by Ian Tregillis
Well, here we are again. For the second time in a row, writing this blog post isn’t the carefree lark that I’d hoped (and assumed) it would be by the time my turn in the Wild-Cards-blog-barrel came around again. As I write this in 2022, we in the consortium continue to reel from the loss of another ace: our friend John Miller has joined the ranks of the fallen, taking with him a great big laugh, a love of baseball and animals and terrible movies, and more than a little Wild Cards history. John’s passing is a painful loss heaped atop a generally difficult and grim time.
And yet, just like last time, I’m still a giant nerd who happens to find escapist solace in the occasional thought experiment. And the foundational premise of Wild Cards offers one doozy of a gedankenexperiment. So what the hell.
This is the third entry in a slowly evolving and somewhat accidental series of posts exploring the physical and mathematical implications of the effect of xenovirus Takis-A, aka the Wild Card virus, on individuals and large populations. I’m playing a game here, and I invite you to join me. Given what’s been established canonically, in-universe, about how the Wild Card virus affects people and populations, what cool or interesting things might we deduce?
Previously, On Mathematical Wankery for Fun and (No) Profit…
In May of 2019, the first post in this series established three axioms:
- The Wild Cards universe simply must be rife with “crypto-jokers” and “crypto-aces.” These terms, coined by Sage Walker, aptly describe people mistakenly thought to be nats, even by themselves. Cryptos are walking cases of unidentified Takis-A expression — individuals for whom the effects of a turned card were so subtle as to go unnoticed, or were ascribed to more mundane causes. (Cryptos differ from those holding an ace up their sleeve.)
The logical inevitability of cryptos has an immediate corollary:
- Our longstanding epidemiological 90:9:1 rule of thumb (fatalities : jokers : aces) for Wild Card outcomes can only apply to known, identified cases of Takis-A expression. But we don’t, and can’t, know the true number of cases, thanks to those pesky cryptos, and therefore we can’t know the true percentages. Those ratios are subjective, not objective. This also means the mortality rate of the virus is not 90%, contradicting a longstanding article of faith.
And, last but not least,
- The effects of the Wild Card virus likely arise from a multivariate probability distribution.
Attempting to define ace-dom and joker-dom as extremes on a single one-dimensional continuum requires more logical contortions and ad-hoc assumptions than a satisfying model can accommodate. It also leads to problems of delineation, as explained below.
You can read the original post in full if you’d like to see the chain of reasoning underlying these conclusions. I know I’ve said this before, but as far as I’m concerned they’re really pretty logical.
Next, in September of 2020, the second post in this series used those axioms to develop what I call the “polar model” for Takis-A expression. In this picture, the outcome from every turned card is the result of sampling from two independent statistical distributions: S and θ. The former represents the “strength” or “severity” of a transformation, while the latter is what’s known as a “mixing angle” that combines ace-dom and joker-dom.
A key feature of the polar model is that every turned card represents a simultaneous combination, in varying amounts, of ace-ness and joker-ness, rather than being an either/or proposition. Though this may seem strange at first, the mixing angle avoids deep conceptual problems that inevitably arise in models dedicated to segregating aces and jokers into disparate, hard-and-fast categories. The Wild Cards universe is chockablock with joker-aces, yet shoehorning them into such a rigorously segregated model immediately presents a challenge: are joker-aces jokers, or are they aces? Where do they belong? Instead, the mixing angle approach tells us joker-aces are the only meaningful category. But sometimes people land closer to joker than ace, and viceversa, and sometimes they land somewhere in-between, so we perceive a variety of categories.
Another compelling feature of the polar model is that fatal outcomes, the so-called black queens, are a natural feature of the model. They don’t require an ad-hoc rule. They’re an inevitable consequence of the mixing-angle distribution. Cryptos are likewise a natural consequence of the model: they’re individuals who have undergone a low-severity (low S value) transformation. The model even accounts for “metapowered” aces: those like Genetrix, or the Sleeper, whose abilities subvert the usual (observed) Takis-A statistics.
As you can probably tell, I happen to think this mental picture for Takis-A expression has quite a lot going for it. And we still haven’t plumbed its depths.
