MATH.1:5.1 - A continuation that needs a permission

Two routes p and q start at S and reach location V; their costs are 1 and 4. A final step r costs 2. Under the first rule, r is available after either route, so p;r and q;r are paths and their costs are 3 and 6.

Now change the rule: only q grants the permission needed for r. Keeping a single endpoint V would still make p;r appear composable.

Construct two intermediate objects, V0=(V,0) and V1=(V,1). Set:

The path q;r exists and costs 6. The expression p;r fails the endpoint test. Selecting the cheaper prefix first would therefore lose the available completion. The useful result is the permitted path and its cost; the refined endpoint explains why it is permitted.

If a further generator a:V0→V1 grants permission at cost 1, a new path p;a;r becomes available at cost 4. The former failure has opened a construction question: what additional arrow would connect the available prefix to the required continuation?