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I'm going to keep on believing there was at least some subconscious influence if you don't mind (I've actually made that exact same mispronunciation multiple times for the math reason).

Is Eric a mathematician? The mixing up of Veil and Weyl sounded like someone who's worried about Weil vs. Weyl a lot before.

Here's an excellent source of fun problems: https://kskedlaya.org/putnam-archive/. There are very substantial jumps in difficulty between the different numbers 1 through 6 given to the problems. Therefore don't try go through every problem for a single year and instead look at those across all years in the appropriate range. Some may require a little college-level math knowledge, so feel free to ignore those if you don't have the background. Finally, if you're not used to these kinds of problems, even 1's may be tricky at the beginning---don't be discouraged if it takes you 1 or 2 hours to figure out how to solve them. 2009 A1 is a good place to start and has a really cute solution: Let f be a real-valued function on the plane such that for every square ABCD in the plane, f(A) + f(B) + f(C) + f(D) = 0. Does it follow that f(P) = 0 for all points P in the plane?

I always try to look for the make-yourself-smarter option with things like this. Therefore, even though I'm not precisely sure what "mental speed" means, zinc-zinc sounds really cool. It also might be really powerful to have extra mental speed together with Rioting. Imagine actually being able to quickly process all the information you see in someone's body language and facial expressions, perfectly understanding all their micro-gestures and tics. You would know exactly what they're feeling. You would be able to subtly push at specific emotions and see precisely how they react, rapidly iterating and perfecting how you Riot them to get some desired reaction. Just this super-skilled Rioting alone seems like one of the most useful powers you can get, ignoring everything else "near-infinite mental speed" might mean. As mentioned by others, chromium-chromium for near-infinite fortune sounds really useful too.

@Friendshipspren The name comes from some mysterious, exceptional cases of objects in math that I think about a lot. There's a classification of "continuous symmetry types" (symmetry types of things like circles and spheres instead of things like squares that only have "discrete" symmetries) that is mostly infinite families that mathematicians expected. However, when they were finishing up the classification, they found 5 bizarre exceptions in specific large dimensions ranging from 7 to 248. I find these fascinating. @Ookla of Truthshapers Ah, I think those two concerns you brought up are related. So there are three ways to think of a wave like sin(x - vt). First, you think of it as a one dimensional moving image plotted against the x-axis and moving as you increase t. Second, you think of it as the opposite: plotted against t and moving as you increase x. Third, you can think of it as a stationary plane wave plotted against x and t. In the first interpretation you are correct: changing the plus to a minus just changes the direction the wave in moving in space. However, in the second interpretation, changing the sign changes the direction the wave is moving in time. The third interpretation reveals what's going on: a right-moving, forwards-in-time wave is indistinguishable from a left-moving, backwards-in-time wave. Both are diagonal plane waves oriented top-right to bottom-left in the (x,t)-plane. Which interpretation is correct depends on which problem you are trying to solve. This is the analogy I am going for: interpreting a positron as an electron going backwards in time is the same. It's just a mathematical trick/change in perspective that's useful for doing some calculations. It does not at all imply all the associations you would normally think of when you hear the English phrase "backwards in time"---you can't mess with causality/kill you grandfather and cause a paradox with an antimatter bullet. I think the wikipedia section is careful to clarify this in the last sentence? Of course this is an analogy that's wrong in all details. Somehow I'm conflating direction of spacial travel with electron vs. positron. Heck, electrons/positrons are all in quantum fields so even thinking of waves in them as functions plotted in spacetime is problematic. However, I think the analogy is correct in the intuition. Definitely let me know if I'm wrong there however---I'm a mathematician, not a physicist. @Serack Thanks for bringing that up about coherent light---I did not know that! Given that Navani did not seem to need anything like perfect pitch to get the tones right, the magic behind Intent might be good enough to not necessitate something as exact as a laser and those lower-tech methods might be good enough. The Boson/Fermion thing is interesting too. I really would not be surprised if Brandon Sanderson meant to go that deeply into physics analogies. His extreme emphasis on symmetries and pure tones is really strong signal that he was thinking a lot about real-world physics.

