Chapter 4

Surtea support and boundary

The Surtea layer supplies the support language behind the visualisation: partition, interior, closure, boundary, class, and interaction. In RSG, this tells us what kind of topological support a surviving history is carried by.

Why This Layer Exists

The core RSG formalism can discuss histories, phase projection, loss, and survival. The Surtea bridge adds a question underneath that: what is the support that a history is carried on? Without a support language, closure, boundary instability, and objecthood remain visually intuitive but formally vague.

Universe As Partitioned Support

The simplified reading begins with a non-empty set M and a partition D. The partition elements are disjoint non-empty pieces whose union is the whole set. 

U = (M, D)
M != empty
D is a partition of M

The visualisation turns this into cells. The cells are not just decoration: they are a readable version of partitioned support.

Interior, Closure, Boundary

The most portable Surtea-style diagnostics are interior, closure, and boundary. These are the terms the live site uses when it colours topology cells. 

int_D(X) = interior of X in the D-topology
cl_D(X)  = closure of X in the D-topology
bd_D(X)  = cl_D(X) \ int_D(X)

Interior

Stable support that belongs cleanly to the object or local region.

Boundary

The active edge where membership, class, or interaction can change.

Closure

Support pulled into the represented object by the topology.

Class

A coarse objecthood type used when histories are grouped by support behaviour.

Topological Survival Loss

RSG can treat boundary change, class change, and interaction instability as survival-relevant. This does not mean topology becomes a force. It means the support grammar contributes to the loss functional when a model chooses to include it. 

S_{n+1} = S_n exp(-L_D Delta t)
L_D = lambda W
    + alpha Delta_bd
    + beta Delta_class
    + chi Delta_int

Visualisation Reading

In the canvas, Surtea cells have three practical meanings:

  • Green interior: stable support.
  • Red boundary: active transition or instability.
  • Amber closure: support drawn into the represented object or basin.

Topology heat makes the partition finer and more restless. Closure bias makes open paths more willing to curl, recur, thicken, or form record-like persistence.

Claim Boundary

The Surtea layer is a support grammar. It helps describe objecthood and boundary behaviour in the RSG site, but it does not by itself validate the survival-weighting formalism or any physical bridge. A model must still specify how topological diagnostics affect measured loss or output.