Union LA x Air Jordan 2 “Future Is Now” PREVIEW
With the help of Virgil Abloh and Union LA‘s Chris Gibbs, it's now clear that Jordan Brand intended to bring the Air Jordan 2 back in 2022.
The “Future Is Now” collection includes two colorways of MJ's second signature as well as an extensive range of apparel and accessories.
“We wanted to juxtapose what some futuristic gear might look like after being worn and patina'd,”
Union stated on the collaboration's landing page.
“You often see people's future visions that are crisp and sterile. We thought it would be cool to wear it in and make it organic...”
The classic co-branding appears on short-sleeve tees, hoodies, and sweat shorts/sweat pants, all lightly distressed at the hems and seams.
Also, a filtered black-and-white photo of MJ graces the adjacent long sleeves, labels stitch into the socks, and the Jumpman logo adorns the four caps.
Liner jackets and flight pants will also be available, adding reimagined militaria to a civilian ensemble.
The Union LA x Air Jordan 2 (Grey Fog and Rattan) shares many of the same beats. Vintage suedes show age, while perforations and detailing reimagine Bruce Kilgore's design for the future.
The “UN/LA” tag across the modified eye stays, the leather patch across the tongue, and the label that wraps over the lateral side of the collar complete the look.
The footwear will also include a Crater Slide in the “Grey Fog” color scheme.
BUYING
On 4/9 and 4/10 from 9am-3pm, Union LA will be giving away a pair of Air Jordan 2s at their La Brea storefront (110 S. LA BREA AVE. LA, CA 90036). The raffle is only open to LA County residents with a valid CA ID. You must enter by 11:59pm on 4/10 to win. Winners will be notified via email.
More on Lifestyle
Joanna Henderson
2 years ago
An Average Day in the Life of a 25-Year-Old -A Rich Man's At-Home Unemployed Girlfriend
And morning water bottle struggles.
Welcome to my TikTok, where I share my stay-at-home life! I'll show you my usual day from morning to night.
I rise early to prepare my guy iced coffee. I make matcha, my favorite drink. I also fill our water bottles, which takes time and effort, so I record and describe the procedure. As you see me perform the unthinkable by putting a water bottle in a soda machine, you'll see my magnificent but unowned condo. My lover has everything, including:
In the living room, a sizable velvet alabaster divan. I was unable to use the words white or sofa in place of alabaster or a divan since they are insufficiently elegant and do not adequately convey how opulent the item is. The price tag on the divan was another huge feature; I'm sure my lover wouldn't purchase any furniture for less than $20k because it would be beneath him.
A plush Swiss coffee-colored Tabriz carpet. Once more, white is a color associated with the underclass; for us, the wealthy, it's alabaster or swiss coffee. Sorry, my boyfriend is wealthy; I'm truly in the same situation. And yet, I’m the one whos freeloading off of him, not you haha!
Soft translucent powder is the hue of the vinyl wallcoverings. I merely made up the name of that hue, but I have to maintain the online character I've established. There is no room for adopting language typical of peasant people; I must reiterate that I am wealthy while they are not.
I rest after filling our water bottles. I'm really fatigued from chores. My boyfriend is skeptical about hiring a housekeeper and cook. Does he assume I'm a servant or maid? I can't be overly demanding or throw a tantrum since he may replace me with a younger version. Leonardo Di Caprio's fault!
After the break, I bring my lover a water bottle. He's off to work with my best wishes. After cleaning the shower, I text my BF saying I broke a nail. He charged $675 for a crystal-topped shellac manicure. Lucky me!
After this morning's crazy choirs, especially the water bottle one, I'm famished. I dress quickly and go to the neighborhood organic-vegan-gluten-free-sugar-free-plasma-free-GMO-free-HBO-free breakfast place. Most folks can't afford $17.99 for a caffeine-free-mushroom-plus-mud-and-electrolytes morning beverage. It goes nicely with my matcha. Eggs Benedict cost $68. English muffins are off-limits. I can't make myself obese. My partner said he'd swap me for a 19-year-old Eastern European if I keep eating bacon.
