Appreciate you showing genuine curiosity here unlike most people I've interacted with on this issue. Part of why I started throwing large wager offers around is it was the only way to call out most of those larpers on their bluff. But since you're interested, I'll say what I can here. Some of my info is whispers from insiders I trust, which I'm not going to repeat here. I'll rely atm on arguments that mostly do not depend on them. First off, the smart money has been that the lab rumors were regarding a new case of the Hodge Conjecture. This has been true at least since the guest post on Terry's blog on 9/11, which seems to vaguely suggest awareness such a result has been achieved. A popular interpretation in the High Math Circles is that it's hinting at Hodge for abelian varieties, and Brockman's claim of substantial progress is consistent with that [I'll explicitly say: no one I've talked to has explicitly confirmed that theory, but conversations I've had have significantly increased my credence on it]. The Information article is unlikely to have new info beyond that. The mole Palazzolo spoke to added two new things beyond the previous public record: 1) It's Hodge (duh), and 2) "We're close to solving the full thing." The latter I very strongly doubt, and sounds like the embellishment of a non-math savvy insider who heard about the substantial progress. I'm going to be very clear: Hodge for abelien varieties has long been seen as more in-reach than full Hodge and the profs I talk do not think that's substantial evidence in favor of full Hodge falling anytime soon. Remember, many OpenAI lab employees believed and spread the originally-quiet rumor that Anthropic had fully solved two Millennium Problems, which we now can be nearly certain is false. It's fair to acknowledge that the new round of OpenAI hype is a touch more credible on the basis of "well they're closer to the source this time." But I maintain that everything we've seen is fully explainable by the same credulous insiders as the ones to blame for the last round of bullshit getting confused on how much progress the new case of Hodge is. And my final argument: Hodge hard