vilhelm_s
This article shows that we can't conclude from the working Lean proof that the NL proof is also correct, because they are different in some places. On the other hand I think it's unlikely that it's "very" broken. OpenAI first spent a lot of computer time to find the NL proof, and then less time to translate it into Lean. So if there were errors in the NL proof it seems they were not fatal, the agent translating it into Lean was able to patch them up as it went along.
> The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT‑6 Astra.
This seems pretty clear that they found the proof first and then mechanized it afterwards.
> 2. If a ‘‘shall’’ or ‘‘shall not’’requirement that appears outside of a constraint is violated, the behavior is undefined. Undefined behavior is otherwise indicated in this International Standard by the words ‘‘undefined behavior’’ or by the omission of any explicit definition of behavior. There is no difference in emphasis among these three; they all describe ‘‘behavior that is undefined’’.
Compilers will not refuse to compile the code, indeed the blog post we are all commenting on reports the results from a bunch of different compilers. Historically the reason the C standard specified a lot of undefined behavior is that the actually existing C compilers at the time compiled the code but disagreed about the output.
> The Fifth Circuit has held that the VWP statute “‘unambiguously’ limits an alien’s means of contesting removal solely to an application for asylum.” McCarthy v. Mukasey, 555 F.3d 459, 460 (5th Cir. 2009) (citation omitted). And once an individual violates the terms of the VWP by remaining in the United States for more than ninety days, the individual is no longer entitled to contest removal on any other basis. Id. at 462. This is true even when an individual has a pending adjustment of status application on the basis of their marriage to a U.S. citizen. Id. at 460, 462.
> Culleton concedes he is removable under the VWP. Reply 10. But he argues that because USCIS accepted and began processing his adjustment of status application, he is entitled to due process protections in its fair adjudication. Id. at 9. The Fifth Circuit has foreclosed this very argument, reasoning that the VWP waiver includes a waiver of due process rights. See Mukasey, 555 F.3d at 462. And “[t]he fact that [Culleton] applied for an adjustment of status before the DHS issued its notice of removal is of no consequence.” Id.
The linked blog post says "Unfortunately, a student once submitted work containing this error; it was almost entirely incorrect and he had no idea." I guess the student probably was aware that not every proof had gone through, and that the that he saw like "99 out of 100 proofs succeeded" and assumed that meant that he had mostly completed the task, not realizing that a false theorem would be used to the give incorrect proofs for the rest of the file.
> Michelangelo's biographers—Giorgio Vasari (1511–1574) and Ascanio Condivi (1525–1574)—tell us that, aside from some drawings, his first work was a painted copy after a well-known engraving by Martin Schongauer (1448–1491) showing Saint Anthony tormented by demons. Made about 1487–88 under the guidance of his friend and fellow pupil Francesco Granacci, Michelangelo's painting was much admired; it was even said to have incited Ghirlandaio's envy. [https://www.metmuseum.org/exhibitions/listings/2009/michelan...]
> Another possible explanation to consider is that the current indings were caused by cross-cueing (one hemisphere informing the other hemisphere with behavioural tricks, such as touching the left hand with the right hand). We deem this explanation implausible for four reasons. First, cross-cueing is thought to only allow the transfer of one bit of information (Baynes et al., 1995). Yet, both patients could localize stimuli throughout the entire visual field irrespective of response mode (Experiments 1 and 5), and localizing a stimulus requires more than one bit of information. Second, [...]
I get the impression that the authors of the paper have some kind of woo (nonmaterialist) view of consciousness. But they also mention this possiblity, which seems more plausible to me:
> Finally, a possibility is that we observed the current results because we tested these patients well after their surgical removal of the corpus callosum (Patient DDC and Patient DDV were operated on at ages 19 and 22 years, and were tested 10–16 and 17–23 years after the operation, respectively). This would raise the interesting possibility that the original split brain phenomenon is transient, and that patients somehow develop mechanisms or even structural connections to re-integrate information across the hemispheres, particularly when operated at early adulthood.
> At this time, we prioritize performance and out-of-the-box functionality over minimal binary size. As is, this issue isn't very actionable, but if you have concrete optimization ideas that don't compromise these priorities, we'd be happy to consider them in a new issue.
Not all leaded gasoline was the same either:
> 206Pb/207Pb ratios commonly found in Pb ores throughout the world range between 16.0–18.5 and 1.19–1.25, respectively (Hansmann and Köppel, 2000). Exception to this rule is the commonly used Pb ore from the Broken Hill deposit, Australia, which is characterised by extremely low 206Pb/207Pb ratios (1.03–1.10). On the other hand, Pb originating from the Mississippi Valley ore deposit, USA, exhibits significantly more radiogenic Pb isotopic composition (206Pb/204Pb N20.0; 206Pb/207Pb= 1.31–1.35) (Doe and Delevaux, 1972). American leaded gasoline reflected therefore significantly higher 206Pb/207Pb ratios compared to European gasoline (Fig. 1). The introduction of the European leaded gasoline around 1945 resulted in a steep decrease of the 206Pb/207Pb ratio of atmospheric Pb (Weiss et al., 1999; data from peat deposits). The isotopic composition of leaded gasoline was to some extent dependent on economical factors, such as the availability and price of Pb ores and has evolved due to the different Pb ores used. For example, Pb used for French leaded gasoline originated from Australian, Moroccan and Swedish ores and the contribution of the separate ores changed during time (Véron et al., 1999). It is therefore indispensable to gather data concerning the origin of gasoline used in studied regions.
[from https://sci-hub.ru/https://doi.org/10.1016/j.envint.2007.10....]
Currently the cheapest non-intermittent energy source is gas; solar costs about half as much, and nuclear costs 50% more than gas [0]. Battery storage is currently competitive with gas for storing around 4 hours of electricity [1].
If we would want to replace the baseload with solar + batteries we would need to store 12 hours instead, during the dark half of the day, so it would cost 3x as much, 200% more than gas.
Maybe we can hope for battery prices to drop, but extrapolating from a Wright's law curve, for them to become cheaper by a factor of 3 we need to produce 32 times as many of them [1, again], it won't happen in the near future.
[0] https://www.eia.gov/outlooks/aeo/electricity_generation/pdf/... [1] https://www.lesswrong.com/posts/mnaEgW9JgiochnES2/2024-was-t...
Also, the fuel for fusion reactors is much more plentiful. If we went all in on fission we might run out of easily minable uranium ore in a century or so, so it would be nice to have fusion reactors ready to take over then.
> The plaintiffs had asked for $5 billion in damages, but the settlement provided no monetary relief. Individuals will be allowed to file claims in state courts, and Boies Schiller lawyers have said they already filed more than 1,000 such cases in California state court, with plans to file many more.