So in 34 seconds at 7000fpm you could achieve a 4000ft increase in ALT. But surely you have to start the climb and then reduce it to 700fpm too, all within 34 seconds, whilst still having a forward speed of M0.68 and all from all from energy contained within the airframe.
Well the pitch input was at 05 and the climb commenced at 16. The peak value was 7000fpm but the mean was about 1400fpm. If we are to believe the the quoted speeds they lost 60kts in this manoeuvre and ended with the same thrust setting and 4°alpha.
How about this scenario:
PF makes his correction for roll and doesn't notice the rate of climb or is confused by the stall warning. When he notices the 7000fpm he corrects with nose down and levels at 375 all without adding thrust. He's now at 4°alpha but having not added thrust his speed is reducing which induces the second stall warning. Now he adds the thrust and pulls up. The aircraft doesn't have much more thrust to give and his alpha is increasing all the time while the speed is reducing. We know that the isis speed comes back at 2:11:06 (15 seconds later ) and shows 185 which matches the PFD.
Doesn't appear unreasonable and doesn't require any external energy input.