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
(Edited)

Hackernoon
3 years ago
👏 Awesome post! When is part 3 coming?

Trent Lapinski
3 years ago
Very complex topic, great explanation
More on Web3 & Crypto

Rishi Dean
3 years ago
Coinbase's web3 app
Use popular Ethereum dapps with Coinbase’s new dapp wallet and browser
Tl;dr: This post highlights the ability to access web3 directly from your Coinbase app using our new dapp wallet and browser.
Decentralized autonomous organizations (DAOs) and decentralized finance (DeFi) have gained popularity in the last year (DAOs). The total value locked (TVL) of DeFi investments on the Ethereum blockchain has grown to over $110B USD, while NFTs sales have grown to over $30B USD in the last 12 months (LTM). New innovative real-world applications are emerging every day.
Today, a small group of Coinbase app users can access Ethereum-based dapps. Buying NFTs on Coinbase NFT and OpenSea, trading on Uniswap and Sushiswap, and borrowing and lending on Curve and Compound are examples.
Our new dapp wallet and dapp browser enable you to access and explore web3 directly from your Coinbase app.
Web3 in the Coinbase app
Users can now access dapps without a recovery phrase. This innovative dapp wallet experience uses Multi-Party Computation (MPC) technology to secure your on-chain wallet. This wallet's design allows you and Coinbase to share the 'key.' If you lose access to your device, the key to your dapp wallet is still safe and Coinbase can help recover it.
Set up your new dapp wallet by clicking the "Browser" tab in the Android app's navigation bar. Once set up, the Coinbase app's new dapp browser lets you search, discover, and use Ethereum-based dapps.
Looking forward
We want to enable everyone to seamlessly and safely participate in web3, and today’s launch is another step on that journey. We're rolling out the new dapp wallet and browser in the US on Android first to a small subset of users and plan to expand soon. Stay tuned!

mbvissers.eth
3 years ago
Why does every smart contract seem to implement ERC165?
ERC165 (or EIP-165) is a standard utilized by various open-source smart contracts like Open Zeppelin or Aavegotchi.
What's it? You must implement? Why do we need it? I'll describe the standard and answer any queries.
What is ERC165
ERC165 detects and publishes smart contract interfaces. Meaning? It standardizes how interfaces are recognized, how to detect if they implement ERC165, and how a contract publishes the interfaces it implements. How does it work?
Why use ERC165? Sometimes it's useful to know which interfaces a contract implements, and which version.
Identifying interfaces
An interface function's selector. This verifies an ABI function. XORing all function selectors defines an interface in this standard. The following code demonstrates.
// SPDX-License-Identifier: UNLICENCED
pragma solidity >=0.8.0 <0.9.0;
interface Solidity101 {
function hello() external pure;
function world(int) external pure;
}
contract Selector {
function calculateSelector() public pure returns (bytes4) {
Solidity101 i;
return i.hello.selector ^ i.world.selector;
// Returns 0xc6be8b58
}
function getHelloSelector() public pure returns (bytes4) {
Solidity101 i;
return i.hello.selector;
// Returns 0x19ff1d21
}
function getWorldSelector() public pure returns (bytes4) {
Solidity101 i;
return i.world.selector;
// Returns 0xdf419679
}
}This code isn't necessary to understand function selectors and how an interface's selector can be determined from the functions it implements.
Run that sample in Remix to see how interface function modifications affect contract function output.
Contracts publish their implemented interfaces.
We can identify interfaces. Now we must disclose the interfaces we're implementing. First, import IERC165 like so.
pragma solidity ^0.4.20;
interface ERC165 {
/// @notice Query if a contract implements an interface
/// @param interfaceID The interface identifier, as specified in ERC-165
/// @dev Interface identification is specified in ERC-165.
/// @return `true` if the contract implements `interfaceID` and
/// `interfaceID` is not 0xffffffff, `false` otherwise
function supportsInterface(bytes4 interfaceID) external view returns (bool);
}We still need to build this interface in our smart contract. ERC721 from OpenZeppelin is a good example.
// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.5.0) (token/ERC721/ERC721.sol)
pragma solidity ^0.8.0;
import "./IERC721.sol";
import "./extensions/IERC721Metadata.sol";
import "../../utils/introspection/ERC165.sol";
// ...
contract ERC721 is Context, ERC165, IERC721, IERC721Metadata {
// ...
function supportsInterface(bytes4 interfaceId) public view virtual override(ERC165, IERC165) returns (bool) {
return
interfaceId == type(IERC721).interfaceId ||
interfaceId == type(IERC721Metadata).interfaceId ||
super.supportsInterface(interfaceId);
}
// ...
}I deleted unnecessary code. The smart contract imports ERC165, IERC721 and IERC721Metadata. The is keyword at smart contract declaration implements all three.
Kind (interface).
Note that type(interface).interfaceId returns the same as the interface selector.
We override supportsInterface in the smart contract to return a boolean that checks if interfaceId is the same as one of the implemented contracts.
Super.supportsInterface() calls ERC165 code. Checks if interfaceId is IERC165.
function supportsInterface(bytes4 interfaceId) public view virtual override returns (bool) {
return interfaceId == type(IERC165).interfaceId;
}So, if we run supportsInterface with an interfaceId, our contract function returns true if it's implemented and false otherwise. True for IERC721, IERC721Metadata, andIERC165.
Conclusion
I hope this post has helped you understand and use ERC165 and why it's employed.
Have a great day, thanks for reading!

Chris
2 years ago
What the World's Most Intelligent Investor Recently Said About Crypto
Cryptoshit. This thing is crazy to buy.
Charlie Munger is revered and powerful in finance.
Munger, vice chairman of Berkshire Hathaway, is noted for his wit, no-nonsense attitude to investment, and ability to spot promising firms and markets.
Munger's crypto views have upset some despite his reputation as a straight shooter.
“There’s only one correct answer for intelligent people, just totally avoid all the people that are promoting it.” — Charlie Munger
The Munger Interview on CNBC (4:48 secs)
This Monday, CNBC co-anchor Rebecca Quick interviewed Munger and brought up his 2007 statement, "I'm not allowed to have an opinion on this subject until I can present the arguments against my viewpoint better than the folks who are supporting it."
Great investing and life advice!
If you can't explain the opposing reasons, you're not informed enough to have an opinion.
In today's world, it's important to grasp both sides of a debate before supporting one.
Rebecca inquired:
Does your Wall Street Journal article on banning cryptocurrency apply? If so, would you like to present the counterarguments?
Mungers reply:
I don't see any viable counterarguments. I think my opponents are idiots, hence there is no sensible argument against my position.
Consider his words.
Do you believe Munger has studied both sides?
He said, "I assume my opponents are idiots, thus there is no sensible argument against my position."
This is worrisome, especially from a guy who once encouraged studying both sides before forming an opinion.
Munger said:
National currencies have benefitted humanity more than almost anything else.
Hang on, I think we located the perpetrator.
Munger thinks crypto will replace currencies.
False.
I doubt he studied cryptocurrencies because the name is deceptive.
He misread a headline as a Dollar destroyer.
Cryptocurrencies are speculations.
Like Tesla, Amazon, Apple, Google, Microsoft, etc.
Crypto won't replace dollars.
In the interview with CNBC, Munger continued:
“I’m not proud of my country for allowing this crap, what I call the cryptoshit. It’s worthless, it’s no good, it’s crazy, it’ll do nothing but harm, it’s anti-social to allow it.” — Charlie Munger
Not entirely inaccurate.
Daily cryptos are established solely to pump and dump regular investors.
Let's get into Munger's crypto aversion.
Rat poison is bitcoin.
Munger famously dubbed Bitcoin rat poison and a speculative bubble that would implode.
Partially.
But the bubble broke. Since 2021, the market has fallen.
Scam currencies and NFTs are being eliminated, which I like.
Whoa.
Why does Munger doubt crypto?
Mungers thinks cryptocurrencies has no intrinsic value.
He worries about crypto fraud and money laundering.
Both are valid issues.
Yet grouping crypto is intellectually dishonest.
Ethereum, Bitcoin, Solana, Chainlink, Flow, and Dogecoin have different purposes and values (not saying they’re all good investments).
Fraudsters who hurt innocents will be punished.
Therefore, complaining is useless.
Why not stop it? Repair rather than complain.
Regrettably, individuals today don't offer solutions.
Blind Areas for Mungers
As with everyone, Mungers' bitcoin views may be impacted by his biases and experiences.
