What: Three curvature indicators: volume-growth distortion, branching asymmetry, geodesic concentration. Flat model: all vanish. Curved bottleneck model: all nonzero. Geodesics concentrate through bottlenecks — the free-fall analogue.
Flatness and Curvature
Flatness is defined as local uniformity of neighborhood profiles. In a flat event geometry, every event sees the same local structure: the same number of neighbours at each distance, the same branching pattern, the same accessibility profile.
Curvature is deviation from this uniformity. When neighborhood profiles vary across the event set — when some events have more accessible neighbours than others, when branching is asymmetric, when paths concentrate through particular regions — the geometry is curved.
Three Curvature Indicators
Volume-Growth Distortion
Ball-size deviation from reference. Count the number of events within distance r of each event. In a flat geometry, this count is uniform. Deviation from uniformity measures volume curvature.
Branching Asymmetry
Degree deviation in the transition graph. In a flat geometry, every event has the same number of immediate successors and predecessors. Asymmetry in branching signals curvature.
Geodesic Concentration
Betweenness centrality deviation. In a flat geometry, geodesics are uniformly distributed. When geodesics concentrate through particular events, those events act as geometric bottlenecks.
The Free-Fall Analogue
In general relativity, free-falling objects follow geodesics — paths determined entirely by the geometry of spacetime. No force is imposed; the trajectory is a consequence of curvature.
Geodesics concentrate through high-accessibility regions. Paths follow the geometry without imposed force — the structural analogue of gravitational free fall.
Where the event geometry is non-uniform, shortest paths preferentially route through regions of high accessibility. This concentration is not imposed by an external force but arises from the geometry itself. The analogy with gravitational free fall is structural: paths are shaped by the geometry they traverse.
Flat vs Curved
Two toy models illustrate the distinction:
Symmetric model (flat)
A regular event graph where every event has identical neighborhood profiles. All three curvature indicators vanish: uniform volume growth, symmetric branching, uniform geodesic distribution. The geometry is flat.
Bottleneck model (curved)
An event graph with a constriction — a region where fewer events connect two larger regions. All three curvature indicators are nonzero: volume growth is distorted near the bottleneck, branching is asymmetric, and geodesics concentrate through the constriction. The geometry is curved.
Comparison with Known Curvature
| VFD Indicator | Known Analogue |
|---|---|
| Volume-growth distortion | Bishop–Gromov volume comparison |
| Branching asymmetry | Ollivier–Ricci curvature |
| Geodesic concentration | Raychaudhuri focusing |
Stated Limitations
- No Einstein equations — curvature is kinematic, not coupled to a source
- No curvature tensor — the indicators are scalar measures, not tensorial
- No continuous limit — all constructions are on finite discrete event sets
- No dynamics — the geometry is static; evolution is introduced in GR-V
Curvature without spacetime. The geometry of accessibility determines where paths concentrate.
Full sequence: GR-I → GR-II → GR-III → GR-IV → GR-V