Check out the full post if you’d like to see the full discussion of the polar model, complete with equations and pretty graphs. For the purposes of the following discussion, we’ll need the sector constraints previously derived from the polar model. These tell us “how much” of each probability distribution is located within particular sub-domains of the overall parameter space. Expressed in the language of calculus, and as functions of the true mortality fraction of the virus, μ, these are:
Here θ0 is the “joker-ace boundary”, and Sc is the subtlety threshold for cryptos.
Viral Expression as Wave Function Collapse
Before we dive back into the polar model, let’s take a moment to think about what happens when a latent carrier’s card turns. Fundamentally, it goes something like this.
Prior to the turn, the carrier has the potential to pull any allowed values from the severity and mixing-angle probability distributions: there are myriad possibilities. But after the turn, the carrier is permanently assigned a single, unchanging, S value and a single unchanging θ value. (Of course, there are very rare special individuals who can re-draw a new card; Croyd Crenson, our old friend The Sleeper, does this on the reg. And going back a few decades, Typhoid Croyd essentially shuffled the deck and dealt new cards to numerous people. Nevertheless, each individual draw always acts as described above.)
Framing things this way gives me déjà vu. Why? If you replace phrases like “card turn” or “viral expression” with “measurement,” then to a physicist this begins to sound like a description of basic quantum mechanics. Frankly, it sounds a hell of a lot like the Copenhagen Interpretation of QM. (One could also cast this discussion in terms of that evergreen darling of science fiction, the Many Worlds Interpretation of QM.)
Quantum systems are described by mathematical entities called wave functions. This wave function is related to the probability distribution of potential outcomes from a future measurement. According to Copenhagen, performing a measurement on that system causes the associated wave function to “collapse” such that a single outcome is chosen.
And a very similar thing happens when a latent carrier’s card turns. When the virus expresses in their body, the uncountable plethora of possible results collapses to a single location in the polar model’s two-dimensional parameter space. In other words, the viral expression randomly selects a single (S,θ) point and declares, “Aha! From now on, you’re right there.”
It may seem like a stretch to start calling manifestations of the Wild Card virus “measurements,” but a professional quantum mechanic wouldn’t bat an eyelash at this idea. Hell. By the standards of quantum mechanics, this doesn’t qualify as even remotely weird.
I’m not the first member of the WC consortium whose mental meanderings have taken them down this path. Not by a very long shot. OG Wild Carder Vic Milán put a quantum spin (that’s a joke for you übernerds out there) on the Takisian virus way back in the appendix to the very first volume of our wild unwieldy series.
A Tiger in the Jungle
Having put a pin in quantum mechanics, let’s turn our attention to the polar model’s probability distributions, P(S) and P (θ). Previously we used those “sector” constraints to derive expressions for the probabilities of jokers, aces, and cryptos in terms of the inherent-butunknown mortality fraction (aka the black queen probability) of the virus, μ, that preserve the 90:9:1 rule.
Crucially, that subjectively observed ratio (what we see happening in the world) is an artifact of the objective distributions (the underlying truth of the virus).
But why are those probability distributions the way they are? What creates them? If they were completely random, it might not be fruitful to speculate. But the sector constraints point to a deeper structure lurking within the apparent randomness. Like the gleam of a tiger’s eyes in a moonlit jungle, they are a hint, telling us, “Pssst. Look closer. There’s something here.”
I propose the probability distributions are an emergent phenomenon arising from the evolution of an abstract dynamical system.
Hey, where are you going? This is cool, I promise.
The Principle of Least Action & Lagrangian Mechanics
Sometimes known more formally as the Principle of Stationary Action, the Principle of Least Action (PLA) is a deep and ubiquitous concept across many different fields of physics.
Here I don’t mean “action” in the colloquial sense; this isn’t a statement about laziness or lethargy. “Action” is also a physics quantity which, in various forms, is applicable to a staggering variety of physical systems: a line-drive baseball, the Moon orbiting the Earth, a ray of light refracting through a lens, a photon bouncing off an electron… A strange yet Deep Truth of the universe is that systems tend to evolve in a way that minimizes the action. In principle, this single fact is the key to understanding innumerable physics scenarios. It’s amazing and weird.