I leave no tip since tipping is too much pressure and math for me, so I go shopping.
My shopping adventures have gotten monotonous. 47 designer bags and 114 bag covers Birkins need their own luggage. My babies! I've never caught my BF with a baby. I have sleeping medications and a turkey baster. Tatiana is much younger and thinner than me, so I can't lose him to her. The goal is to become a stay-at-home wife shortly. A turkey baster is essential.
After spending $955 on La Mer lotions and getting a crystal manicure, I nap. Before my boyfriend's return, I can nap for 5 hours.
I wake up around 4 pm — it’s time to prepare dinner. Yes, I said “prepare for dinner,” not “prepare dinner.” I have crystals on my nails! Do you really think I would cook? No way.
My husband's arrival still requires much work. I clean the kitchen, get cutlery and napkins. I order UberEats while my BF is 30-45 minutes away.
Wagyu steaks with Matsutake mushroom soup today. I pick desserts for my lover but not myself. Eastern European threat?
When my BF gets home from work, we eat. I don't believe in tipping UberEats drivers. If he wants to appreciate life's finer things, he should locate a rich woman.
After eating, we plan our getaway. I requested Aruba's fanciest hotel for winter and expect a butler. We're bickering over who gets the butler. We may need two.
Day's end, I'm exhausted. Stay-at-home girlfriends put in a lot of time and work. Work and duties are never-ending.
Before bed, I shower and use a liquid gold mask in my 27-step makeup procedure. It's a French luxury brand, not La Mer.
Here's my day.
Note: I like satire and absurd trends. Stay-at-home-girlfriend TikTok videos have become popular recently.
I don't shame or support such agreements; I'm just an observer. Thanks for reading.
Marcus Lu
2 years ago
The Brand Structure of U.S. Electric Vehicle Production
Will Tesla be able to maintain its lead in the EV market for very long?
This is one of the most pressing issues in the American auto sector today. One positive aspect of Tesla is the company's devoted customer base and recognizable name recognition (similar to Apple). It also invests more in research and development per vehicle than its rivals and has a head start in EV production.
Conversely, established automakers like Volkswagen are actively plotting their strategy to surpass Tesla. As the current market leaders, they have decades of experience in the auto industry and are spending billions to catch up.
We've visualized data from the EPA's 2022 Automotive Trends Report to bring you up to speed on this developing story.
Info for the Model Year of 2021
The full production data used in this infographic is for the 2021 model year, but it comes from a report for 2022.
Combined EV and PHEV output is shown in the table below (plug-in hybrid electric vehicle).
It is important to note that Toyota and Stellantis, the two largest legacy automakers in this dataset, only produced PHEVs. Toyota's first electric vehicle, the bZ4X, won't hit the market until 2023.
Stellantis seems to be falling even further behind, despite having enormous unrealized potential in its Jeep and Ram brands. Stellantis CEO Carlos Tavares said in a recent interview that the firm has budgeted $36 billion for electrification and software.
Legacy Brands with the Most Momentum
In the race to develop electric vehicles, some long-standing manufacturers have gotten the jump on their rivals.
Volkswagen, one of these storied manufacturers, has made a significant investment in electric vehicles (EVs) in the wake of the Dieselgate scandal. The company plans to roll out multiple EV models, including the ID.3 hatchback, ID.4 SUV, and ID. Buzz, with the goal of producing 22 million EVs by 2028. (an electric revival of the classic Microbus).
Even Ford is keeping up, having just announced an EV investment of $22 billion between 2021 and 2025. In November of 2022, the company manufactured their 150,000th Mustang Mach-E, and by the end of 2023, they hoped to have 270,000 of them in circulation.