OK.
But Munger has always advocated classic value investing and may be wary of investing in an asset outside his expertise.
Mungers' banking and insurance investments may influence his bitcoin views.
Could a coworker or acquaintance have told him crypto is bad and goes against traditional finance?
Right?
Takeaways
Do you respect Charlie Mungers?
Yes and no, like any investor or individual.
To understand Mungers' bitcoin beliefs, you must be critical.
Mungers is a successful investor, but his views about bitcoin should be considered alongside other viewpoints.
Munger’s success as an investor has made him an influencer in the space.
Influence gives power.
He controls people's thoughts.
Munger's ok. He will always be heard.
I'll do so cautiously.
You might also like

Davlin Knight
3 years ago
2 pitfalls to stay away from when launching a YouTube channel
You do not want to miss these
Stop! Stop it! Two things to avoid when starting a YouTube channel. Critical. Possible channel-killers Its future revenue.
I'll tell you now, so don't say "I wish I knew."
The Notorious Copyright Allegation
My YouTube channel received a copyright claim before I sold it. This claim was on a one-minute video I thought I'd changed enough to make mine, but the original owner disagreed.
It cost me thousands in ad revenue. Original owner got the profits.
Well, it wasn't your video, you say.
Touché.
I've learned. Sorta
I couldn't stop looking at the video's views. The video got 1,000,000 views without any revenue. I made 4 more similar videos.
If they didn't get copyrighted, I'd be rolling in dough.
You've spent a week editing and are uploading to YouTube. You're thrilled as you stand and stretch your back. You see the video just before publishing.
No way!
The red exclamation point on checks.
Copyright claim!
YouTube lets you publish, but you won't make money.
Sounds fair? Well, it is.
Copyright claims mean you stole someone's work. Song, image, or video clip.
We wouldn't want our content used for money.
The only problem with this is that almost everything belongs to someone else. I doubt some of the biggest creators are sitting down and making their music for their videos. That just seems really excessive when you could make a quick search on YouTube and download a song (I definitely don’t do this because that would be stealing).
So how do you defeat a copyright defense?
Even copyright-free songs on YouTube aren't guaranteed. Some copyrighted songs claim to be free.
Use YouTube's free music library or pay for a subscription to adobe stock, epidemic sound, or artlist.io.
Most of my videos have Nintendo music. Almost all game soundtracks are copyright-free and offer a variety of songs.
Restriction on age
Age restrictions are a must-avoid. A channel dies.
YouTube never suggests age-restricted videos.
Shadow banning means YouTube hides your content from subscribers and non-subscribers.
Keeping your channel family-friendly can help.
I hear you complaining that your channel isn't for kids. I agree. Not everyone has a clean mouth or creates content for minors.
YouTube has changed rapidly in recent years. Focusing on kids. Fewer big creators are using profanity or explicit content in videos. Not YouTube-worthy.
Youtube wants to be family-friendly. A family-friendly movie. It won't promote illegal content. Yes, it allows profanity.
Do I recommend avoiding no-no words in videos? Never. Okay. YouTube's policies are shaky. YouTube uses video content to determine ad suitability.
No joke. If you're serious about becoming a content creator, avoid profanity and inappropriate topics.
If your channel covers 18+ topics, like crime or commentary, censor as much as possible.
YouTube can be like walking on eggshells. You never know what is gonna upset the boss. So play it safe and try to avoid getting on their bad side.
Mr. Beast, Dream, Markplier, Faze Rug, and PewDewPie are popular creators. They maintain it family-friendly while entertaining fans.
You got this.

Todd Lewandowski
3 years ago
DWTS: How to Organize Your To-Do List Quickly
Don't overcomplicate to-do lists. DWTS (Done, Waiting, Top 3, Soon) organizes your to-dos.
How Are You Going to Manage Everything?
Modern America is busy. Work involves meetings. Anytime, Slack communications arrive. Many software solutions offer a @-mention notification capability. Emails.
Work obligations continue. At home, there are friends, family, bills, chores, and fun things.
How are you going to keep track of it all? Enter the todo list. It’s been around forever. It’s likely to stay forever in some way, shape, or form.
Everybody has their own system. You probably modified something from middle school. Post-its? Maybe it’s an app? Maybe both, another system, or none.
I suggest a format that has worked for me in 15 years of professional and personal life.
