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2. Maduro wasn't even the president. He was someone who took the country illegally with cartel people.
3. Why? Maduro was smuggling drugs in USA. Huge operations. And I guess there must be geopolitical reasons. You want China and Russia be there? And people from Venezuela were the biggest migration wave in the World last decades. You want millions of refugees?
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Being two separate tunnels, it also needs twice as much excavation work. It's also ~25x deeper than Toei Oedo (4000ft vs 157ft). At 4000ft the rock itself is 45-50C!
Overall an amazing achievement, and unsurprising it took this long to figure out!
Thanks for pointing out the elephant in the room with LLMs.
The basic design is non-deterministic. Trying to extract "facts" or "truth" or "accuracy" is an exercise in futility.
I don't think using deterministic / stochastic as a diagnostic is accurate here - I think that what we're really talking is about some sort of fundamental 'instability' of LLMs a la chaos theory.
With OpenAI that part takes up about 2/3 if the proof even with its fragmented prose. I don't think it does much better.
"Since there are 3k - 3 points on the perimeter of the triangle to be covered, and any sunny line can pass through at most two of them, it follows that 3k − 3 ≤ 2k, i.e. k ≤ 3."
Gemini writes:
Let Tk be the convex hull of Pk. Tk is the triangle with vertices V1 = (1, 1), V2 = (1, k), V3 = (k, 1). The edges of Tk lie on the lines x = 1 (V), y = 1 (H), and x + y = k + 1 (D). These lines are shady.
Let Bk be the set of points in Pk lying on the boundary of Tk. Each edge contains k points. Since the vertices are distinct (as k ≥ 2), the total number of points on the boundary is |Bk| = 3k − 3.
Suppose Pk is covered by k sunny lines Lk. These lines must cover Bk. Let L ∈ Lk. Since L is sunny, it does not coincide with the lines containing the edges of Tk. A line that does not contain an edge of a convex polygon intersects the boundary of the polygon at most at two points. Thus, |L ∩ Bk| ≤ 2. The total coverage of Bk by Lk is at most 2k. We must have |Bk| ≤ 2k. 3k − 3 ≤ 2k, which implies k ≤ 3.
I know o3 is far from state of the art these days but it's great at finding relevant literature and suggesting inequalities to consider but in actual proofs it can produce convincing looking statements that are false if you follow the details, or even just the algebra, carefully. Subtle errors like these might become harder to detect as the models get better.
I gave it a bunch of recent, answered MathOverflow questions - graduate level maths queries. Sometimes it would get demonstrably the wrong answer, but it not be easy to see where it had gone wrong (e.g. some mistake in a morass of algebra). A wrong but convincing argument is the last thing you want!
As this is based on detection of the child, what happens on Halloween when kids are all over the place and do not necessarily look like kids?