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Well yes, the hands where you would like to make a responsive double, but unfortunately RHO refused to make a raise are indeed a problem. However, don't you have exactly the same problem in standard?
In standard you can just cue, promising - usually - a semibalanced hand that MIGHT have -or not- 3 card support (or, alternatively, a huge game/slam going hand).
Often the 5-3 fit will play well even in NT.
The 4 card raise can go via 2NT (and various other kind of raises- FJS and Robson-Segal stuff...)
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However, I would not be as rigorous as you about 1N promising a stopper.
Well, the 1NT bid can be shaded, so I thought of biddding 1NT without a stopper but, couple of issues:
1) after all, rightsiding the contract seems to be a major issues for lots of people (last time I played vs you and Han you were playing a xfer advance system whose most useful feature seemed to be exactly that, e.g. putting on lead the most dangerous opp...)
2) 1NT is a limited bid; many play it as 8-11, but, no matter what rang you choose (according to the overcall style), there will be hands too good for a 1NT call, hands that would love to be able to stop in 2NT or 2M if pard is broke.
These hands need a 1RF bid
3) rare enough, but still to be considered, especially if opps open light: slamgoing hand in a minor
there will be hands without a fit in the major that might foresee slam in a selfsufficient minor.
Hence the need for a forcing bid without a fit for overcaller's major
In "standard" (is there any??) the forcing bid is the cuebid.
"Bridge is like dance: technique's important but what really matters is not to step on partner's feet !"