Try it out and see if it works for you. If not, no worries. You do you! Hopefully though you can learn a thing or two, and I from you too.
It is merely a Google Doc, yes.
It's a giant list. One task per line. Indent subtasks on a new line. Add or move new tasks as needed.
I recommend using Google Docs. It's easy to use and flexible for structuring.
Prioritizing these tasks is key. I organize them using DWTS (Done, Waiting, Top 3, Soon). Chronologically is good because it implicitly provides both a priority (high, medium, low) and an ETA (now, soon, later).
Yes, I recognize the similarities to DWTS (Dancing With The Stars) TV Show. Although I'm not a fan, it's entertaining. The acronym is easy to remember and adds fun to something dull.
What each section contains
Done
All tasks' endpoint. Finish here. Don't worry about it again.
Waiting
You're blocked and can't continue. Blocked tasks usually need someone. Write Person Task so you know who's waiting.
Blocking tasks shouldn't last long. After a while, remind them kindly. If people don't help you out of kindness, they will if you're persistent.
Top 3
Mental focus areas. These can be short- to mid-term goals or recent accomplishments. 2 to 5 is a good number to stay focused.
Top 3 reminds us to prioritize. If they don't fit your Top 3 goals, delay them.
Every 1:1 at work is a project update. Another chance to list your top 3. You should know your Top 3 well and be able to discuss them confidently.
Soon
Here's your short-term to-do list. Rank them from highest to lowest.
I usually subdivide it with empty lines. First is what I have to do today, then week, then month. Subsections can be arranged however you like.
Inventories by Concept
Tasks that aren’t in your short or medium future go into the backlog.
Eventually you’ll complete these tasks, assign them to someone else, or mark them as “wont’ do” (like done but in another sense).
Backlog tasks don't need to be organized chronologically because their timing and priority may change. Theme-organize them. When planning/strategic, you can choose themes to focus on, so future top 3 topics.
More Tips on Todos
Decide Upon a Morning Goal
Morning routines are universal. Coffee and Wordle. My to-do list is next. Two things:
As needed, update the to-do list: based on the events of yesterday and any fresh priorities.
Pick a few jobs to complete today: Pick a few goals that you know you can complete today. Push the remainder below and move them to the top of the Soon section. I typically select a few tasks I am confident I can complete along with one stretch task that might extend into tomorrow.
Finally. By setting and achieving small goals every day, you feel accomplished and make steady progress on medium and long-term goals.
Tech companies call this a daily standup. Everyone shares what they did yesterday, what they're doing today, and any blockers. The name comes from a tradition of holding meetings while standing up to keep them short. Even though it's virtual, everyone still wants a quick meeting.
Your team may or may not need daily standups. Make a daily review a habit with your coffee.
Review Backwards & Forwards on a regular basis
While you're updating your to-do list daily, take time to review it.
Review your Done list. Remember things you're proud of and things that could have gone better. Your Done list can be long. Archive it so your main to-do list isn't overwhelming.
Future-gaze. What you considered important may no longer be. Reorder tasks. Backlog grooming is a workplace term.
Backwards-and-forwards reviews aren't required often. Every 3-6 months is fine. They help you see the forest as often as the trees.
Final Remarks
Keep your list simple. Done, Waiting, Top 3, Soon. These are the necessary sections. If you like, add more subsections; otherwise, keep it simple.
I recommend a morning review. By having clear goals and an action-oriented attitude, you'll be successful.
Daniel Clery
3 years ago
Twisted device investigates fusion alternatives
German stellarator revamped to run longer, hotter, compete with tokamaks
Tokamaks have dominated the search for fusion energy for decades. Just as ITER, the world's largest and most expensive tokamak, nears completion in southern France, a smaller, twistier testbed will start up in Germany.
If the 16-meter-wide stellarator can match or outperform similar-size tokamaks, fusion experts may rethink their future. Stellarators can keep their superhot gases stable enough to fuse nuclei and produce energy. They can theoretically run forever, but tokamaks must pause to reset their magnet coils.
The €1 billion German machine, Wendelstein 7-X (W7-X), is already getting "tokamak-like performance" in short runs, claims plasma physicist David Gates, preventing particles and heat from escaping the superhot gas. If W7-X can go long, "it will be ahead," he says. "Stellarators excel" Eindhoven University of Technology theorist Josefine Proll says, "Stellarators are back in the game." A few of startup companies, including one that Gates is leaving Princeton Plasma Physics Laboratory, are developing their own stellarators.
