research
Each point can have a relational centre
A local neighbourhood can define a pull without a supplied global centre—but that alone does not preserve a structure.
Checked consensus and geometric invariance examples. Physical gravity remains a specific bridge rather than a consequence of graph adjacency.
Define the local relation
Assign nonnegative weights from point i to its neighbours. Their weighted average is its local relational centre. The declared law ṗᵢ=Σⱼwᵢⱼ(pⱼ−pᵢ) moves the point toward that centre.
Every point receives its own neighbourhood contribution. A common centre can emerge from those interactions rather than being inserted as an external leader.
Attraction alone can erase the geometry
On a fixed connected symmetric graph, that first-order attraction law brings the positions to one centroid. The separations collapse. To preserve a patterned structure, the model needs something further, such as rest distances, repulsion, circulating phase or another specified constraint.
Similarly, pure gradient descent cannot sustain a nonstationary closed orbit: its potential decreases whenever the state moves. A persistent cycle requires a law that allows circulation or continued input.
For ṗ=−∇V, dV/dt=−||∇V||²
Movement and strain are different
A rigid rotation and translation preserve internal distances. Moving a body through fixed surroundings can nevertheless change its external constraints and forces. Breakdown belongs to those actual interactions and material limits, not to motion alone.
A geometric centre, a phase attractor and physical gravity have different mathematical jobs. A gravitational application must specify its source quantity, field law, units and resulting acceleration.
Questions to carry forward
Can relational degree simply be renamed mass?
That would be an additional physical model mapping. It needs a defined quantity, law and evidence; the graph count alone does not provide them.
Where this belongs in the journey
- Explore geometry →geometry
- Research and questions →research
- Continue with shared circles →The teaching journey