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

Overview

Q5b addresses a central question of the Cosmochrony programme: which geometric conclusions follow from the BFS structure of a supplied Heisenberg carrier, and which require an additional continuum-limit hypothesis?

Current status (version 2.1). Q5b works conditionally on a supplied Heisenberg carrier: neither the finite Heisenberg group nor its identification with the admissible fibre is selected by the admissibility axioms. Q5a version 3.2 does not derive the spatial limit operator $L_\Pi = -A\partial_x^2$. The spatial input of Q5b is now the explicit, unestablished hypothesis [H-L], and every result consuming it (metric extraction, coefficient closure, Lorentzian signature) is conditional on [H-L]. The geometric convergence results (Carnot limit, $D_{\mathrm{hom}}=4$) are independent of [H-L] and stand. Establishing or replacing [H-L] is the open content of Q5.

The BFS (Breadth-First Search) shell stratification of $\mathrm{Heis}_3(\mathbb{R})$ unconditionally has a Carnot limit of homogeneous dimension four. Under [H-L], this stratification is read as one temporal direction — BFS depth — and a three-dimensional spatial sector. Identifying the measured admissible space $H_\mathrm{eff} \simeq \mathbb{C}^3$ with that sector is a hypothesis, which Q7 version 2.0 records as supplied by no source; the horizontal distribution of $\mathrm{Heis}_3$ has rank two and is not that sector.

Under [H-L], the stratification defines a four-velocity and the co-metric takes the Lorentzian signature $\mathrm{diag}(-2, 2, 2, 2) \propto \eta^{\mu\nu}$, whose three spatial entries come from Q8 and Q10 and are pending revision. The temporal co-metric coefficient $A_\tau$ is identified by Q11.

Central result. BFS stratification of $\mathrm{Heis}_3(\mathbb{R})$ has a Carnot limit with $D_{\mathrm{hom}}=4$. Conditional on [H-L], the associated co-metric is read as $\mathrm{diag}(-2,2,2,2) \propto \eta^{\mu\nu}$; those values are pending revision since Q7 version 2.0 withdrew support for their derivation.

Quick facts

Author

Jérôme Beau

Work type

Preprint

Focus

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

Key result

Carnot limit with $D_{\mathrm{hom}}=4$; under [H-L], and pending revision of its coefficient values, 4D Lorentzian co-metric $\mathrm{diag}(-2,2,2,2) \propto \eta^{\mu\nu}$

Status

Version 2.1: supplied Heisenberg carrier; conditional on the unestablished spatial limit hypothesis [H-L] (Q5a 3.2); Carnot geometry structural; Q9 2.0 leaves [H-lift] open

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 under [H-L]: conditional on [H-L], and on the identification Q7 version 2.0 records as supplied by no source, the three spatial directions are read as the measured admissible space $H_\mathrm{eff} \simeq \mathbb{C}^3$, inheriting the admissibility constraints from the O-series (O23–O29).
Four-velocity under [H-L]: conditional on [H-L], 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 under [H-L]: conditional on [H-L], the co-metric $\mathrm{diag}(-2,2,2,2)$ follows from the asymmetry between the BFS depth direction and the three admissible directions, with the temporal co-metric coefficient $A_\tau$ identified separately in Q11.
Q9 2.0 status: under [K] and [R], it gives only a free kinetic Mosco limit. The geometric lift [H-lift] does not follow and remains open.

BFS stratification and spacetime

The BFS shell stratification supplies an unconditional Carnot limit with homogeneous dimension four. Its interpretation as a Lorentzian spacetime mechanism is conditional on [H-L]. Under that hypothesis, no background metric is inserted independently: the metric structure is read from the combinatorial properties of the Heisenberg group under BFS exploration.

Under [H-L], the key insight is that the Heisenberg group $\mathrm{Heis}_3(\mathbb{R})$ has an intrinsic asymmetry: the central direction $\tilde Z$, generated by the commutator of the two horizontal generators, behaves differently from them under the admissibility filter. It is a separate direction from the BFS depth, which supplies the temporal co-ordinate. This asymmetry is precisely the origin of the Lorentzian signature.

Under [H-L], the horizontal distribution of $\mathrm{Heis}_3$, which is two-dimensional, together with the central direction makes up the spatial sector, while the BFS depth provides the temporal co-ordinate. The reading of the resulting co-metric as $\mathrm{diag}(-2,2,2,2)$, proportional to $\eta^{\mu\nu}$, takes its coefficient values from Q8, Q10 and Q11, which are pending revision: Q5b's own Theorem 5.2 puts $0$ in the central slot.

Conditional structural observation. Under [H-L], the 1+3 split of spacetime dimensions is read from the internal structure of $\mathrm{Heis}_3(\mathbb{R})$: the BFS depth as the temporal co-ordinate, and the two horizontal generators together with the central direction as the three spatial ones. Identifying the measured $H_\mathrm{eff} \simeq \mathbb{C}^3$ with that spatial sector is the hypothesis Q7 version 2.0 records as supplied by no source.

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: states the requirements an equivariant bridge to the three-dimensional spatial sector would meet; version 2.0 determines no coefficient and no ratio. The horizontal distribution of $\mathrm{Heis}_3$ is two-dimensional and is kept distinct from that sector.
Q8: under the frozen, unestablished hypothesis [H-F], Casimir rigidity gives $A_Z = 2$, assuming the Q7–Q9 bridge non-obstruction; any co-metric reading additionally requires [H-L] and the identification Q7 version 2.0 records as supplied by no source. Pending revision.
Q9: under [K] and [R], gives the free kinetic form; [H-lift] and [H-L] remain open.
Q10–Q11: assert the remaining coefficient values; pending revision in the cascade Q7 version 2.0 opens.
Q12: its fixed-base Yang–Mills theorem does not depend on Q5b; only its interpretation on the Cosmochrony emergent base uses the Q5b geometry under [H-L].

The O-series papers O23–O29 provide the admissibility analysis of the measured admissible 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.

Ancillary status. Q5b-O1 remains open: Q9 2.0 does not establish [H-lift]. Q5b-O2 ($A_z=2$, Q8) is also scoped to [H-F], while every co-metric reading and Q5b-O3 remain conditional on their stated inputs, including [H-L]. Under [H-L], the co-metric $g^{\mu\nu}=2\,\eta^{\mu\nu}$ was presented as uniquely determined with no free parameter; that absence rests on coefficient values whose derivation Q7 version 2.0 withdraws support for, those values depending on an identification no source supplies. [H-L] itself is not established (Q5a 3.2), and Q5 is open.

References

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