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I was recently playing a GK v Vanilla game and was shot at by a turbo penetrator. The GK player claimed that he gained an extra d3 on any and all of the 4d6 that rolled a six for armor penetration (via rending rules). Is this true?

Rending says that a roll of a 6 on armor penetration grants an additional D3. So I would say yes, each 6 would grant an additional roll of a D3.

If there is ever a situation where he gets 2 rending die, then he's penning regardless so it really doesn't matter.

Example: 1+1+6 (+1 on the rending die) +6 (+1 on the rending die)=16

Its true, although he will rarely need the extra D3. The only time a Vindicare would need a D3 from rending would be if he rolled three 1s and one 6 and was shooting at AV13 or higher. the average roll on 4D6 is 14, add the Str3 from the Sniper rifle itself and you are already penning a Landraider with 2 points to spare.

Where do you find the rules explaining all of this..? So I can prove it later to other players.

BRB FAQ, whcih talks about multiple rending dice.