Some further thoughts on the 372 results released by OpenAI today, across 722 manuscripts. If I were to classify theorems that mathematicians prove and publish according to their groundbreaking nature, I would (very roughly) divide them into four categories: A) Non-breakthrough results. This is the overwhelming majority of published mathematics. Such results can range from solid to excellent, and some represent genuine advances in a field. But they are not hugely surprising, and would not normally be described as “breakthroughs.” B) Exceptional advances within an existing programme. These are spectacular results, but where there was nonetheless an existing credible route to the theorem, and some expectation that sufficient work would get you there. Completing the programme may require a lot of ingenuity and deep work, but mathematicians would not be shocked that the theorem had finally been proved. C) Surprising breakthroughs. These are results that clear a major barrier and substantially change the state of a field. Before the proof there was no convincing roadmap to the full result. Yet, while mathematicians would find the theorem remarkable and surprising, they would not find it completely shocking: if you asked them beforehand if it was plausible such a theorem could be proved today, most would say yes. Note: The very best mathematicians prove only a small number of results in categories B and C in a lifetime; many mathematicians never prove even one. Such results would normally belong in the very top journals, such as Annals, Inventiones, etc., and there are only a handful of them each year in any given area. Several results of this calibre by a single person would make a very strong case for a Fields Medal. D) Shock breakthroughs. These are results that, before their announcement, leading experts would have regarded as *extremely* unlikely to be proved with the current mathematical technology available. So the theorem itself would come as a shock. These are extraordinarily rare, and instant-Fields medal variety. (There is one further category I have deliberately left out, because I suspect it is empty: a correct proof of a problem for which the overwhelming consensus of top experts, until the proof came, was that a proof was so far beyond existing mathematics that a claimed solution should, on prior grounds alone, be regarded as almost impossible. I would put the Riemann Hypothesis today in that category) My current impression is that some of OpenAI's announcements today lie in A, but most fall into categories B or C. There is exactly one example in D (the Quasi-Riemann Hypothesis). It is a very big day for mathematics.

Captured this drone footage yesterday. Alot of activity at the newly acquired 444 acres SpaceX site along Hwy 4 near Starbase.

The nature of human civilization, viewed as an ecosystem, is that it’s mostly chickens.

Ok, we now have the actual text of the Greenland deal (whitehouse.gov/briefings-sta…), so we can say how it compares to the existing 1951 arrangement and who, between Trump and Denmark, lied the most in their framing of it. Long story short - which may surprise some - the Danes lied the most: Trump is pretty much correct in saying that this new deal grants the US control of Greenland. Let's look into what changed compared to the 1951 deal. 1) This is now a permanent arrangement: whereas the existing 1951 deal only lasted as long as NATO existed, this one "does not have an end date and may only be amended by mutual consent." It even survives a hypothetical future independence of Greenland: in that event, Denmark and Greenland must ensure that the independent Greenlandic state "affirmatively assume[s] all of the rights and obligations of the Kingdom of Denmark specified in this Agreement." 2) In the 1951 agreement, the US could set up bases only where both governments agreed, and only on the basis of NATO planning. The new deal locks in three bases: the US can "modernize and expand its activities" at Pituffik, its only base in Greenland until now, and "shall be allowed to establish an additional Defense Area at Narsarsuaq and Mestersvig." Further bases beyond these 3 can be justified by the defense of "the American continent," a US interest rather than a NATO one. Also, quite subtle but this is interesting: Article IV(iii) says that "the United States may establish additional defense areas in Greenland" and that consultations are then held "to decide implementation details based on mutual agreement." What this means is that the principle of a new base is automatically granted and only the implementation details require agreement. Article VII also gives the US "the right to raise concerns regarding construction, or change of use" of any building near a US base, after which the parties decide together what to do. Which means that the US has a say over civilian planning around its bases. 3) On US military movement across Greenland, the wording change between the 1951 agreement and this new one is quite funny. The 1951 text said (Art. V(3)): the US "may enjoy... the right of free access to and movement between the defense areas through Greenland, including territorial waters, by land, air and sea," but only "in accordance with general rules mutually agreed upon and issued by the appropriate Danish authority in Greenland." The new one says: "the United States of America shall enjoy... the right of free access to and movement between the Defense Areas through Greenland, including the Territorial Waters, by land, air and sea." That's it: "May" becomes "shall," and the US military movement no longer has to follow rules "issued by the appropriate Danish authority in Greenland." The US also gets something new on top: "undersea access to and movement within the Territorial Waters," i.e. submarines. 4) Non-NATO exclusion: non-NATO states are barred from military installations or a persistent presence in Greenland unless all three parties approve, giving the US an effective permanent veto. The 1951 deal had no equivalent clause. 5) Investment screening: this is entirely new. "States or investors from a state that is not a member of NATO, a NATO partner, or an EU member state" (i.e. Russia, China, India, the Gulf States, etc.), "shall not be allowed to have (i) control, (ii) significant influence, or (iii) access to non-public information...within Particularly Sensitive Sectors or Activities in the territory of Greenland (including the Territorial Waters), unless agreed between the Parties." Again, it basically means that the US has a permanent veto over who can invest in strategic sectors in Greenland, as far as non-NATO or EU states are concerned. 6) Lastly, in stark contrast with the 1951 deal, the framing of the deal changes significantly: this new deal is largely framed around the US. The 1951 agreement existed only "for the benefit of the North Atlantic Treaty Organization." The new one lists the defense of "the American continent" as an objective, celebrates "the irreplaceable role" of the United States military as well as "the United States' indispensable historical and ongoing contributions to the security and defense of Greenland," and names the establishment of Trump's "Golden Dome" missile defense system as a key goal. So, to sum up: - The US secured a permanent military foothold that even an independent Greenland would inherit whether it likes it or not - The US now has veto power over any non-NATO military presence in Greenland, and over who can invest in its strategic infrastructure and resources (beyond NATO and the EU) - Denmark signed away its say over the principle of new US bases, with only the right to discuss the "implementation details" - Danish authorities no longer set the rules for US military movement across the island - And all of it is now justified by the defense of "the American continent" I don't know in which universe this even remotely "recognizes the sovereignty and territorial integrity of the Kingdom and the Greenlandic people's right to self-determination," as Danish PM Mette Frederiksen put it (stm.dk/en/press/press-releas…). Yes, the preamble of the text cynically "reaffirms" Danish sovereignty but, in my book, when you let a foreign country put bases on your soil, move its troops without following your rules and veto who else can come in, in perpetuity, after it spent two years threatening to take you by force, the word for that is anything but "recognized sovereignty" 🤷

BREAKING: the UK government publishes official rules on how to use AI: 1. Ask first if you really need to use AI 2. Check if a spreadsheet can do the job before using AI 3. Choose the worst model possible so it uses less energy 4. Keep prompts short to reduce the environmental impact 5. In general, use AI only when necessary, as a climate-saving measure With a mindset like this, we should accept the UK back into the European Union

Lots of people discuss immigration & crime in Europe, so I compiled what I think is ALL the data on the topic, which is hard because many countries hide it. Do they commit more crime? Is it because of age? Sex? Education? Poverty? Culture? 1. Sexual offenses & violent crime:

it's starting to feel like the early days of covid to me where everyone is still going about their day normally, but at some level you know it's all not real anymore and things will change very soon