Wherever one invokes the PLA, one often finds another concept lurking nearby: the Lagrangian.
In short, a Lagrangian is a special mathematical expression that, when combined with the Principle of Least Action, automagically generates the equations defining the dynamical evolution of the system of interest. (That automagical step involves solving something known as the Euler-Lagrange differential equation, but for the purposes of this post we needn’t worry about the details.) And just as the PLA is applicable to a myriad of physical systems, so, too, is the Lagrangian. One can define Lagrangians for systems mundane as a baseball hurled from center field to second base, or esoteric as the orbit of neutron stars in relativistic gravity and photons scattering off electrons in quantum field theory. A wonderful thing about the Lagrangian is that a practiced eye can read it and immediately deduce several physical properties of the associated system: its symmetries, its conservation properties, and more. So, if you know the Lagrangian, you know a lot. Obtaining a system’s Lagrangian is a bit like sequencing an organism’s DNA.
Ok. But what the hell does any of this have to do with latent carriers of xenovirus Takis-A? If you flip back to my previous post, you’ll see that a turned card in the polar model can be represented as an arrow pointing to the associated (S,θ) location. But until a latent carrier’s card turns, we don’t know where that arrow will point. Why not? Because the tip is constantly moving around. As time passes it spins, shrinks, expands, rocks back and forth, slows down, speeds up…
Each unturned card traces a continually evolving trajectory in the polar model’s configuration space. (It’s tempting to say this evolution is the defining characteristic of a latent carrier.) The moment a latent carrier’s card turns, that trajectory freezes and the arrow tip’s location in configuration space at that instant locks down the unique severity and mixing angle combination defining that person’s Wild Card fate.
If this abstract trajectory is sufficiently complex, and fast, it will be effectively unpredictable — thereby randomizing latent card turns.
Furthermore, if the arrow tip’s trajectory “fills” the configuration space in a finite amount of time, then the average fraction of time it spends in a given subdomain of the space determines the probability that we’ll find it there. For example, say that on average the mixing-angle component of the trajectory spends more time rattling between 117 degrees and 125 degrees than it does between 0 and 90 degrees. This means that when the latent card turns — whether that be 15 minutes from now or 30 years hence — its probability of drawing a mixing angle with 117 < θ < 125 is higher than the probability of drawing an angle with 0 < θ < 90.
In this picture, the probability sector constraints emerge from the phantom trajectory’s “dwell time” in the associated portions of configuration space.
The dwell time is easiest to define mathematically if we imagine a system that is periodic. If, for instance, the mixing angle component of the trajectory always repeats itself after a time Tθ, then the dwell time in a given range of theta values can be thought of as a fraction of Tθ. Similarly for the transformation severity, which may have its own period, TS. (As far as we know, the severity and mixing angle are completely uncorrelated during random draws from the bivariate distribution. But the polar model could easily accommodate such a correlation, were such a thing detected.) If the two dynamical periods differ by an irrational factor, then they’ll never sync up, and the resulting trajectory can fill the configuration space. The technical term for this is ergodicity. (Nerdier readers might point out the dynamical system could even embody deterministic chaos, though that’s not the case for the candidate solutions presented here.)
What are those dynamical timescales? We don’t know. But we do know that the very first people exposed to the Wild Card virus started turning cards, with wildly random outcomes, within minutes of Jetboy’s failure in the New York sky. This might indicate the dynamical randomization process gets underway well before somebody even gets exposed to a spore. Alternatively, if the configuration space trajectory kicks in only after somebody becomes a latent carrier, the dynamical randomization process is fast. The poor souls Croyd Crenson witnessed on his walk home that afternoon had been latent carriers for mere minutes, perhaps even seconds! Yet already there was no rhyme or reason to the horrors he witnessed. So Tθ and TS are probably very short compared to human timescales. (And this makes sense, if there’s something quantum mechanical going on. In physics, a good rule of thumb is smaller = faster.)
The Mixing Angle Lagrangian
This is all fine and good. But let’s dispense with pie-in-the-sky speculation and turn this into something concrete.