Additionally, over 200,000 F-150 Lightnings have been reserved since Ford announced the truck. The Lightning is scheduled to have a production run of 15,000 in 2022, 55,000 in 2023, and 80,000 in 2024. Ford's main competitor in the electric pickup truck segment, Rivian, is on track to sell 25,000 vehicles by 2022.
Josh Chesler
3 years ago
10 Sneaker Terms Every Beginner Should Know
So you want to get into sneakers? Buying a few sneakers and figuring it out seems simple. Then you miss out on the weekend's instant-sellout releases, so you head to eBay, Twitter, or your local sneaker group to see what's available, since you're probably not ready to pay Flight Club prices just yet.
That's when you're bombarded with new nicknames, abbreviations, and general sneaker slang. It would take months to explain every word and sneaker, so here's a starter kit of ten simple terms to get you started. (Yeah, mostly Jordan. Does anyone really start with Kith or Nike SB?)
10. Colorways
Colorways are a common term in fashion, design, and other visual fields. It's just the product's color scheme. In the case of sneakers, the colorway is often as important as the actual model. Are this year's "Chicago" Air Jordan 1s more durable than last year's "Black/Gum" colorway? Because of their colorway and rarity, the Chicagos are worth roughly three pairs of the Black/Gum kicks.
Pro Tip: A colorway with a well-known nickname is almost always worth more than one without, and the same goes for collaborations.
9. Beaters
A “beater” is a well-worn, likely older model of shoe that has significant wear and tear on it. Rarely sold with the original box or extra laces, beaters rarely sell for much. Unlike most “worn” sneakers, beaters are used for rainy days and the gym. It's exactly what it sounds like, a box full of beaters, and they're a good place to start if you're looking for some cheap old kicks.
Pro Tip: Know which shoes clean up nicely. The shape of lower top sneakers with wider profiles, like SB Dunk Lows and Air Jordan 3s, tends to hold better over time than their higher and narrower cousins.
8. Retro
In the world of Jordan Brand, a “Retro” release is simply a release (or re-release) of a colorway after the shoe model's initial release. For example, the original Air Jordan 7 was released in 1992, but the Bordeaux colorway was re-released in 2011 and recently (2015). An Air Jordan model is released every year, and while half of them are unpopular and unlikely to be Retroed soon, any of them could be re-released whenever Nike and Jordan felt like it.
Pro Tip: Now that the Air Jordan line has been around for so long, the model that tends to be heavily retroed in a year is whichever shoe came out 23 (Michael Jordan’s number during the prime of his career) years ago. The Air Jordan 6 (1991) got new colorways last year, the Air Jordan 7 this year, and more Air Jordan 8s will be released later this year and early next year (1993).
7. PP/Inv
In spite of the fact that eBay takes roughly 10% of the final price, many sneaker buyers and sellers prefer to work directly with PayPal. Selling sneakers for $100 via PayPal invoice or $100 via PayPal friends/family is common on social media. Because no one wants their eBay account suspended for promoting PayPal deals, many eBay sellers will simply state “Message me for a better price.”
Pro Tip: PayPal invoices protect buyers well, but gifting or using Google Wallet does not. Unless you're certain the seller is legitimate, only use invoiced goods/services payments.
6. Yeezy
Kanye West and his sneakers are known as Yeezys. The rapper's first two Yeezys were made by Nike before switching to Adidas. Everything Yeezy-related will be significantly more expensive (and therefore have significantly more fakes made). Not only is the Nike Air Yeezy 2 “Red October” one of the most sought-after sneakers, but the Yeezy influence can be seen everywhere.
Pro Tip: If you're going to buy Yeezys, make sure you buy them from a reputable retailer or reseller. With so many fakes out there, it's not worth spending a grand on something you're not 100% sure is real.
