There’s a new math result which is a milestone for AI mathematics. It’s a human readable and insightful result on a conjecture of some renown. It improves on a previous construction of Erdos to make a set of points in the plane with a relatively large number of unit distances between them.

Where the AI got its inspiration from can be as ineffable as it is for humans, but there’s a plausible narrative that it got direct inspiration from the Erdos construction. A proof tells a story, and the moral of the story belongs to the reader not the storyteller. To some the Erdos construction is a story about square grids. But it can also be read as a story about taking an algebraic construction, finding a projection onto geometric space which preserves unit distances, and then solving a number theory problem in the algebraic space to have lots of unit distances. Instead of using the straightforward grid structure the new construction uses a more esoteric algebraic construction, involving pulling in a powerful theorem from a completely different place. In a funny detail the underlying number theory problem it relies on is fairly trivial while the Erdos one requires some work. That is not coincidental with there being a lot more edges: the requirements for them to work are much less stringent.

The obvious question is: What does it look like? The papers and articles contain no pictures of the new construction and there’s a reason for that but another reason one should be included anyway. The construction used for small examples produces some very tesseract-looking things and at larger scales looks like a point cloud without any obvious nice geometric properties. At the smaller scale where the structure can be gleaned it looks actively counterproductive, producing fewer distance coincidences than the Erdos construction. You have to crank up the number of dimensions and the radius of the ball up quite a bit before it starts getting favored, and by then the number of points has become huge.

But that doesn’t mean there can’t be a picture! You can have a density plot where regions with more points points are darker, and having the picture may yield geometric insights which the algebraic construction was obfuscating. Does it look like the shadow of a sphere? A disc? A Gaussian plot? Whatever the shape is, the next question is: How big is the unit distance compared to the width of the shape? Here is where it gets interesting: It appears to be that the distance is quite small. For me that starts raising alarm bells. Didn’t we already crop to within a ball in the algebraic construction? Yes we did, but that was to make the number of points finite, not to reduce the geometric range. The projection between the algebraic and geometric space makes many things look very different with the one exception that certain exactly unit distances stay unit. Other distances get scrambled. So that raises the next question: Why can’t we just crop geometrically to some small constant factor of the unit distance at the end, thus making a much better result by reducing the denominator? This might actually work! It depends on just how much smaller the cropping is and how sparse of a region can be found. I honestly don’t know if it works out, and don’t have the tools to analyze this because it’s a bizarre jump back into geometric space from algebraic but it’s plausible and the benefits might be big, so it’s certainly worthy of further analysis.

The concrete bounds now stand at there being a lower bound on the polynomial exponent of 1.014, up from the previously conjectured to be optimal value of 1. The known upper bound is 4/3. That range of possibilities is very interesting and we most definitely have not heard the last word on this. The AI construction just showed 1+e and the 1.014 is a later explicit improvement. Maybe there will be a polymath project on it.

Talking to AI (specifically Opus 4.7) about this is very interesting. It can read through the whole construction no problem, and talk about it fluently. But then when it gets into discussing geometric insights its intuition is garbage. With some prodding I can get it to understand basic points, and it readily understand after they’re pointed out that these are very basic things, but it just can’t wrap its brain around anything without having it explained. It seems like the new construction is exactly the thing it happens to be super good at: Tackling something purely symbolically, pulling in outside theorems and constructions from seemingly totally unrelated areas, following a roadmap which had already been laid out for it. Drawing from geometric intuitions is something which it simply can’t do. The contrast is very bizarre in this particular case where it’s going from genius to idiot talking about the exact same problem with the perspective shifted only slightly. I haven’t, and probably won’t, grok the full new construction, but it was able to explain the basics outline of the construction to me and construct some basic examples, which was fun and interesting.

The other notable thing about the AI strength here is that this is a constructive proof. AI seems to be better at that than proofs of nonexistence, which is consistent with it being fast and not having much insight. Constructions require fiddling around until you find something, with much clearer partial results along the way, where with proofs of non-existence you have to intuit a roadmap or you don’t make any obvious headway until the very end. The proof of the Robbins conjecture is similar: The core insight is up front realizing that you can find a counterexample to Modus Tollens and then do proof by contradiction. After that it looks a lot more like finding a solution to a post substitution problem than a meaningful proof.

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Previously, previously, previously, previously, previously, previously.

I thought: enough is enough, I need to figure out what clown service these are coming from and start blocking whole networks.