The sector constraints tell us the mixing-angle component of the hypothetical trajectory must spend a fraction μ of each cycle in the domain 90∘ < θ < 360∘, where μ, the unknown “true” mortality fraction of the virus, is a number greater than 0 but less than 0.9. This means that the mixing angle must spend the remainder of that period, 1 − μ, in the “nonlethal quadrant,” 0∘ ≤ θ ≤ 90∘. The simplest and most natural way to build a function with built-in periodicity is by combining trigonometric functions, sines and cosines. So as a thought experiment, let’s see if we can construct a mixing angle trajectory of the generic form
θ (t) = A ⋅ cos[x (t)] + B ⋅ sin[x (t)] + C
which has the requisite “dwell time” behavior in various portions of the circle 0∘ ≤ θ (t) ≤ 360∘ such that it generates the requisite probability sector constraints.
This may seem like an arbitrary claim to make without slogging through every step of the algebra (I’ll spare you), but with a bit of scratch paper one finds a natural approach is to devise a companion function, θ˜ (t) = θ (t) − 90∘, that:
- Starts at its lowest allowed negative value at the beginning of the cycle, t=0;
- Stays negative until time t = (1 − μ) ⋅ Tθ, at which point the function value is 0; then
- Stays positive until the end of the period, reaching its highest allowed value at t = Tθ.
With a little more (ok, maybe a lot more) scratch paper, one can derive a candidate solution that meets these criteria:
This formula for x is expressed in terms of radians rather than degrees, as radians are the natural unit for these kinds of musings. If you’d like to pull out a calculator and play around, replacing π with 180 degrees will get you from radians to degrees. But such details aside, this is really pretty obvious in retrospect, right?
I’m kidding. It’s ghastly. But it works.
Here are the candidate solutions for a few representative values of the mortality fraction:
The dotted lines represent our proposed solutions for θ˜ at μ = 0.25, 0.50, and 0.75; compare them to the criteria I listed a moment ago, and you’ll see they perform as advertised. The solid lines are our proposed solutions for the mixing angle trajectory itself, θ (t). The nature of the proposed solution depends on the inherent mortality fraction of the virus.
Because we’re talking about an angle, it’s more natural to plot the trajectory in the coordinate system of our polar model. When mapped to a circle, the above mixing angle trajectories look something like this:
Here, for the sake of readability, I’ve forced the radius of each trajectory (the colorful lines) to increase linearly from 0 at time t=0 to 1 at time t=Tθ. In reality, the radial part of the full trajectory will correspond to the transformation severity, S(t) (see below). Notice how the proposed solution begins to oscillate when the mortality fraction gets sufficiently small (the blue line). We’ll come back to that in a bit.
We’re off to the races! We now have enough information to start deriving solutions for previously inaccessible aspects of the model.
For instance, if you look closely at the preceding plot, you’ll notice dotted lines marking a special angle θ0 (very very close to 0) for each μ value. Remember that the polar model defines θ0 as the special mixing angle that marks the boundary between aces and jokers. (As I said above, the polar model considers everybody a joker-ace. But in the world of Wild Cards, we talk and think so much in terms of aces and jokers as distinct categories that any proposed model must have a means of delineating those categories, regardless of how arbitrary they are.) Previously, we couldn’t say where that boundary landed, only how it was related to the sector constraints. But armed with a candidate solution for θ (t), we can calculate the location of that boundary directly.
And it turns out that boundary angle is very tiny indeed: so-called “pure” aces comprise a minuscule fraction of the overall configuration space.
And we’re barely getting started. The cumulative (integrated) mixing angle probability distributions, P (θ), corresponding to the above solutions look like this:
They start at 0 and rise to 1 − μ at exactly ninety degrees: precisely what the sector constraints require. Better still, using a little bit of calculus, we can transform those “cumulative” probability distributions into candidates for the actual probability density functions (PDFs, in stats jargon). They look like this:
(The probability density functions give us the relative likelihood of a given value across the entire domain, which is why we see values exceeding unity.) These plots are a little bit noisy, owing to the low-order numerical differentiation I used to derive them. But, looking past that minor calculational artifact, these distributions are eye-opening. My naive expectation, based on nothing more than the generally fatal nature of the virus, was that the probability distribution would be higher in the fatal quadrants (the “Black Queen Badlands”) and lower in the nonlethal sector. Instead, our candidate solution — which is certainly not the only possible solution, but probably one of the simplest — is cusped at the boundaries. (Remember that 0 degrees and 360 degrees are the same here; we’re walking around a circle.)
