Betterball Stableford Scores for Handicap

And there are no consequences for recording both scores. It is up to the Committee to decide which score will count for the side. Rules 23.2b(1) and (2).
From the Rules of Handicapping as applied in GB&I

The standard approach to Fourball competition should continue to be used – as soon as the
best score has been recorded, the other player should pick up in order not to impact pace of
play. Handicap Committees will be justified in disregarding scores for handicapping purposes where scores for both players are recorded on a significant number of holes on a regular basis.


As you can see there can be consequences for recording both scores.
 
From the Rules of Handicapping as applied in GB&I

The standard approach to Fourball competition should continue to be used – as soon as the
best score has been recorded, the other player should pick up in order not to impact pace of
play. Handicap Committees will be justified in disregarding scores for handicapping purposes where scores for both players are recorded on a significant number of holes on a regular basis.


As you can see there can be consequences for recording both scores.
That's the rules of handicapping, not the Rules of golf (which are clear).
 
And only if the Handicap Committee (if one exists) decides to act.
For your information, it is a condition of affiliation (to the Handicapping Authority) that a Handicap Committee must exist otherwise the club risks not being able to administer handicaps, so their members would not be able to have one therefore they would not be able to run a 4BB competition and enforce any possible consequences of a pair returning both scores repeatedly on every hole.
 
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