September 2, 2025 – Philadelphia









Incomprehensible
Incomprehensible
Incomprehensible, let me be
Highway 17, cotton candy rain
Drivin’ with my lover, we missed our plane
So we added on the hours to see the lupine flowers
Way out past the border, we blew through Thunder Bay
The pine trees are narrow, a billion broken arrows
The ravens and the crows, robins and the sparrows
All across Ontario, static on the stereo
Went swimmin’ in the lake, Old Woman Bay
Travelin’ with some stuff I left when I was a kid
Mr. Bear and the wooden box I hid
Full of broken gadgets that mean nothin’ now
The only thing I’ll keep are the letters and the photographs
In two days it’s my birthday and I’ll be 33
That doesn’t really matter next to eternity
But I like a double number, and I like an odd one too
And everything I see from now on will be somethin’ new
“I’m afraid of getting older, ” that’s what I’ve learned to say
Society has given me the words to think that way
The message spirals, “Don’t get saggy, don’t get grey”
But the soft and lovely silvers are now fallin’ on my shoulder
My mother and my grandma, my great-grandmother too
Wrinkle like the river, sweeten like the dew
And as silver as the rainbow scales that shimmer purple blue
How can beauty that is livin’ be anything but true?
So let gravity be my sculptor, let the wind do my hair
Let me dance in front of people without a care
Let me be naked alone with nobody there
With mismatched socks and shoes and stuff stuffed in my underwear
Incomprehensible, let me be
Incomprehensible, let me be
As a street photographer, I absolutely thrive in the mundane. I can walk the same lane every single day, embracing the mundane, but still find infinite ways to create something from nothing. This to me is the superpower of being a photographer. It’s the ability that the camera provides you, the excuse, the key that unlocks the door to possibility, the sword that strikes through the heart of chaos, putting order to the world that is our canvas. Now these thoughts may sound lofty and grandiose and ridiculous, but actually, it’s quite profound in the most mundane and simple way.
So I returned to the garden, a life of simplicity, spending my days in the park, tending the land as a horticulturalist. I voluntarily decided to opt out of this game that we call modern living. I live a simple life, pulling weeds, trimming trees, planting things, lugging logs, chopping, lifting, digging, crawling in the dirt, because I absolutely find zero meaning in the menial tasks and labor involved in modern productivity. However, by returning to simplicity, to the most fundamental way of life, tending the garden, living the most monotonous and mundane repetitive loop of showing up, watering plants, doing this, doing that—literally I’m doing the same thing every single day—but I absolutely thrive in the mundane!
When I think about Adam and Eve in the garden, God gave us this beautiful place to play, to run around, to eat, sleep under trees, and he just gave us one thing to do: tend the land and live in Paradise. That’s how I feel every day. I’m just tending the land and living in paradise. But paradise isn’t some place that is all that interesting. It’s pure bliss. You hear birds chirping, butterflies flying, insects crawling on your skin sometimes, you might get bitten, you might cut yourself, you might bleed, but there’s something about the simplicity and the boredom of being in paradise that provides me with the opportunity to have creative breakthroughs.
So in this boredom of being in paradise, there are possibilities to have creative breakthroughs. I’ll go on the same mundane walk around in a loop, read a book, maybe lay under a tree, which may seem like I’m deep in thought, but I’m actually trying to do the opposite. By returning to the garden and living a monotonous lifestyle, I’m purposely trying to embrace boredom, to empty my mind, and to become a child again. I’m purposely trying to shut my brain off. I’m trying to return to being a child so that I can have more creative thoughts and ideas that I can utilize in my photography and creative practices.
I seriously think we think too much, and when I’m walking around, it may seem like I’m deep in thought but it’s really the opposite. I’m actually trying not to try. Another way to put it is: I’m trying my best to keep myself focused on the moment so profoundly that all of my thoughts shut off and I’m in a complete flow state throughout the entirety of my day.