W7-X has been running at the Max Planck Institute for Plasma Physics (IPP) in Greifswald, Germany, since 2015, albeit only at low power and for brief runs. W7-X's developers took it down and replaced all inner walls and fittings with water-cooled equivalents, allowing for longer, hotter runs. The team reported at a W7-X board meeting last week that the revised plasma vessel has no leaks. It's expected to restart later this month to show if it can get plasma to fusion-igniting conditions.
Wendelstein 7-X's water-cooled inner surface allows for longer runs.
HOSAN/IPP
Both stellarators and tokamaks create magnetic gas cages hot enough to melt metal. Microwaves or particle beams heat. Extreme temperatures create a plasma, a seething mix of separated nuclei and electrons, and cause the nuclei to fuse, releasing energy. A fusion power plant would use deuterium and tritium, which react quickly. Non-energy-generating research machines like W7-X avoid tritium and use hydrogen or deuterium instead.
Tokamaks and stellarators use electromagnetic coils to create plasma-confining magnetic fields. A greater field near the hole causes plasma to drift to the reactor's wall.
Tokamaks control drift by circulating plasma around a ring. Streaming creates a magnetic field that twists and stabilizes ionized plasma. Stellarators employ magnetic coils to twist, not plasma. Once plasma physicists got powerful enough supercomputers, they could optimize stellarator magnets to improve plasma confinement.
W7-X is the first large, optimized stellarator with 50 6- ton superconducting coils. Its construction began in the mid-1990s and cost roughly twice the €550 million originally budgeted.
The wait hasn't disappointed researchers. W7-X director Thomas Klinger: "The machine operated immediately." "It's a friendly machine." It did everything we asked." Tokamaks are prone to "instabilities" (plasma bulging or wobbling) or strong "disruptions," sometimes associated to halted plasma flow. IPP theorist Sophia Henneberg believes stellarators don't employ plasma current, which "removes an entire branch" of instabilities.
In early stellarators, the magnetic field geometry drove slower particles to follow banana-shaped orbits until they collided with other particles and leaked energy. Gates believes W7-X's ability to suppress this effect implies its optimization works.
W7-X loses heat through different forms of turbulence, which push particles toward the wall. Theorists have only lately mastered simulating turbulence. W7-X's forthcoming campaign will test simulations and turbulence-fighting techniques.
A stellarator can run constantly, unlike a tokamak, which pulses. W7-X has run 100 seconds—long by tokamak standards—at low power. The device's uncooled microwave and particle heating systems only produced 11.5 megawatts. The update doubles heating power. High temperature, high plasma density, and extensive runs will test stellarators' fusion power potential. Klinger wants to heat ions to 50 million degrees Celsius for 100 seconds. That would make W7-X "a world-class machine," he argues. The team will push for 30 minutes. "We'll move step-by-step," he says.
W7-X's success has inspired VCs to finance entrepreneurs creating commercial stellarators. Startups must simplify magnet production.
Princeton Stellarators, created by Gates and colleagues this year, has $3 million to build a prototype reactor without W7-X's twisted magnet coils. Instead, it will use a mosaic of 1000 HTS square coils on the plasma vessel's outside. By adjusting each coil's magnetic field, operators can change the applied field's form. Gates: "It moves coil complexity to the control system." The company intends to construct a reactor that can fuse cheap, abundant deuterium to produce neutrons for radioisotopes. If successful, the company will build a reactor.
Renaissance Fusion, situated in Grenoble, France, raised €16 million and wants to coat plasma vessel segments in HTS. Using a laser, engineers will burn off superconductor tracks to carve magnet coils. They want to build a meter-long test segment in 2 years and a full prototype by 2027.
Type One Energy in Madison, Wisconsin, won DOE money to bend HTS cables for stellarator magnets. The business carved twisting grooves in metal with computer-controlled etching equipment to coil cables. David Anderson of the University of Wisconsin, Madison, claims advanced manufacturing technology enables the stellarator.
Anderson said W7-X's next phase will boost stellarator work. “Half-hour discharges are steady-state,” he says. “This is a big deal.”