5. GR/Limited
Regardless of how visually repulsive, uncomfortable, and/or impractical a sneaker is, if it’s rare enough, people will still want it. GR stands for General Release, which means they're usually available at retail. Reselling a “Limited Edition” release is costly. Supply and demand, but in this case, the limited supply drives up demand. If you want to get some of the colorways made for rappers, NBA players (Player Exclusive or PE models), and other celebrities, be prepared to pay a premium.
Pro Tip: Limited edition sneakers, like the annual Doernbecher Freestyle sneakers Nike creates with kids from Portland's Doernbecher Children's Hospital, will always be more expensive and limited. Or, you can use automated sneaker-buying software.
4. Grails
A “grail” is a pair of sneakers that someone desires above all others. To obtain their personal grails, people are willing to pay significantly more than the retail price. There doesn't have to be any rhyme or reason why someone chose a specific pair as their grails.
Pro Tip: For those who don't have them, the OG "Bred" or "Royal" Air Jordan 1s, the "Concord" Air Jordan 11s, etc., are all grails.
3. Bred
Anything released in “Bred” (black and red) will sell out quickly. Most resale Air Jordans (and other sneakers) come in the Bred colorway, which is a fan favorite. Bred is a good choice for a first colorway, especially on a solid sneaker silhouette.
Pro Tip: Apart from satisfying the world's hypebeasts, Bred sneakers will probably match a lot of your closet.
2. DS
DS = Deadstock = New. That's it. If something has been worn or tried on, it is no longer DS. Very Near Deadstock (VNDS) Pass As Deadstock It's a cute way of saying your sneakers have been worn but are still in good shape. In the sneaker world, “worn” means they are no longer new, but not too old or beat up.
Pro Tip: Ask for photos of any marks or defects to see what you’re getting before you buy used shoes, also find out if they come with the original box and extra laces, because that can be a sign that they’re in better shape.
1. Fake/Unauthorized
The words “Unauthorized,” “Replica,” “B-grades,” and “Super Perfect” all mean the shoes are fake. It means they aren't made by the actual company, no matter how close or how good the quality. If that's what you want, go ahead and get them. Do not wear them if you do not want the rest of the sneaker world to mock them.
Pro Tip: If you’re not sure if shoes are real or not, do a “Legit Check” on Twitter or Facebook. You'll get dozens of responses in no time.
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Thomas Smith
2 years ago
ChatGPT Is Experiencing a Lightbulb Moment
Why breakthrough technologies must be accessible
ChatGPT has exploded. Over 1 million people have used the app, and coding sites like Stack Overflow have banned its answers. It's huge.
I wouldn't have called that as an AI researcher. ChatGPT uses the same GPT-3 technology that's been around for over two years.
More than impressive technology, ChatGPT 3 shows how access makes breakthroughs usable. OpenAI has finally made people realize the power of AI by packaging GPT-3 for normal users.
We think of Thomas Edison as the inventor of the lightbulb, not because he invented it, but because he popularized it.
Going forward, AI companies that make using AI easy will thrive.
Use-case importance
Most modern AI systems use massive language models. These language models are trained on 6,000+ years of human text.
GPT-3 ate 8 billion pages, almost every book, and Wikipedia. It created an AI that can write sea shanties and solve coding problems.
Nothing new. I began beta testing GPT-3 in 2020, but the system's basics date back further.
Tools like GPT-3 are hidden in many apps. Many of the AI writing assistants on this platform are just wrappers around GPT-3.
Lots of online utilitarian text, like restaurant menu summaries or city guides, is written by AI systems like GPT-3. You've probably read GPT-3 without knowing it.
Accessibility
Why is ChatGPT so popular if the technology is old?
ChatGPT makes the technology accessible. Free to use, people can sign up and text with the chatbot daily. ChatGPT isn't revolutionary. It does it in a way normal people can access and be amazed by.
Accessibility isn't easy. OpenAI's Sam Altman tweeted that opening ChatGPT to the public increased computing costs.