But numbers don’t lie. These candidate probability distributions for the mixing angle conform exactly to the sector constraints we derived previously. Better still, they were generated by a hypothetical trajectory in an abstract configuration space.
Speaking of which, and getting back to the point of all this, we can reverse-engineer a Lagrangian that generates these trajectories via the Principle of Least Action. Without going into the details of the derivation (it’s not a terrible slog, but it is a tossed salad of calculus, algebra, and scratch paper), I present our candidate for the Mixing Angle Lagrangian:
, the dots represent the instantaneous rate of change of the time-dependent quantities (time derivatives, in calculus jargon, with two dots denoting a second derivative), and the function x (t) is as defined above. Conveniently, the letter L stands for both Lagrangian and Latent Carrier.
This Lagrangian has some interesting properties. Without going into a long digression about how one deciphers these things (isn’t this post discursive enough already?) I’ll just mention that its functional form is effectively equivalent to a special case of a mass attached to a spring (for the übernerds reading this, it’s a forced harmonic oscillator with a time-dependent spring constant). More significantly, the fact that the Lagrangian is explicitly time dependent (via x) indicates this is not a closed system. In other words, as the mixing angle trajectory evolves it continually exchanges energy with an external reservoir.
Holy macaroni. We’ve just established — by accident — a beachhead for attacking the most inconvenient question haunting any attempt to put Wild Cards on a defensible scientific footing: what is the energy source that drives ace powers? (And yes. I do understand this is all comicbook physics and pesky ideas like conservation of energy got defenestrated on the very first Wild Card Day.) When Tesseract opens a hole in space, or Popinjay teleports people across the city, or Tinker drives around town in a car fueled by nothing more than a cupful of water, there’s clearly a lot of unaccounted-for energy at work. We might not know exactly how Tesseract does what she does, but our current understanding of physics tells us that, at the very least, her portals must be incomprehensible energy hogs. All Wild Cards powers raise this question, to varying degrees.
We don’t (yet) know what this phantom energy source is, but we can see it at work in the Lagrangian. The flow of that energy, whatever the hell it is, controls the mixing angle trajectory.
So far, we only have half of a solution. We also need the severity trajectory, S(t). But before we take aim at that, we have to confront a rather icky caveat.
Remember how I pointed out that our proposed solution for the mixing angle trajectory begins to oscillate at low values of μ? This is a hint that our simplest-imaginable functional form for θ (t) is too simple. We constructed our proposed solution based on a companion function θ˜ (t) that must rise from a value of 0 at time (1 − μ)Tθ to a value of 270 degrees at time Tθ, staying positive throughout that interval. But the lower μ gets, the shorter that interval of time becomes, and the sharper the required transition. There comes a point at which the required transition is simply too sharp for our simplistic combination of a single cosine and a single sine function. When that happens, the oscillations will grow until θ˜ (t) crosses the zero line multiple times, and our conceptual solution fails. (The übernerds in the audience might recognize that I’m effectively describing the Gibbs phenomenon in Fourier analysis.) The good news is that there’s a way around this: it’s an extremely well-known fact that any periodic function, no matter how intricate, can be represented as the sum of a (possibly infinite) series of sines and cosines of varying frequencies. In principle, we could rewrite our solution to accommodate multiple frequency components. The bad news is that the requisite equations rapidly become impossible to solve by hand. It could be done numerically via computer, but that isn’t so satisfying. Long story short, our proposed ultra-simplistic solution is really only valid for mortality fractions in the range 0.195 < μ ≤ 0.9. (The same issue rears its head if the mortality fraction approaches 100%. But this isn’t a problem because, as we discovered previously, the inevitable existence of that nonzero crypto population caps the mortality fraction at 90%.)
