There are many high speed / Intercity trains in this country and can be used on tickets marked 'Any Permitted' ( or any other routeing that does not forbid their use).
There is nothing special about South Eastern's high speed trains whatsoever (they don't even have first class or a buffet!) and they are not timetabled to go any faster than other high speed trains on various other operators.
Unfortunately ATOC have decided to use their definition of "simple" by making the system more complicated.
In the case of some routes around the country, there are tickets that are cheaper if you avoid the high speed routes (e.g. York to Middlesbrough route NOT Darlington, Leeds to Carlisle route Appleby etc) but the tickets that are valid on the high speed routes are marked Any Permitted.
ATOC decided this was too simple for the South Eastern route and decided that the Any Permitted tickets for some flows (generally flows that do not go beyond London) would be abolished and replaced with "Plus High Speed" or "Not HS1"
however there are still Any Permitted tickets for longer journeys, and they are of course valid via any permitted route so there is no problem using them on HS1.
ATOC could have kept things genuinely simple (rather than their laughable ideas of simple) by continuing to use the term Any Permitted.
ATOC have confused people, and this isn't the first time and it won't be the last. They seem to delight in doing that. Even some South Eastern guards are confused (which is inexcusable!)
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I looked at a return from Eastbourne to Chafford Hundred and it shown that i could use HS1 for a couple of journeys (to connect with the limited c2c train at Stratford).
A ticket from Eastbourne to Chafford Hundred is valid for a journey via all the permitted routes between Eastbourne & London, then all the permitted routes between London & Chafford Hundred. If you use HS1 there are various options including taking NXEA from Stratford to Romford then using the train to Upminster, or you can go via West Ham (which is perhaps the most sensible route) or even via Gospel Oak (which is a bit bonkers, but valid).
If you are returning another day, you will find it cheaper to split the ticket at Upminster. (Chafford Hundred is £10.70 more than Upminster for an SVR!)