@Friendshipspren I think "backwards in time" brings up a lot of associations that aren't correct. It's only true in a very narrow sense as described here. As an very simplified analogy, if an electron is represented by a wave like sin(x - vt), a positron would be something like sin(x + vt). There's a change of sign in an equation that you can interpret in many different ways, but the english words people attach to the interpretations---"one electron" or "backwards in time"---tend to connote things that are way stronger than what the equations say. I think investiture and the various lights is some sort of new thing where, importantly, it may not matter what the really technical physics details are. In the real world, when we have a popular-science discussion like this, everything we say can only be an analogy not at all approaching any true description. However, something magical in Intent makes these actually relevant in the Investiture case. You can actually manipulate Investiture playing around with pop-science analogies like sound tones and light colors without having to deal with any complicated physics. Some part of the magic system latches onto your Intent and does the translation for you. Ok, now for some more wild speculation: it could be possible that as you understand more about the actual physics, you get more Connection to a more powerful "language" that you can use to state you Intent and make more precise Commands. We know that magic like Awakening requires you to speak a Command in your native language. Maybe knowing more math/physics concepts would be like adding "words" to this language---if you understand what a gauge boson is, you might be able to use analogies related to them to manipulate investiture. I guess what I'm trying to say is that the focus should be more on what physics analogies we can match to Investiture phenomena instead of what exactly Investiture is at the really technical physics level. I guess my return question then would be what would it mean for the magic if the various lights analogized to a gauge boson instead of something else? (I don't know enough to speculate here as you can see by the edits to fix the physics details).

Oh wow, this sounds super plausible. However, Awakening similarly seems to use color as a filter/catalyst and does use it up even when there's an another source of investiture in breath. Maybe it sometimes takes investiture to move investiture (like a refrigerator needs to use energy to move heat energy)? On the other hand, this same sort of filtering seems to happen with gemstones in fabrials without using up the color, so I'm a bit confused on what the general principle is. If you like the physics analogies, this field theory perspective maybe clarifies what's going on with anti-investiture also. In real-world physics, there's an electron field (that I guess you hope to find as some component of a larger field in some hypothetical grand unified theory) and electrons are a certain kind of wave traveling through it. However, the equations governing the field allow waves traveling in the "opposite" direction (there's a technical sense in which "opposite" means "backwards in time", but this creates very misleading and overly dramatic associations---it is only true in a very specific technical sense) that turn out to be anti-electrons, or positrons. Similarly anti-whatever-light is would come from the same field piece as whatever-light. It is just "inverted" or "traveling in an opposite direction" and the precise details of how this is set up are worked out automatically by the magic system as long as you have the right Intent.

