Gérard TAILLE wrote:
The definition [of a double sente] should be as near as possible as the common understanding of go players
Since not all players use values but players with a weak understanding of endgame only use an informal understanding, the common go players' understanding of double sente would be informal. However, some players have not reflected yet that local versus global considerations of double sente differ. Therefore, the common go players' understanding of double sente does not exist. Concerning global considerations, some players are aware that one should not always play a double sente immediately because it might be relatively small while other players (with a weak understanding of endgame) are not aware of that and instead believe overly simplistic traditional advice to play in double sente as early as possible. Only for local considerations, we can identify some common go players' understanding of double sente: that either player's local play is sente meaning an immediate reply by the opponent. In only informal terms, we cannot better characterise why an immediate local reply should be necessary.
In terms of values, we can characterise why an immediate local reply should be necessary: after either player's local play, the reply is more valuable. That is, the move value in the initial local endgame position is smaller than both replies' follow-up move values. Let us use these variables:
M := the move value in the initial local endgame position.
Fb := the move value in the follow-up position created after Black's start.
Fw := the move value in the follow-up position created after White's start.
Now, we can characterise a local double sente endgame be these value conditions:
M < Fb, Fw.
(This annotation abbreviates "M < Fb and M < Fw".)
However, simply speaking, the mathematically proven theorem says:
A local double sente endgame with M < Fb, Fw does not exist.
The common go players' understanding did not know this yet:)