As a more substantive comment: You may find the thesis "Propagation networks : a flexible and expressive substrate for computation" by Alexey Radul interesting. https://dspace.mit.edu/handle/1721.1/54635

> The test equipment can’t possibly cost more than $100K. That leaves $26.9M of “you’re doing something obviously wrong”.

It seems clear from the original text ("It costs $100,000 per certification") that it's the certification FEE that is $100k. For example, https://ww2.arb.ca.gov/sites/default/files/2024-08/mac202403... includes an individual base fee of $126,358.

Amusingly, there are at least 3 different definitions of commutative ring.

The principal issue is whether it must have a 1 (unity, ie a multiplicative inverse). Wikipedia https://en.wikipedia.org/wiki/Commutative_ring as well as most modern sources insist on this.

Britannica https://www.britannica.com/science/ring-mathematics#ref89421... as well as many older sources (such as Noether's original definition and van der Waerden) do not insist that the ring have a 1. Even first-edition Bourbaki didn't have 1!

Finally, if you do have a 1, then sometimes people include the condition that 0 != 1, ie the trivial/zero ring is deemed not a [commutative] ring. This is somewhat hard to find, but is relatively common among people who specifically define the concept of "ring with identity" (eg Zariski+Samuel). I have also found it unqualified (ie, just in the definition of "commutative ring") in the wild, eg in "Handbook of Mathematical Logic" by Barwise or "The Math You Need" by Mack.

(I agree with people like Conrad and Poonen that rings should have a 1. And I guess that the zero ring is in fact a [commutative] ring.)

(My understanding is "no" to both questions.)

The promise to seek UN Security Council action is only in case Ukraine is attacked with nuclear weapons.

In example 3, the NBBO for stock ABC is 495--500, but there is also an odd lot offer for 497 on exchange. If a Robinhood customer sends a market buy order, then Citadel is allowed to fill it for 499.999 even though it's better to send to the exchange. (And if they then pick up the odd lot themselves, it's easy arbitrage.)

By the way, while you're correct about some of your claims, odd lot executions definitely have to occur within the NBBO. (How could it be otherwise?) Otherwise, in the example above, Citadel would give an even worse price!

For stocks like SPY (those over $500 per share!), the vast majority of orders are odd lots.

This article is many years old and already has data strongly in that direction: https://www.nasdaq.com/articles/odd-facts-about-odd-lots-202...

Why can't you just get 2 shares on an exchange?

I got the following small margins of victory smaller than the 1.5% in 2024, five of which are positive and five of which are negative.

-7.8% in 1824, 1.4% in 1844, -3% in 1876, 0% in 1880, 0.6% in 1884, -0.8% in 1888, 0.1% in 1960, 0.7% in 1968, -0.5% in 2000, -2.1% in 2016.

https://www.google.com/search?q=no%20soot%20in%20the%20pyram...

I used a random estimate online for computing cost which had 5.6e17 Flops per dollar on A100s gives about a dollar every 4.4 seconds or ~$7 million per year.

Sadly, I do not vouch for the correctness of any part of this, though I did try.

Culberson, Joseph. "Sokoban is PSPACE-complete." (1997). https://era.library.ualberta.ca/items/f551dfd8-c8e6-4e78-883...

See also https://erikdemaine.org/papers/NCL_TCS/