As I’ve said, our proposed solution for the mixing angle trajectory is about as simple as one could imagine. In particular, we’re only considering solutions with the lowest possible oscillation frequency. But it’s easy to imagine higher-frequency solutions, too. (Perhaps you were wondering why the subtitles on the preceding plots make cryptic references to “n = 1.”) In fact… since the number of half-cycles in any proposed solution component must be a positive integer — let’s call it n — there are solutions for n=1, n=2, n=3, and so on. In other words, solutions for the mixing angle trajectory are quantized. And this takes us right back to quantum mechanics: in that world, n would be called the principle quantum number.
If the spirit moves you, feel free to explore the spectrum of n > 1 solutions. Go nuts. Perhaps I’ll revisit this in a future blog post.
The Transformation Severity Lagrangian
OK. We’re halfway there. But what about the transformation severity, S?
Candidate solutions for the S(t) trajectory present their own peculiar challenges. Recall that the polar model places no upper limit on S, allowing it to theoretically approach infinity in extraordinarily rare cases. (Think of it this way: the Wild Cards canon has never definitively ruled out the possibility that a card turn could transform somebody into an omnipotent being with essentially unlimited abilities. Maybe it happens only once in a very great while, but when it does the lucky mega-ace rapidly loses interest in petty humanity and Dr. Manhattans off to do gods-know-what gods-know-where, so we never hear or see them in canon.) Thus, viable candidates for the transformation severity trajectory have to be mathematical functions that diverge to infinity in finite time: they must be asymptotic. Furthermore, to keep things “simple” and periodic, they have to return from infinity also in finite time. The (finite) timescale for S to go from 0 (the origin of our polar coordinate system) to arbitrarily close to infinity — let’s call this Tm — need not be the same as the (finite) timescale for S to drop down from arbitrarily close to infinity back to 0, which we’ll call Tp. The overall period for the transformation severity, Ts , will be the sum of those two timescales. (As with the mixing angle trajectory, S(t) need not be truly periodic. This simplifying assumption keeps the math tractable.)
Just as with the candidate solutions for the mixing angle, θ(t), there are infinite candidates for S(t). Here I’m jotting down the simplest example I can devise. Because it’s necessarily cleaved in two parts by a vertical asymptote, any solution for S(t) will have two parts, one for times earlier than the asymptotic time, and one for times later than that. Given arbitrary constants η and ζ > 0,
The plot shows one example of what such a function for S(t) might look like. The location of Tm is a free parameter; here I’ve arbitrarily chosen a value for the sake of illustration. Notice how this trajectory starts at S=0 at t=0, shoots off toward infinity as t approaches Tm, and then drops back down to S=0 at t=Ts. I capped the upper edge of this plot at S=15, just to make it legible, but in reality that black line for S(t) would keep rising no matter how large we made the plot.
Notice one big difference between this simplest proposed solution for S(t) and that for θ(t): the latter was explicitly a function of the mortality fraction, μ, but this is not. It could be, of course, if we chose to define η or ζ as functions of μ, but as far as I can tell there’s no need to do this. This differs from θ(t) where the mortality fraction crowbarred itself into even the simplest imaginable solutions, whether we wanted it to or not.
The dashed horizontal lines define the crypto subtlety threshold, Sc, which is where the mortality fraction enters the picture for S(t). With a bit more algebra, one can prove that in order for Sc to satisfy the required sector constraints, it must be the solution to a quadratic equation:
The cumulative (integrated) transformation severity probability distribution, P (S), corresponding to the above solution looks like this:
Again, I’ve capped this plot at S=15 for legibiltiy’s sake, but the cumulative probability only reaches unity as the severity goes to infinity. We can also compute the probability density function for P(S), by numerically differentiating the above plot. The result isn’t particularly interesting; we get a function that is highest near S=0 and drops sharply as S increases. This tells us the severity trajectory in this proposed solution spends a good portion of its time near the origin, occasionally shooting off on, and returning from, excursions to infinity.