When I’m pruning plants repetitively and I’m looking at these little brown tips on the leaves, and I have to trim like 1000 of them, I’m completely laser focused, not thinking at all. But through the non-thinking, through the simply being part, when you enter the flow state, you have these Eureka moments.
So one thing I’ve been thinking about is how to not think, hahahah. For instance, when I’m gardening, I’m not thinking. I’m looking at blades of grass wiggling, listening to the sounds of birds, water flowing from the creek, and I’m empty.
Similarly, when I’m photographing and moving through the world slowly, being very observant of every little detail—from the buildings to the people to the streets to the textures—I’m completely at peace amongst the chaos. But it’s the chaos that I thrive in. It’s embracing the boredom of a walk and finding novelty and intricate details in the mundane world around me.
There’s such a profound experience of clicking the shutter that I can’t really describe with words. When you click the shutter, it’s like having that little Eureka moment, and it perpetually keeps you in this flow state of non-action. Like yes, I’m moving and I’m walking, but I really am putting zero effort into anything that I’m creating right now. I’m simply a vessel. I’m just there and prepared with my camera, and I press the shutter when life flows towards me.
When I’m walking extremely slowly, I’m existing outside the passage of time, and the moments that come to me—fleeting, a fraction of a second away—are just an instinct away from making or breaking a frame. That moment when I click the shutter is such a profound feeling because I’m so immersed in the moment, to the point where I’m not thinking at all, and it feels like a transcendental experience.
It may sound kind of crazy to say on paper right now, but realistically this is, to me, the most beautiful way to be. It’s literally returning to the Garden of Eden, where you’re just a vessel for God again and there’s no pressure. There’s no society, there’s nothing bogging you down. It’s just you and the world and the canvas in front of you, and you can create. But it only happens in a flow state, where you stop thinking and you just move from your gut, from your instincts, your intuition, your Thumos!

When people think of the most influential composers of the modern era, names from film and classical music often come to mind. Yet one of the most impactful composers of the 21st century has worked quietly in the world of video games: C418, also known as Daniel Rosenfeld.
C418 is the artist name of Daniel Rosenfeld, a German musician, sound engineer, and composer. Born in East Germany in 1989, Rosenfeld began experimenting with music and sound design in his teens, teaching himself how to craft soundscapes that would later change the world of gaming forever.
C418 is best known as the composer of the Minecraft soundtrack. His ambient, minimalist, and deeply emotional pieces have become inseparable from the experience of exploring the blocky worlds of Minecraft.
For millions of players, these soundtracks are not just background music—they are the sound of childhood, creativity, and nostalgia.
While Minecraft made him a household name, C418’s discography extends far beyond it. He has released numerous independent albums that explore ambient, experimental, and electronic sounds, including:
These works reveal a composer unafraid to push boundaries, crafting soundscapes that are as introspective as they are innovative.
C418’s style is marked by minimalism, mood, and atmosphere. His music is less about catchy melodies and more about creating an emotional environment. This approach has inspired a generation of game developers and composers to view music not just as accompaniment, but as a vital part of immersion.
In many ways, C418’s influence mirrors that of great classical composers—except his stage is the digital world, and his orchestra is the limitless imagination of players.
C418 is one of the best composers of all time because his music transcends its medium. He gave sound to creativity itself, making the act of building, wandering, and dreaming in Minecraft feel timeless. His work proves that video game music can be as profound, moving, and culturally significant as any symphony or film score.
Simply put: C418 created the soundtrack to a generation’s imagination.
A structured, in-depth study guide to the 2-hour conversation. Organized for fast review and long-term study.
Core thesis: Winogrand insists that photographs are not narratives or opinions in words; they are records of what a piece of time and space looked like to a camera. The photographer’s job is to make pictures that are more dramatic (more compelling) than the thing photographed—without loading the frame with self-conscious “artiness.”
“They show you what a piece of time and space looked like to a camera.”
“The work has to be more dramatic than what was photographed.”
“I would like not to exist [in the picture].”
Walker Evans — “Transparent”
“Evans… is as close to the absence of a strategy as I know of.”
