Also, in case this wasn't clear, they produced a counterexample but proved it was a counterexample via contradiction. So the paper did proceed by contradiction. I should have spent more time editing in anticipation of someone seizing on the smallest error to avoid defending their flawed and mathematically incorrect position.

Actually, the fact I flipped contradiction and counterexample when you said counterexample, and the media keeps saying counterexample, implies just the opposite. You will see I did eventually mention the counterexample explicitly in my post. Most people aren't going to accidentally flip counterexample and contradiction when they mean counterexample, unless they're aware counterexample is a special case of contradiction, and just came off reading through a clear contradiction conclusion in a paper. Again, that's a top-down processing slip, that only happens when someone is using top-down processing. If I had been using AI to write my posts, it wouldn't have done that, so this is also evidence I wrote my own post. That was the mathematical equivalent of a typo. And I didn't name drop the paper because the folders are numbered, and I assume most readers won't know much about what it's saying anyway.

Again, my point stands. The transfer principle had to be setup in order to establish the contradiction they wanted, and it applies to a broader class of R-modules than just the one that was targeted. The bridge can probably be milked for more results than just that. Do you have any objections of mathematical substance or do you just want to continue with ad-hominens?

This many results appearing this close together is a genuine statistical anomaly. This is not "an ordinary few years of of results that got scooped by AI." Any one of these would have been a watershed moment for its respective field. The fact they all hit at once cannot reasonably be explained by "it stole the ideas mathematicians were about to share." Because normally there aren't this many solutions for major problems within the same decade, let alone the same year. Moreover, you cannot possibly think that every one of these major results was obtained by "rearranging puzzle pieces" because that would imply 300+ landmark theorems were really just jigsaw puzzles waiting to be rearranged. That's not how theorems to major problems work at all.

You really need to sit down and think over what just happened. You were wrong about AI, you need to just accept that. It's understandable -- the metaverse, the cloud, etc., were all utopian pitches from tech companies that everyone swooned over, and they all turned out to be oversold. This dump is hard evidence the wolf is real this time.

Your assessment of proof by contradiction is flawed.

I've been working on verifying the proof of result #195, which proceeds by contradiction. It uses a fairly standard proof technique, but in a way that is novel as far as I can tell. Basically, they want to show a certain class of modules cannot exist. They do this by showing each module has a bridge to a related object that is easier to study, and then use the bridge to transfer data between them. This is pretty standard. What is novel is how the bridge and related class of objects are obtained: they pass from a certain class of modules to coherent sheaves on a related object, and then perform some other formal operations. Next, they establish numerical constraints that any hypothetical module would have to satisfy through this bridge. Then they construct a specific ring whose geometry makes those constraints impossible. Ergo, no module of the specified type can exist over that ring.

But this isn't like it just produced a contradictory object that it can plug into a calculator to check if it violated some known constraint. The proof technique is standard "M sucks, so build a bridge to N that doesn't suck, then transfer data between them." But actually pulling that off is not so straightforward. So result #195 did produce a counterexample, but it did so by creating a transfer principle, and then it used it. It's not like it produced an object with three angles, summed them, saw they didn't add up to 180 degrees, and concluded it isn't a triangle. The mathematical machinery that produces the contradiction in #195 is actually pretty impressive. Thus, I think it's a mistake to classify the contradiction results as easy somehow. Also, contradiction is the most straight forward way to disprove something, so if you feed an AI model a false conjecture it's naturally going to gravitate to contradiction. Not because it's somehow less creative than a person, but just because that's the most reasonable way to complete the task.

I should also remark that a lot of #195 is out of my area of expertise, so this assessment may be flawed. Still, this is what I've gathered from it so far.

I went to Best Buy to try a couple of models, just assuming I'd buy one. I was shocked to realize the "Copilot" was just "crappy skinned ChatGPT in a web app." I sort of just assumed basic agentic features would be available, like modifying PC settings. But nope. It was mind-bogglingly shockingly bad. Worse, I think it's pretty clear that "agent manager" is the correct interface for AI -- it separates "the user interface a human uses" and "the interface for interacting with an AI." For some reason, Microsoft is still shoveling out "embed a chat window in every app" as opposed to having a proper agent manager built into Windows. It's a big missed opportunity, because they could have just rolled out the red carpet for Anthropic, Google, and OpenAI to have unified OS integration built directly into Windows, but nope.

Shhh! Don't let them find out that x^2 + 1 = 0 has no solutions at all. It's our best kept secret. You have to learn, like, at least three secret handshakes and get an A in the entire calculus series before you can learn about that.

You suggested a few techniques that work in a cooler environment, but trust me, Texas sized heat will laugh at your European passive cooling strategies, and probably your improperly sized mobile AC.