Ok. So what about the Lagrangian for the severity trajectory? Once again, I’ll spare you the details of the derivation (another tossed salad of algebra, calculus, scratch paper, and the abovementioned Euler-Lagrange equation) and present the severity Lagrangians corresponding to our proposed solution (again, we have one for each portion of the bifurcated time domain):
As before, the dots represent time derivatives. Notice that once again, the proposed Lagrangian carries an explicit time dependence: as with the mixing angle Lagrangian, this tells us that as the severity trajectory evolves, it continually exchanges energy with an external reservoir. Is it the same reservoir that controls the mixing angle trajectory? Maybe, maybe not. But once again, we’re getting a glimpse of the dark beast lurking at the very heart of the number one question about xenovirus Takis-A: what is the energy source that enables ace powers?
Putting it All Together
The full Lagrangian encoding the complete time-dependent trajectory of the arrow tip through the configuration space of the polar model is simply the sum of the individual Lagrangians:
And there we have it! A detailed model underlying the emergent behavior of the Wild Card virus, based on Lagrangian mechanics and the principle of least action.
Of course, now that we have proposed solutions for both pieces of the time-dependent trajectory, both the mixing angle and the severity, it’d be a shame, maybe even a crime, not to glue them together and see what we get. This is very very cool.
First, let’s consider the lowest-mortality scenario peppered through many of our example calculations, μ = 0.25.
When the mixing angle and severity trajectories share the same period, Tθ = TS, then clearly the trajectory will form a closed loop in the configuration space, returning to its initial state after each cycle. Here’s what a single cycle looks like when we zoom in near the origin of our polar model coordinate system:
Here I’ve color-coded things such that the line color is deep red at t=0 and transitions smoothly to deep blue at t=Ts = Tθ. The complete loop doesn’t fit in the plot window because, of course the severity trajectory — the radial component of this line — includes a brief excursion where it shoots off to infinity. We can get a better sense of that excursion if we zoom out a bit:
The dotted line is our old friend the joker/ace boundary, θ0, and the dashed line is the crypto subtlety threshold, Sc. (The spacing between the trajectory points increases at large severity values because I computed this trajectory image at a set of evenly spaced times, but the tip of the arrow is moving very quickly near the severity asymptote.)
Because both components share the same period, this scenario will never fill the configuration space in the way is must if the outcomes from card turns are to be truly unpredictable. For instance, after 10 cycles, we haven’t gained any new ground:
All that’s changed is that the trajectory line has gotten bluer and bluer as it keeps writing over itself at later times. This confirms that if we want card turn outcomes to be wildly unpredictable, the severity and mixing angle trajectories can’t share the same period.
Well, OK. What if they’re not equal, but instead differ by some simple ratio? Say, 3/2? Here’s what that looks like after 1, 2, 3, and 25 periods:
It takes longer before both trajectories sync up but, because they differ by a simple ratio, they share what’s called a resonance. Run the trajectory long enough and it will still repeat, just like before, because you start picking up common integer multiples of both components. In this case, every two cycles of the mixing angle trajectory exactly equal three cycles of the severity trajectory. A similar relationship would result from any integer ratio. So, once again, this fails to give us the configuration-space-filling trajectory that we desire for truly unpredictable card turns.
But we already know how to get that result. Or, at least, I claimed there was a way to do it. Remember that earlier I said irrational numbers were the key to ergodicity. So, what if the mixing angle and severity trajectory periods differ by an irrational factor? Say, the square root of two? Well, that’s when things get very interesting.
So far, not terribly different from the other scenarios. But here’s what we get after letting the trajectory evolve for 25 cycles of the severity period:
Yikes! Suddenly the trajectory through the polar model’s configuration space starts looping all over the place. Feast your eyes on the result after 250 cycles:
Gorgeous, right? We’ve already hit pretty much every location within S < 1.
(Speaking of which, the units on the severity calculation here are arbitrary “wild cards” units, something probably only the Takisian engineers behind the virus truly understand. But in the grand scientific tradition of naming units after prominent scientists or discoverers, I propose that we name the base unit of transformation severity the Turtle, after the Great and Powerful one. A deuce like Father Henry may have a transformation severity measured in milliTurtles [mTu], while a powerful ace like Tesseract might enjoy an S value on the order of a Turtle [Tu] or two. Mark Meadows may have had an S value approaching a decaTurtle [dTu].)
