How to handle opener's rebid after the Kaplan/Granville Inversion and the related subject of how to handle 1[SP]-1NT forcing (in particular how to combine Bart and Gazzilli - one attempt is in ETM Gold Premium) holds great interest for me - however a recent rec.games.bridge posting asking about Bart/Gazzilli had zero replies so most do not find it so absorbing, so I'll understand that most will skip the rest of this or skim unread it.

To avoid repetition below, all sequences that begin 1[HE]-1[SP] are using the Kaplanville Inversion, where 1[SP] acts like a proxy "forcing notrump". In this style, 1[HE]-1NT shows 5+[SP]s.

Most of the design work in this area involves mapping hand types to opener's rebid, and deciding what works or not. For example I considered 1[HE]-1[SP]--2[CL] to show either 4[SP]s or 6[HE]s (opener's 2[HE] rebid shows [CL]s - thus a remapping). Now responder can signoff in 2[HE] or can bid 2[DI] to ask. However some hands want to invite opposite 6[HE]s but pass opposite 5 (i.e. if 4[SP]s), so the scheme has flaws.

Canadian experts Vincent Demuy and Gavin Wolpert play (or played?) a method where 1[HE]-1[SP]--1NT shows a balanced hand or [DI]s. This allows them to play 1[HE]-1[SP]-2[DI] to show 4[SP]s (sort of a delayed Flannery). The problem with this mapping is that the range of 1NT is considerable, forcing responder to move on when a pass would be better. Also the distributional [HE]s & [DI]s hands might place a notrump contract by the wrong side.

Currently I favour this approach:

After 1[HE]-1[SP]:

1NT: Maximum of 14. Balanced or with exactly a four card minor (not 0-5-4-4). Now 2m is to play, 2[SP] is an artificial game force, 2[HE] and invite with 2[HE]s and (or poor invite with 3[HE]s), 3X natural invite.

2[CL]: Four or longer minor and 14/15+, or five or longer minor, or any 18/19+. Structure below.

2[DI]: 4+[SP]s, up to 16 (not 5[SP]s if 14-16).

2[SP]: 4+[SP]s, 16/17-18 if 4[SP]s, 14-16 if 5+[SP]s.

3[CL]/[DI]: 5-5+, 13/14-16.

Rest fairly standard.

After 1[HE]-1[SP]--2[CL]:

2[DI]: Up to 9, 4-5[DI]s or 3[DI]s and short [HE]s. Now:
**** Pass: 4+[DI]s, no game interest.
**** 2[HE]: 4[CL]s and 15-18 or 5+[CL]s up to 13. 3[CL] to play. 2[SP] asks, 2NT shows 15-16, 3[CL] 5-5+, above 3[CL] if 17-18
**** 2[SP]: 4+[DI]s and game interest, forcing just to 2NT.
**** 2NT: 18/19+ Puppet to 3[CL] or [CL] signoff with 6+[CL]s. After responder bids 3[CL], 3[DI] shows 4[DI]s, 3[HE] and 3[SP] show 4[SP]s (the latter with 3/4[DI]s), 3NT is flat with 6[HE]s.
**** 3[CL]: 18/19+ with 4+[CL]s. 3[DI] asks hand type.
**** 3[DI]: 17/18+ with 5+[DI]s.
**** 3[HE] and higher: Hands with 6+[HE]s and a singleton/void.

2[HE]: Up to 9, 2[HE]s & 4-5[CL]s. Now:
**** Pass: 4+[DI]s, no game interest.
**** 2[SP]: 4+[DI]s and game interest, forcing just to 2NT.
**** 2NT: 18/19+ Puppet to 3[CL] or [CL] signoff. As 2NT puppet above.
**** 3[CL]: 17/18+ with 4+[CL]s. 3[DI] asks hand type.
**** 3[DI]: 18/19+ with 5+[DI]s.
**** 3[HE] and higher: Hands with 6+[HE]s and a singleton/void.

2[SP]: Most hands with 9/10+ (see 2NT alternative). Now:
**** 2NT transfer to 3[CL] with [CL]s – opener can pass 3[CL] if 5-5 but will bid again if 14/15+ to show shape. 3[SP] over 3[CL] shows 4[SP]s and short [DI]s. Responder does not have to accept the transfer - in particular can bid 3NT with 12 and can bid 3[DI]s with 6+[DI]s.
**** 3[CL] shows 5-5 in the reds, not forcing - responder can pass with long [CL]s.
**** 3[DI] is natural and forcing.
**** 3[HE] and 3[SP] are natural, 18/19+.
**** 3NT to play.
**** 4[CL]/[DI] with big 6-5+.

**** 3NT to play.
**** 4[CL]/[DI] with big 6-5+.

2NT: 10+ with 4-2-2-5 exactly and a hand that wants to play 2NT opposite 5[DI]s up to 13.
**** 3X is now natural and forcing to 3NT or 4X.

3[CL]/[DI]: Six or longer suit, up to bad 9. 3X by opener is now forcing to 3NT.

3[HE]: Game invite with 3[HE]s.

3[SP]: 5-5+ in the minors, 10+.