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In this replay, I managed to paralyze a greedy Blaziken. According to the calculator, a jolly Blaziken that is paralyzed and with two speed boosts is supposed to have a speed stat of 284 or basically the value before it gets any speed boosts while a jolly Garchomp is supposed to have a speed stat of 333. Yet in that replay on turn ten, Blaziken was supposed to have gone first but got paralysis even though Garchomp should be faster. Why is that? Is the calcs wrong or is this a showdown bug?

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Maybe it was trying to use Protect? I don't play OU, so I don't really know if Blaziken even uses Protect.
It does use protect but usually not on life orb sets. Those things usually prefer to go with a coverage move to hit as hard as possible
Given that the person had a rocky helmet Ferrothorn, he/she/they/it probably also has an unusual Blaziken moveset. Also protect has more than 68% usage on Blaziken, so even with life orb, the chance of seeing a protect Blaziken shouldn't be that much lower.
Rocky helmet Ferrothorn isn't that uncommon these days. It's one of the best ways to brutally punish a Weavile axel or Urshifu strikes and these usually go max defense. What is the most common item on Blaziken? Protect is common but those sets are mostly found on lefties set, maybe even with bulky spreads

Although that wouldn't be too unreasonable given that the guy used a smooth rock Tyranitar which is basically non existent at this point
I don't see why you should doubt that Protect is the cause when the alternatives are all very unlikely.
Because it's rather hard to believe. Protect on Blaziken is used with lefties and it is usually to rack up the recovery or to ensure that scarf Lele doesn't immediately revenge kill it. Life orb Blaziken has fallen out of favor like five months into the crown tundra and there's a good reason

I did conceede in my last comment that protect might have been the case. That's the life of laddering on showdown I guess. You run into guys who run crazy sets

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(The explanation to this was discussed in the comments, however I thought I'd add in an answer anyways to make things easier to read for anyone else who finds this question.)

As the Blaziken was holding a Life Orb instead of a speed-boosting item, the only possible explanation to this scenario is that the Blaziken used a priority move, likely protect, as unusual as that seems. Several people (myself included, I'm not amazing at comp) will sometimes wind up using off meta sets that are otherwise sub-par, and this person is likely no exception.

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