But that’s just a small window on the vast space of the polar model. What does this look like if we zoom out to, say, S=40? This:
That’s just 250 cycles. And remember, we’ve speculated that the dynamical timescale might align with the microscopic length scales (we’re talking about a virus and molecules, after all). So a thousand cycles, or even a million cycles, could still be on the order of one second of human wall-clock time. This trajectory will have no trouble filling the space. Pretty great, right?
Here’s what 250 cycles looks like in the middle case we’ve considered, μ = 0.50, for S < 1:
and zoomed out to S < 40:
And here are the analogous results for the highest mortality example we’ve included in our example calculations, μ = 0.75, for S <1
and S< 40:
I don’t know about you, but I find these plots extremely beautiful.
There’s an interesting trend to these zoomed-out plots as the mortality fraction, μ, increases. Notice that when the mortality fraction is relatively low, 25%, the trajectory is predominantly concentrated — more predominantly blue — in the polar model’s nonlethal quadrant, between 0 and 90 degrees. That’s what we’d expect if the wild card virus is predominantly benign. When the mortality fraction is exactly 50%, the predominant concentration is smeared a little more evenly around the entire range of mixing angles. Not entirely evenly, but noticeably so. And when the virus has a high mortality fraction, 75%, the trajectory is predominantly concentrated outside the nonlethal quadrant. Qualitatively speaking, a trend like this is exactly what we’d expect to see, based on how we defined the polar model in the first place. That’s a nice, reassuring sanity check.
What A Long, Strange Trip It’s Been
This is a very long blog post… built upon two earlier, similarly long blog posts. It’s been a long walk to get here. So maybe, before wrapping up the current discussion, we should take a step back and look at the big picture. Where are we now, and what have we learned?
Rather than recapping things in the order we deduced them over the course of 3 blog posts, let’s instead arrange things more logically. Let’s start at the most fundamental level, the deep-down physics and math that we postulate lie at the heart of xenovirus Takis-A, and then see how the usual results we know and love grow out of that foundation.
- When the Wild Card virus infects a person, it initially lies dormant. The dormancy period may be very short, a few seconds, or very long — many years.
- While inside that latent carrier, the virus is fundamentally defined by two things: an unknown mortality fraction value, μ, and a mathematical expression known as the Lagrangian, L. The latter encodes, in a deep way, the vast spectrum of potential effects the virus, if it expresses, may have on its host.
- Via the physics Principle of Least Action, the Lagrangian establishes a unique time-dependent trajectory of that virus through an abstract two-dimensional parameter space. Every location in this space is associated with a “severity”, S, and a “mixing angle,” θ. This is not a physical motion by the virus; it is motion through an abstract mathematical space.
- The trajectory rapidly fills that parameter space in a finite time that is short relative to human timescales. In other words, at any given time, the virus could be anywhere in that space. Furthermore, and as dictated by the Lagrangian, this trajectory spends very specific amounts of time (the “dwell time”) in very specific sub-portions (“sectors”) of this parameter space.
- When the latent carrier’s card turns, the virus’s transit along that abstract trajectory instantly stops. Its location in the parameter space at that moment chooses a specific (S, θ) pair that permanently defines that person’s Wild Card fate: black queen, joker, ace, or crypto.
- These randomly established values of S and θ each conform to a probability distribution controlled by the sector dwell times (the sector constraints).
- These emergent probability distributions are structured such that, regardless of the unknown mortality fraction, the subjectively observed ratios of wild card outcomes in the population at large conform to the well-known and canonical 90 : 9 : 1 rule for fatalities : jokers : aces.
- The resulting population automatically includes an unknown but nonzero number of crypto-aces and crypto-jokers.”
And there we have it: a path from abstract Lagrangian mechanics to the defining characteristic of the Wild Card virus, the empirically observed 90 :
9 : 1 rule.
Of course, the specific example solutions I’ve explored here are merely possibilities, and relatively simple ones at that (simple enough to be derivable with pencil and paper, anyway). Given the information we have on hand, the number of possible solutions is infinite. But some of these could be quite interesting. For instance, I’ve already pointed out that possible solutions for the mixing angle will be inherently quantized. And although I’ve deliberately sidestepped the possibility in this post, a similarly quantized spectrum of solutions might also exist for the transformation severity. I leave those explorations to interested readers, or perhaps a future blog post.