Each chat costs "low-digit cents" to process. OpenAI probably spends several hundred thousand dollars a day to keep ChatGPT running, with no immediate business case.
Academic researchers and others who developed GPT-3 couldn't afford it. Without resources to make technology accessible, it can't be used.
Retrospective
This dynamic is old. In the history of science, a researcher with a breakthrough idea was often overshadowed by an entrepreneur or visionary who made it accessible to the public.
We think of Thomas Edison as the inventor of the lightbulb. But really, Vasilij Petrov, Thomas Wright, and Joseph Swan invented the lightbulb. Edison made technology visible and accessible by electrifying public buildings, building power plants, and wiring.
Edison probably lost a ton of money on stunts like building a power plant to light JP Morgan's home, the NYSE, and several newspaper headquarters.
People wanted electric lights once they saw their benefits. By making the technology accessible and visible, Edison unlocked a hugely profitable market.
Similar things are happening in AI. ChatGPT shows that developing breakthrough technology in the lab or on B2B servers won't change the culture.
AI must engage people's imaginations to become mainstream. Before the tech impacts the world, people must play with it and see its revolutionary power.
As the field evolves, companies that make the technology widely available, even at great cost, will succeed.
OpenAI's compute fees are eye-watering. Revolutions are costly.
Tim Denning
3 years ago
Bills are paid by your 9 to 5. 6 through 12 help you build money.
40 years pass. After 14 years of retirement, you die. Am I the only one who sees the problem?
I’m the Jedi master of escaping the rat race.
Not to impress. I know this works since I've tried it. Quitting a job to make money online is worse than Kim Kardashian's internet-burning advice.
Let me help you rethink the move from a career to online income to f*ck you money.
To understand why a job is a joke, do some life math.
Without a solid why, nothing makes sense.
The retirement age is 65. Our processed food consumption could shorten our 79-year average lifespan.
You spend 40 years working.
After 14 years of retirement, you die.
Am I alone in seeing the problem?
Life is too short to work a job forever, especially since most people hate theirs. After-hours skills are vital.
Money equals unrestricted power, f*ck you.
F*ck you money is the answer.
Jack Raines said it first. He says we can do anything with the money. Jack, a young rebel straight out of college, can travel and try new foods.
F*ck you money signifies not checking your bank account before buying.
F*ck you” money is pure, unadulterated freedom with no strings attached.
Jack claims you're rich when you rarely think about money.
Avoid confusion.
This doesn't imply you can buy a Lamborghini. It indicates your costs, income, lifestyle, and bank account are balanced.
Jack established an online portfolio while working for UPS in Atlanta, Georgia. So he gained boundless power.
The portion that many erroneously believe
Yes, you need internet abilities to make money, but they're not different from 9-5 talents.
Sahil Lavingia, Gumroad's creator, explains.
A job is a way to get paid to learn.
Mistreat your boss 9-5. Drain his skills. Defuse him. Love and leave him (eventually).
Find another employment if yours is hazardous. Pick an easy job. Make sure nothing sneaks into your 6-12 time slot.
The dumb game that makes you a sheep
A 9-5 job requires many job interviews throughout life.
You email your résumé to employers and apply for jobs through advertisements. This game makes you a sheep.
You're competing globally. Work-from-home makes the competition tougher. If you're not the cheapest, employers won't hire you.
After-hours online talents (say, 6 pm-12 pm) change the game. This graphic explains it better:
Online talents boost after-hours opportunities.
You go from wanting to be picked to picking yourself. More chances equal more money. Your f*ck you fund gets the extra cash.
A novel method of learning is essential.
College costs six figures and takes a lifetime to repay.
Informal learning is distinct. 6-12pm:
Observe the carefully controlled Twitter newsfeed.
Make use of Teachable and Gumroad's online courses.
Watch instructional YouTube videos
Look through the top Substack newsletters.
