let’s say I’ve chosen Take Action for the session maneuver, same goes for the opposition.
We test independently.
It’s Infiltration, I want to use Rethoric which I haven’t opened yet. So I use Beginner’s Luck…
However there’s not base Obstacle to double (no “special rule” as for Assess which is Ob 2 when tested Independently)
I roll 7 dices and get 4 Success :
Do I use half of my Success to substract to Opposition’s disposition ?
or do I get away with it, as a lucky bastard who scripted wisely (and got lucky) ? 
Half successes would be my answer.
Half successes, just like any other beginners luck test.
This is precisely my point:)
As far as I know you never ever half successes in Burning Wheel or Burning Empires (in Mouse Guard you half exponent dice)
When using Beginner’s Luck, you double the Obstacle for the test, be it a fixed number or the total number of successes the opposition obtained (in case of a versus test)
And here, in an indepedent test, no obstacle to double…
EDIT: browsing the forum, I found that actually “half successes rounded-down” is the official recommendation in this case. As it’s not written in the books, I just wanted to make sure.
Thanks guys ! 
Yeah, it’s an extrapolation but since halving successes should give basically the same margins of success/failure as doubling the obstacle it works. I think doubling the ob is the only possibility mentioned in the book because it’s generally cleaner to multiply a static number than divide something dependant on the dice. Also because you only multiply the base ob and division has the effect of doubling the effectiveness of both the base ob and any situational penalties.
I also find the Ob doubling very élégant.
Patching the system by halving obtained successes won’t happen often though, I can live with it.
thx