In the United States, the AC build out reduced heat related deaths by 75% since 1960 [1]. At the same time, you can also have a situation where increased demand for AC creates an increase in CO2 production, which creates a feedback loop [2]. So you're not entirely off your rocker. Still, evidence from the United States establishes pretty clearly that ACs are effective for heat adaptation. That said, you say it's "for the vulnerable," but where do you confine the vulnerable for months at a time when the world heats up? You're basically saying your society, as a whole, is going to be off-limits for literally months for large portions of your own population. Proper long-term planning means you need to expand out renewables enough to intercept increased heat related AC energy demand, while considering equitable access to habitable environments for everyone. It also means you need to de-stigmatize AC use, because that's ultimately going to shame people away from AC that actually need it. At the moment, you're already seeing 25,000 in excess mortality due to the heat. That is a clear signal that "too hot" is already arriving.

Incidentally, I froze my butt off at work today, so you're right that Americans have an issue with abusing AC settings.

[1] https://ancillary-proxy.atarimworker.io?url=https%3A%2F%2Fwww.journals.uchicago....
[2] https://ancillary-proxy.atarimworker.io?url=https%3A%2F%2Fwww.nature.com%2Farticle...

You don't need to buy or run the AC now. But you do want to think about how climate control needs will change in 30 years. Judging from this thread, a lot of yall couldn't get an AC even if you wanted one. That's the sort of thing people need to start planning for now. But if planning for the future is too American for you, well, carry on.

Climate change is not going to stop even if the world hits the most optimistic targets. It is going to keep getting hotter, a lot hotter. In 30 years, today's peak heats will happen on 5x as many days, and the new peaks will reach 10 degrees higher. Your "a couple of weeks out of the year" is going to turn into months, and the new peaks are going to be lethal. You need to start getting your houses and culture ready now.

An insecure server is the street equivalent of parking a car with the window rolled down and valuables in sight. Someone *will* steal your crap if given a clear enough opportunity. The catch is, actually finding and exploiting bugs requires a high level of talent, time, and dedication, so it mostly suffices to make the bugs that do exist sufficiently hard to exploit. That is, at its most fundamental level, security through obscurity. The problem is, AI changes the talent and time issue.

The issue is there is a large class of bugs that were never patched out, they were just hidden because they were slightly beyond the human attention horizon. Now that AI has started to improve, that horizon is effectively machine-extended, so bugs that sat unnoticed for 8 years are suddenly surfacing. It isn't that they weren't important, it's that the bug was effectively secured through obscurity, and now that obscurity is gone. Therefore, we need to patch the bugs the AI can spot, because an attacker using an AI can spot them now. The evidence is pretty clear now that AI can find these bugs and exploit them end-to-end, so it's only a matter of time before criminals obtain models with such capabilities.

Starbucks more or less wanted to be an online order fulfillment hub. People order online, pick up, leave immediately. Ultimately, they realized this was a mistake and rolled the changes back. I have a feeling T-Mobile may end up doing the same.

Granted, I use Google Fi, so I actually *prefer* not dealing with a cell phone carrier's store. So maybe it'll work out for them.

Gambling arises from using a variable schedule of positive reinforcement based on the exchange of goods to derive a dopamine hit. Pokemon cards can be used as gambling, so can the stock market. The question is how it's being used. Someone who invests in the stock market long-term is not getting a dopamine hit from a variable schedule of positive reinforcement, so they're not gambling. Someone who plays Magic the Gathering at a gathering by buying boosters, then assembling a deck (a common play mode, both players end up with a random spread) is not seeking a dopamine hit derived from a variable schedule of positive reinforcement. But once those activities become centered around seeking the next hit, it becomes gambling. Of course there is a gradient -- someone might actually like poker, but when a major factor in the enjoyment is the risk/reward pattern, it's gambling. It becomes an addiction when it interferes with one or more major activities of daily living.

Hypothetically, someone could use a "prediction market" completely detached from the variable schedule of positive reinforcement. In practice, I highly doubt that is how it's being used, and I think deep down you know it.

You mean a tax on young and impulsive men who need guidance on making good choices? Because it's mostly young guys who are still shaking off puberty who get their credit scores ruined by stuff like this. Isn't nature beautiful? The little chick just barely waddled out of mommas nest and an alligator up and eats it. After all, the little jackass should've known about alligators!

The risk equation is:

risk = offensive_capability * jailbreakability

According to Anthropic's own doom-saying, Mythos could literally end the whole-wide world overnight with a series of catastrophic, hypercane level cyberattacks. Currently, Fable 5 and Mythos are the only Mythos-class models available publicly. Thus, even if other models have high jailbreakability, the risk level is down-regulated by having lesser offensive_capability. Again, this is according to *Anthropic.* Obviously, jail-breaking GPT 3.5 isn't going to end the world. According to Anthropic, a jailbroken Fable 5 or Mythos is the apocalypse. Since those are the ONLY apocalypse models available, jailbreakability is uniquely severe for those models in particular. This makes Anthropic's attempt to say "but GPT-5.5!" particularly moot, since no one has claimed GPT-5.5 can usher in the cybersecurity apocalypse.

Anthropic made the claims, not me. I'm just following them to their logical conclusion.