there is only one valid route, as stated above, via Hexham
The ticket for Carlisle to Newcastle is routed 'Any Permitted':
Routeing Guide Instructions said:
HOW TO DETERMINE PERMITTED ROUTES
Where the timetable offers journey opportunities over a longer route which is not covered by a through train service, the alternative options are included in the Routeing Guide. This may offer the customer a choice of routes for the same overall journey. If a choice of routes is available and the fare for the journey is described as "Any Permitted" route or is unrouted, you are free to choose any of the routes listed in the Guide. Where the fare specifies a particular route, you may choose only those routes listed in the Guide which pass through the station shown in the route description. The Routeing Guide may have to be used to find out how to reach the station shown in the route description. Where there are other permitted routes, these may also be used for the same journey provided the same or a lower fare applies.
From the Routeing Guide, the available mapped routes are:
(On maps, from example 2 step 7 in
http://www.atoc.org/rsp/_downloads/Routeing_Guide/pdf_docs/sectionf.pdf if travel between the stations on each map is shown as having 2 branches then either is valid provided it does not double back)
- via Hexham (map GN)
- via Leeds and York (maps MM and TP)
- via Leeds, Selby, and York (maps MM, WY, LY, and NE)
- via Warrington, (Runcorn,) Crewe, Stoke-on-Trent, Stockport, Manchester, Leeds, (Selby,) and York (maps NC, MH, and TP)
- via Preston, (Wigan,) Bolton, Manchester, Huddersfield, Leeds, (Selby), and York (maps NW, GM, WY, LY, NE - reverse of that shown in the RG, which is permitted provided they are used in the exact reverse order per page F8 in the above link).
- via Preston, (Wigan,) Bolton, Manchester, Halifax, (Hellifield), Leeds, (Selby), and York (maps NW, GM, WY, LY, NE)
As yorkie has said before, an EXTREMELY flexible ticket.
Per example 3 (step 4) in section F, the fare check rule does NOT apply to this situation
Routeing Guide Instructions said:
Step 4: It is unnecessary to discover if the routeing points are appropriate because both stations are routeing points and there is no choice.
The only query you can have about fares in such situations is when there are two ticket types available (e.g. 'Any Permitted' and 'via xxx'). If the ticket via xxx costs more than the 'Any Permitted' then you cannot travel through xxx on your journey.
As only the 'Any Permitted' fare exists here then all routes are valid on this ticket.
All info taken from the online Routeing Guide which, as it tells me when I open it, was last updated November 2007.