Nope, they're almost all from cable modems, not from hosting facilities:

49.43.169.105	"55836 | IN | apnic | 2010-10-27 | RELIANCEJIO-IN Reliance Jio Infocomm Limited, IN" 49.36.222.165	"55836 | IN | apnic | 2010-10-27 | RELIANCEJIO-IN Reliance Jio Infocomm Limited, IN" 49.207.199.34	"24309 | IN | apnic | 2004-12-08 | CABLELITE-AS-AP Atria Convergence Technologies Pvt. Ltd. Broadband Internet Service Provider INDIA, IN" 60.254.88.85	"17488 | IN | apnic | 2000-11-28 | HATHWAY-NET-AP Hathway IP Over Cable Internet, IN" 14.176.188.15	none 113.181.115.52	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 222.252.180.57	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.191.68.81	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 203.162.75.165	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.166.25.40	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.234.181.48	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.232.16.94	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.226.156.45	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.186.56.201	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 123.27.88.193	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 123.24.165.158	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 123.23.84.13	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.173.75.211	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.254.59.191	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.247.93.88	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 134.122.23.167	"14061 | US | arin | 2012-09-25 | DIGITALOCEAN-ASN - DigitalOcean, LLC, US" 113.184.185.188	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.170.114.205	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.236.40.167	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.170.244.63	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.165.28.93	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 163.223.48.0	"138797 | IN | apnic | 2019-02-11 | COASTAL-AS-IN Coastal Broadband And Online Services Pvt. Ltd., IN" 58.249.137.74	"17622 | CN | apnic | 2001-01-18 | CNCGROUP-GZ China Unicom Guangzhou network, CN" 39.68.1.38	"4837 | CN | apnic | 2001-09-17 | CHINA169-BACKBONE CHINA UNICOM China169 Backbone, CN" 14.191.63.98	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.168.114.54	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.161.135.60	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.228.183.221	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.177.130.34	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.236.8.100	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 144.31.35.211	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 14.230.66.236	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.166.3.113	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 143.20.253.201	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 14.172.46.240	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.191.87.248	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.240.145.79	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 171.250.165.14	"7552 | VN | apnic | 2002-10-08 | VIETEL-AS-AP Viettel Group, VN" 14.230.54.62	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.180.27.31	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 177.137.214.144	"263087 | BR | lacnic | 2012-04-18 | Rawnet Informatica LTDA, BR" 120.140.8.61	"4788 | MY | apnic | 1996-10-20 | TTSSB-MY TM TECHNOLOGY SERVICES SDN. BHD., MY" 49.37.213.156	"55836 | IN | apnic | 2010-10-27 | RELIANCEJIO-IN Reliance Jio Infocomm Limited, IN" 216.75.132.173	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 144.31.35.77	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 216.75.132.173	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 216.75.132.173	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 144.31.35.77	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 144.31.35.77	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 216.75.132.173	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 216.75.132.173	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 143.20.253.246	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 143.20.253.246	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 14.169.142.16	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.190.135.174	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.165.212.210	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.165.39.79	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.254.166.139	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.170.190.94	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 123.31.236.195	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.175.2.197	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.245.142.32	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.168.130.126	none 14.234.120.102	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.187.7.237	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.169.200.172	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.187.7.158	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 222.253.132.0	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.179.243.6	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.189.84.206	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.189.125.112	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 123.27.175.238	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.179.139.90	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 216.75.132.167	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 216.75.132.167	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 14.191.192.133	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 216.75.132.212	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 207.180.11.239	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 207.180.11.239	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 113.186.16.134	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.170.25.114	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.229.9.235	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.188.28.166	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 123.16.224.122	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.173.194.120	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.239.74.25	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.239.32.210	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 14.177.207.17	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 113.173.255.127	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 144.31.35.232	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US" 181.213.41.191	"28573 | BR | lacnic | 2003-11-27 | Claro NXT Telecomunicacoes Ltda, BR" 113.182.199.97	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 88.235.232.128	"9121 | TR | ripencc | 1998-12-29 | TTNET, TR" 181.191.153.5	"267488 | BR | lacnic | 2017-08-17 | Pronet Construcoes LTDA, BR" 211.241.113.62	"9316 | KR | apnic | 1998-06-03 | DACOM-PUBNETPLUS-AS-KR DACOM-PUBNETPLUS, KR" 152.58.183.66	"55836 | IN | apnic | 2010-10-27 | RELIANCEJIO-IN Reliance Jio Infocomm Limited, IN" 59.184.62.53	"9829 | IN | apnic | 2000-01-19 | BSNL-NIB National Internet Backbone, IN" 14.164.166.200	"45899 | VN | apnic | 2009-08-28 | VNPT-AS-VN VNPT Corp, VN" 152.57.115.213	"55836 | IN | apnic | 2010-10-27 | RELIANCEJIO-IN Reliance Jio Infocomm Limited, IN" 49.34.210.1	"55836 | IN | apnic | 2010-10-27 | RELIANCEJIO-IN Reliance Jio Infocomm Limited, IN" 1.246.164.57	"9318 | KR | apnic | 1998-06-03 | SKB-AS SK Broadband Co Ltd, KR" 31.223.92.91	"12735 | TR | ripencc | 1999-10-18 | ASTURKNET, TR" 103.66.177.245	"135578 | BD | apnic | 2016-06-18 | DIT-AS-AP Muhammad Nasir ta Dhaka Information Technology, BD" 152.42.242.171	"14061 | US | arin | 2012-09-25 | DIGITALOCEAN-ASN - DigitalOcean, LLC, US" 143.20.253.31	"401560 | US | arin | 2025-01-03 | ONECABLE - OneCable Network LLC, US"

Previously, previously, previously, previously, previously.

