New Post: Search old tweets

Twitter is not using the Public REST API.
Twitter have access to way more endpoint than the one available to public developers.

For example Twitter can access to all the Tweets of a user. The public API allows to get 3200 tweets from a specific user Timeline.
Same goes for User Timeline, Settings, Streams, and Search.

Being able to do something on Twitter does not mean it can be done with the public API.

But I might be wrong but as far as I can read in the documentation there is no way to do what you are asking with the PUBLIC REST API.

Cheers,
Linvi