If it does not, then we should read "Unrelenting Momentum" as "when Tachyon takes an actionability for the second time in a turn, she may take that action a third time for free"?
No. I was trying to find the ruling on this. And what i found in that second link is saying that LSB is not an action, since it is a health ability, then unrelenting momentum doesnt affect it.
Whereas since Hud Goggles affects all abilities not just action abilities should unrelenting momentum be the same? If i use LSB twice would UM allow me to use it a third time without costing me the health?
My opinon of it is great let her do her worst. It gives rise to that old addage "hell hath no fury like a womens scorn".
Keep in mind the date on those links. Paul's response in the first link you posted is not only super official (since it's directly from Paul obviously) it's also more current than the 2nd link by a few good months.
Unrelenting Momentum will allow a free usage of LSB, but the timing of the free use renders it useless.
As Luther stated, and I can't find anything overruling it, when you use an ability the second time you immediately have the option to do it a third time for free. You can choose to do it again or not, but it must be right then. You cannot use any ability inbetween.
So your free LSB will doubly ensure that your next attack uses dice equal to your movement.
It will have no actual effect, the next attack will be affected by both LSB uses, which is no different than being affected by one.
When you're sprinting for free attacks, you can Sprint/Free attack, then Sprint a second time/make a free attack. Unrelenting Momentum allows a 3rd Sprint and a free attack, even though there was an attack made inbetween the 2nd and 3rd Sprints.
That is a legit interaction and doesn't jive with what you described.
It jives just fine. Her Sprint works very much differently than Lightspeed Barrage.
Accelerated Assault
Whenever Tachyon Sprints, She may then make one of her active attacks for free.
Unrelenting Momentum
When Tachyon takes an action for the second time in a turn, she may take that action a third time for free.
Lightspeed Barrage
Tachyon's next attack this turn rolls dice equal to her move value instead of the printed attack dice.
Sprint can work one of two ways. The free attack is part of the Sprint Actions, or it follows the Sprint action (which would give it the same timing as Unrelenting Momentum), I have no idea if it has been discussed and which way it works. The differences between them isn't important in this situation, as you'll still be able to Sprint 3 times. Lightspeed Barrage on the other hand does not grant an free use of an active attack, so immediately after using it for the second time, you are immediately given the option to use it again.
You can find Luther's post here: https://greaterthangames.com/comment/83994#comment-83994
HUD Goggles was specifically ettata'ed, as rules as written did not jive with what was intended. I don't see any reason why that would generalise to Unrelenting Momentum.
The point is you can't use LSB, then sprint/attack, LSB, sprint/attack, free LSB, free sprint/attack.
You could LSB, sprint/attack, LSB, free LSB, sprint/attack, free sprint/attack.
The second case works because you don't do anything between the second use and the free use.
However, because LSB affects the next attack the free sprint/attack will not be affected by LSB. Both the second and free LSB would be used up with the second sprint/attack.