BP+EM is invalid as it requires travelling back from Coventry through Bedworth to Nuneaton.
I don't agree that the Routeing Guide restricts travel in a way that makes this invalid.
As
collybs says, Coventry passes the fares check for a Bedworth - Peterborough journey, so it is an appropriate Routeing Point for that journey.
Step 5 of the Routeing Guide Instructions states that
If one station is a routeing point and the other one is a related station, the permitted route is the shortest route to the routeing point plus the permitted routes between routeing points. That is the position for this journey where Peterborough is a Routeing Point and Coventry and Nuneaton are both appropriate Routeing points for Bedworth for the journey.
There are limitations on
doubling back on the part of the journey between the origin and its associated Routeing Point - the shortest route must be used. There are also limitations on
doubling back on the part of the journey between the Routeing Points - Step 7 states that
the route is via any route on that map from the first routeing point to the final routeing point without doubling back.
But I can't find any limitation on doubling back where that is created by putting the two parts together to create, in this instance, a
Bedworth - Coventry - Bedworth - Nuneaton - Leicester - Peterborough route. In the absence of a limitation, this appears to be a Permitted Route.
This has limited value for a Bedworth - Peterborough journey but is useful, and is used, for an Edinburgh Park - Glasgow Queen Street journey. The most convenient journey is often
- Edinburgh Park - Haymarket (a member of Edinburgh Group Routeing Point), followed by
- Haymarket - Glasgow Queen Street (which doubles back through Edinburgh Park).
This route is recommended by both the Atos Origin and thetrainline families of booking engines.