the Lorentzian signature.

The three horizontal directions correspond to the spatial co-ordinates of Minkowski space, while the BFS depth provides the temporal co-ordinate. The resulting co-metric $\mathrm{diag}(-2,2,2,2)$ is proportional to the Minkowski metric $\eta^{\mu\nu}$.

Structural observation. The 1+3 split of spacetime dimensions is not imposed but derived from the internal structure of $\mathrm{Heis}_3(\mathbb{R})$: one central direction (BFS depth) plus three horizontal directions ($H_\mathrm{eff} \simeq \mathbb{C}^3$).

Relation to the Cosmochrony programme

Q5b occupies a central position in the dependency graph of the Q-series. It is the geometric foundation upon which Q7–Q12 build:

Q7: studies the spatial bridge problem — how the three horizontal co-efficients relate to each other.
Q8: establishes $A_Z = 2$ unconditionally via Casimir rigidity, using Q5b as input.
Q9: proves [H-lift], closing the main conditional of Q5b.
Q10–Q11: complete the metric co-efficient determination.
Q12: uses the Lorentzian geometry of Q5b as the base manifold for Yang–Mills dynamics.

The O-series papers O23–O29 provide the admissibility analysis of the horizontal sector that Q5b relies on. The results are used directly in identifying $H_\mathrm{eff} \simeq \mathbb{C}^3$.

Open directions

Curved geometry: Q5b establishes the flat Lorentzian case; the extension to curved effective geometry (relevant for the gravity paper) requires further analysis.
Matter fields on the BFS foliation: how quantum fields are defined on the emergent spacetime in terms of the BFS stratification remains open.

All ancillary questions closed. Q5b-O1 ([H-lift], Q9), Q5b-O2 ($A_z=2$, Q8), and Q5b-O3 ($A_H=2$ via Q10, $A_\tau=2$ via Q11) are fully resolved. The co-metric $g^{\mu\nu}=2\,\eta^{\mu\nu}$ is uniquely determined with no free parameter.

References

Jérôme Beau. BFS Shell Stratification and the Emergence of Four-Dimensional Lorentzian Geometry, 2026. doi:10.5281/zenodo.20277381