"AI" "scribes" used by some doctors to generate medical records make significant errors in their output, fairly often.

European countries are seeking excuses to deny asylum to true refugees that have reason to fear being tortured, or wish to live where their close relatives live.

Kash Patel faces a new criticism — for snorkeling in the vicinity of the USS Arizona National Monument.

Patel has done a series of cavalier, unjust, and repressive actions, for which he ought to be impeached and removed, and in some cases perhaps jailed. But snorkeling near a sunken battleship is not one of them. It is of no real significance — only a symbolic meaning which anyone might attribute or not.

I observe a tendency to reproach, excessively, alleged failures to "show respect" is a distraction from the real injustice that really matter. It plays into the hands of right-wing extremists, who just love "showing respect" for dead heroes as an excuse for persecuting or murdering the living heroes who campaign for freedom today.

US deportation thugs may retain enough of an idea of common humanity and an idea of right and wrong to feel a moral conflict between that idea and the cruelty of their job. Various organizations are offering them paths to repentance and rehabilitation.

The persecutor seems to be threatening to kidnap Raúl Castro, the aged former president of Cuba.

In Cuba there are right-wing dissidents, that want to allow some people to get rich subjugating the rest, and there are left-wing dissidents, such as the late Oswaldo Payá, who want to preserve the educational and medical achievements of the Cuban revolution while establishing freedom of speech. The persecutor would surely despise Payá just as he despises the Americans funds for good education and medical treatment he is eliminating.

Two thugs from police departments miles away visited several Cincinnati schools and tried to conduct "wellness checks" of a list of students, with no warrants to justify this and having no jurisdiction there anyway. It turns out they were working on behalf of the deportation thugs, perhaps seeking to make a few unlucky children's life drastically worse. This follows the usual gross basic emotional dishonesty of the deportation thugs.

The US history of prosecuting reporters and their sources (whistleblowers) runs through Nixon and Obama, and merges into the persecutor's current efforts to totally subjugate the news media.

What the incredibly expensive failed effort to save one stranded humpback whale says about people's irrational priorities.

US citizens: call on your congresscritter and senators to oppose the Earned Wage Access Consumer Protection Act. It is a roundabout way of exempting payday loans from existing regulations that protect the public.

See the instructions for how to sign this letter campaign without running any nonfree JavaScript code--not trivial, but not hard.

US citizens: Join with this campaign to address this issue.

To phone your congresscritter about this, the main switchboard is +1-202-224-3121.

Please spread the word.

Among various kinds of dangerous pollution produced by burning wood in fireplaces, one shocking kind is lead.

I weep for the future.

Previously, previously, previously.

This document establishes a policy for how LLMs can be used when contributing to rust-lang/rust. [...]

No comment on this PR may mention the following topics:

• Long-term social or economic impact of LLMs
• The environmental impact of LLMs
• Anything to do with the copyright status of LLM output
• Moral judgements about people who use LLMs

We have asked the moderation team to help us enforce these rules.

Previously, previously, previously, previously, previously, previously, previously, previously.

Penile implant specialist with history of far-right comments led hantavirus presser:

Before he joined the Trump administration last year, Dr. Brian Christine was an Alabama-based urologist who specialized in penile implants. He has little public health experience and a history of far-right commentary and promoting conspiracy theories. He's said the Covid pandemic led to a wider government plot to control people, compared the Biden administration to Nazi Germany and suggested the Covid vaccine had little effect in stopping the pandemic.

He once hosted a YouTube show called "Erection Connection," a professional YouTube series on erectile dysfunction for fellow urologists.

A CNN review of archived podcast episodes, social media posts and radio appearances found that Christine repeatedly framed public health institutions, the federal government and pandemic-era policies as tools used to target conservatives and religious Americans.

Previously, previously, previously.

The post was advertising exactly what it sounds like: A database of poop images, collected from an AI poop analyzing app that he had launched several years ago. Basically, 25,000 people had been taking images of their poop and uploading them to his app. He'd been collecting, analyzing, and annotating these images and now wanted to sell access to them. [...]

The poop database comes from an app called PoopCheck, an app made by a company called Soft All Things that purports to use AI to analyze images of one's stool in order to give you a "daily gut health score." [...]

The app also features a "community," of 151,317 "shared stools" at the time of this writing and a "leaderboard," where people can share images of their poop for commentary from other users and earn points for participating.

Previously, previously, previously, previously, previously, previously.

At 1Password, we applied agentic tooling to a multi-million-line Go monolith.

(No I'm not linking to it.) Ok first, how in the everloving fuck did you allow a password manager to reach multiple millions of lines??

(I assume they're not even counting the entire embedded copy of Chrome that lurks inside 1Password.)

And now they sloppified it, so more people are jumping ship.

Welp, I warned you about 1Password's self-immolation in 2023 and posted a migration guide. You're welcome.

Please note: If you have come here to recommend a different password manager, I implore you to not do that, as I just do not care. This is what I did. If something else works for you, good for you. Write it up on your own blog.

Previously, previously, previously, previously, previously.