I was looking on BRFares, for stations on the BML between Hassocks and (at least) Redhill to Portsmouth.
There are fares routed both "via Barnham" and "Not via London", but the only possible valid route as per the Routeing Guide is via Barnham no matter what you choose. (Edit: as per post #2 and #3, tickets from Redhill and Earlswood are valid via Guildford due to the shortest route rule.)
Not via London is consistently priced higher (and also has equal off-peak and anytime pricing?) than the equivalent Via Barnham fares.
My assumption is that the "Not via London" fare is intended for use via North Downs Line to Guildford, then SWR to Portsmouth as well as via Clapham Junction.
Looking at the Yellow Pages, it seems that for all the stations in this section, the only maps are CW (+LB where applicable) and/or LP, both require travelling via Barnham and then either (CW) across to Hove/Brighton and up the BML, or (LP) via Horsham and Three Bridges.
These extra fares add no additional validity, and simply serve to confuse passengers or overcharge them.
Here's an example of the fares from Wivelsfield to Portsmouth & Southsea:

And the Yellow Pages extract:

And the map in question (cropped):

There are fares routed both "via Barnham" and "Not via London", but the only possible valid route as per the Routeing Guide is via Barnham no matter what you choose. (Edit: as per post #2 and #3, tickets from Redhill and Earlswood are valid via Guildford due to the shortest route rule.)
Not via London is consistently priced higher (and also has equal off-peak and anytime pricing?) than the equivalent Via Barnham fares.
My assumption is that the "Not via London" fare is intended for use via North Downs Line to Guildford, then SWR to Portsmouth as well as via Clapham Junction.
Looking at the Yellow Pages, it seems that for all the stations in this section, the only maps are CW (+LB where applicable) and/or LP, both require travelling via Barnham and then either (CW) across to Hove/Brighton and up the BML, or (LP) via Horsham and Three Bridges.
These extra fares add no additional validity, and simply serve to confuse passengers or overcharge them.
Here's an example of the fares from Wivelsfield to Portsmouth & Southsea:

And the Yellow Pages extract:

And the map in question (cropped):

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