I posted a theory in the stormlight subreddit some weeks ago. I've discovered this forum since then and realized that it is a much more appropriate place for long theorizing. I'll therefore post the theory here too if that's ok. While reading the Navani/Raboniel chapters, I was shocked to see what sounded like a really good analogy to a fundamental concept in higher-level math and physics that usually does not get good popular-science explanations. I don't know if this was just a coincidence that I'm reading too much into or if someone in Brandon Sanderson's orbit learned the concept somewhere. Either way, I had to make an account to write about this. Field Theory: This section is going to be pop-science, so every detail is going to be a lie. However, the overall ideas and intuitions should be correct. Modern physics theories think of the universe as a made up of a bunch of things called fields. A field can be thought of as a bunch of abstract waves that permeate space. Physicists then apply some standard mathematical concepts for studying waves and eventually derive all the normal particles and laws that we hear about. To go further, let's discuss two intuitive and familiar versions of waves: light and sound. The first step when studying a wave it to break it up into a combination of basic, "pure waves" that, when combined in the right proportions, can make up any sound or color. For sound, these are the pure tones and for light, these are pure colors that a prism breaks it into. The list of pure waves depends only on particular symmetries that the field satisfies. For light and sound the pure waves are sine waves which behave very nicely when translated around (slightly more precisely for those who know, they satisfy the angle-addition formulas from high school trigonometry). Translation symmetry is important because that's the symmetry satisfied by the laws sound and light waves follow. The more complicated fields in physics satisfy more complicated symmetries. For example, rotational symmetry gives you "pure tones" that look like the electron orbital pictures you might see in chemistry class. Mathematicians call these generalized pure tones "irreducible representations". Just like breaking up sound into pure tones gives you the notes that make up everything sound-wise, breaking up the more complicated physics fields into irreducible representations give you the pieces of the universe that make up everything---all the fundamental particles: electrons, quarks, etc. While you don't need so many fancy words to just talk about the concept of decomposing into pure tones, field theory suggests that this is the fundamental idea needed to understand reality. The Fantasy Version: Investiture: In the real world, this analogy between sound/light and the physics fields that make up the universe is just that, nothing more than an analogy. However, something with Intent seems to make it much more than that with investiture in the Cosmere. Navani's chapters suggest that Investiture breaks up into different "pure tones" corresponding to different shards. Instead of just being complicated irreducible representations of the symmetries of some "investiture field", they also correspond to analogous and much more intuitive things: literal tones/rhythms or colors. Somehow, having the right Intent when creating the corresponding literal sound tone turns it into the correct tone of investiture. It lets you turn "base-state" investiture into investiture of the type you want. Because she was born on Roshar, Navani had been hearing the tones of Honor, Cultivation, and Odium her whole life and knew them almost from memory, so this was possible for her to do. Some Wild Speculation: Now, here's some wild theorizing if thinking about investiture in this way is actually correct. First, you should be able to use light instead of sound to imprint an investiture type onto base-state investiture. Light is another human-intuitive form of decomposing a wave into pure tones so it should work similarly if you have the right Intent. However, making pure colors is harder than pure sounds and might require a laser or something similar. Much more interestingly, you should not be restricted to tones you know from memory. In real physics, the "tones" correspond to irreducible representations of the symmetries of the field. If you know the symmetries, it's not too difficult for modern, Earth mathematics to enumerate and describe all the irreducible representations. It should therefore be possible for scholars on Roshar to mathematically derive all the rhythms, tones, and colors for all the other shards. In total, they would then be able to create, for example, devotion-light or ambition-light without having heard the tones for devotion or ambition before. Maybe Connecting(?) light to other shards in this way is the method for moving it off Roshar. As one possible path to this, the equations stormwardens use to predict highstorms should have something to do with Honor's "pure tone"/irreducible representation. As anyone who's worked with difficult equations knows, the best tricks for dealing with them involve finding hidden symmetries, so I think the stormwardens might already know some. These hidden symmetries would be the symmetries of the "investiture field". Some stormwarden might, just out of curiosity, decide to figure out which other equations satisfy these hidden symmetries and suddenly discover the patterns governing other shards. This same type of story has played out a lot in the history of real-world math. Finally, this might give some insight into how Adonalsium was shattered. A good analogy might be splitting light into its pure colors by sending it through a prism. I'm going to run into an issue here of having lied in the pop-science section, but in some sense (that's an even worse lie), light in a vacuum has some extra symmetries that make all frequencies the same. In a prism, this symmetry is ruined and different frequencies travel at different speeds, separating them out. Maybe Adonalsium was shattered in a similar way: put into some medium where some symmetry of investiture was broken and then refracted into all the shards. Why RoW is amazing science fiction This fundamental idea--breaking up waves into pure tones--is something that I really don't see that much in even popular science, no matter how important it is to actual physics. Imagine then my shock then when the first mass media I see it in is, of all things, an ostensibly fantasy novel. It's not just a side-note; it's a critical piece of the magic system that I can use above to build wild theories involving real science concepts. This is what science fiction should be: deep concepts and ideas, not just the superficial trappings of the genre, like robots and spaceships or whatever. It is the pure opposite of technobabble. I didn't even talk here about how Navani's chapters present a more accurate view of what it feels like to work on a science problem than anything I've read before. She sees some strange phenomena that confuse her: the voidsphere explosion and the Thaylen guild's fabrial methods. She thinks deeply about all the implications of what they must mean. She then puts together some simple ideas derived from those in complicated ways, experiments with them, and figures out something amazing. This is what science is at its most basic level.