Informal learning is more effective because it's not obvious. It's fun to follow your curiosity and hobbies.
The majority of people lack one attitude. It's simple to learn.
One big impediment stands in the way of f*ck you money and time independence. So often.
Too many people plan after 6-12 hours. Dreaming. Big-thinkers. Strategically. They fill their calendar with meetings.
This is after-hours masturb*tion.
Sahil Bloom reminded me that a bias towards action will determine if this approach works for you.
The key isn't knowing what to do from 6-12 a.m. Trust yourself and develop abilities as you go. It's for building the parachute after you jump.
Sounds risky. We've eliminated the risk by finishing this process after hours while you work 9-5.
With no risk, you can have an I-don't-care attitude and still be successful.
When you choose to move forward, this occurs.
Once you try 9-5/6-12, you'll tell someone.
It's bad.
Few of us hang out with problem-solvers.
It's how much of society operates. So they make reasons so they can feel better about not giving you money.
Matthew Kobach told me chasing f*ck you money is easier with like-minded folks.
Without f*ck you money friends, loneliness will take over and you'll think you've messed up when you just need to keep going.
Steal this easy guideline
Let's act. No more fluffing and caressing.
1. Learn
If you detest your 9-5 talents or don't think they'll work online, get new ones. If you're skilled enough, continue.
Easlo recommends these skills:
Designer for Figma
Designer Canva
bubble creators
editor in Photoshop
Automation consultant for Zapier
Designer of Webflow
video editor Adobe
Ghostwriter for Twitter
Idea consultant
Artist in Blender Studio
2. Develop the ability
Every night from 6-12, apply the skill.
Practicing ghostwriting? Write someone's tweets for free. Do someone's website copy to learn copywriting. Get a website to the top of Google for a keyword to understand SEO.
Free practice is crucial. Your 9-5 pays the money, so work for free.
3. Take off stealthily like a badass
Another mistake. Sell to few. Don't be the best. Don't claim expertise.
Sell your new expertise to others behind you.
Two ways:
Using a digital good
By providing a service,
Point 1 also includes digital service examples. Digital products include eBooks, communities, courses, ad-supported podcasts, and templates. It's easy. Your 9-5 job involves one of these.
Take ideas from work.
Why? They'll steal your time for profit.
4. Iterate while feeling awful
First-time launches always fail. You'll feel terrible. Okay. Remember your 9-5?
Find improvements. Ask free and paying consumers what worked.
Multiple relaunches, each 1% better.
5. Discover more
Never stop learning. Improve your skill. Add a relevant skill. Learn copywriting if you write online.
After-hours students earn the most.
6. Continue
Repetition is key.
7. Make this one small change.
Consistently. The 6-12 momentum won't make you rich in 30 days; that's success p*rn.
Consistency helps wage slaves become f*ck you money. Most people can't switch between the two.
Putting everything together
It's easy. You're probably already doing some.
This formula explains why, how, and what to do. It's a 5th-grade-friendly blueprint. Good.
Reduce financial risk with your 9-to-5. Replace Netflix with 6-12 money-making talents.
Life is short; do whatever you want. Today.
Vitalik
3 years ago
An approximate introduction to how zk-SNARKs are possible (part 2)
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.
Polynomials
Polynomials are a special class of algebraic expressions of the form:
- x+5
- x^4
- x^3+3x^2+3x+1
- 628x^{271}+318x^{270}+530x^{269}+…+69x+381
i.e. they are a sum of any (finite!) number of terms of the form cx^k
There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.
Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:
- A(0)+B(0)=C(0)
- A(1)+B(1)=C(1)
- A(2)+B(2)=C(2)
- A(3)+B(3)=C(3)
And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:
- 12+1=13
- 10+8=18
- 15+8=23
- 15+13=28
You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:
- A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
- B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
- C(x)=5x+13
Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.
Comparing a polynomial to itself
You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)
As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.
Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).
Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!
Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.
see part 3