Robert Frank — Casual Strategy
“The picture I made was made; the picture he made happened.”
Edward Weston — “Arty”
Bottom line: Arty vs. anti-arty are both strategies; Evans is exceptional for minimizing the sense of strategy while maintaining an unmistakable voice.
“You’re not a camera… You don’t see the way a camera sees.”
“They function like puns… They make you question what you think you know.”
“How do you make a photograph that is more dramatic than what was photographed? That’s the problem.”
“The tools you use are responsible for how the pictures look.”
“I don’t want to buy all that mechanism I don’t use.”
Working rule of thumb:
“It depends on what kind of murder you can get away with.”
“If I had to pick 20 today, 10 would be different tomorrow.”
“Only be edited by my equals… otherwise pay a fortune.”
“You don’t learn from teachers. You learn from work.”
“The camera doesn’t know what you understand… your understanding may catch up with a frame.”
Classroom ethic: If he says something you can’t see in the picture, nail him—“If it isn’t physical, it’s rhetoric.”
“Sander was making a catalog… Avedon is absolutely interested in these people as people.”
*(Note: His critique of Bruce Davidson’s *East 100th Street* is extremely harsh—he argues it reflects condescending liberal stereotypes. Summarized here without repeating harmful phrasing.)*
“I photograph where I am.”
“I would like not to exist.”
“Evans… is as close to the absence of a strategy as I know.”
“The picture I made was made; the picture he made happened.”
“It’s only pictures we’re getting.”
“Technique is easy… You learn from work.”
“If I had to pick 20 today, 10 would be different tomorrow.”
“Only be edited by my equals; otherwise pay a fortune.”
“If I say something you don’t see in the picture, nail me… If it isn’t physical, it’s rhetoric.”
Q: Do you strive to be transparent?
“I’d like not to exist… In the end, all I can do is wrestle and whatever comes out.”
Q: Are Leicas archaic vs. auto SLRs?
Calls that “stupid”; with small lenses, SLRs force focusing aids and push design; prefers rangefinders’ simplicity.
Q: Why call your early rodeo frames ‘junk action’?
Relative to the arena photographers’ peak moments; his access/lens limited him—different problem, different pictures.
Q: Are photos ambiguous?
They’re specific yet can’t settle narrative facts; they work like puns, upsetting assumptions.
Q: Are you a bad editor?
“Horse****.” Also: he *defers to equals* (Papageorge) and accepts that selections shift.
Q: Teaching value?
Still interesting because he’s still learning how to talk about pictures; but learning comes from work.

The source is below, but it stretches above,
Roots planted in the ground, yet flying like a dove.
A good fountain must be dug deep underground,
Through suffering and strife, the living God is found.
When it’s connected to the source it can endlessly feed others,
Even the hounds, the lost, and the brothers.
Playing in the garden, just you and some trees,
Trimming the weeds, getting stung by the bees.
Riding tigers through the forest, freedom everlasting,
When you pick up the flaming swords, and you start fasting.
You can ascend the material plane, return to Paradise,
For it has not been lost — it’s always been inside.
Somebody may strike you, knock you on the chin,
But you turn the other cheek, maybe even grin.
Because love comes from above, and you need not sin,
Laughing in the face of chaos, floating through the unknown,
Nothing can break your spirit, for you’re never alone.
For God is love, the divine source,
We’re all created in His image, set on this course.
But the world turns us astray —
Hatred, bitterness, greed etched on the face,
We consume until we grow fat, our bones give way.
Our bodies six feet under will return to the clay,
But while you’re here on this earth, be a child and play.
The smoke screens, distractions, the clamor, the noise,
May pollute your body, may steal your joys.
But like a good gardener, pruning what’s dead,
New growth will spring forth, fresh life instead.
Good fruits from the vine — you shine with true joy,
For your connection to Him they can never destroy.
Download full video lecture here:

What’s poppin’ people?
It’s Dante — walking outside of City Hall here in the city of brotherly love, Philadelphia.
Today I’m thinking about something deeper than the bricks and fountains around me.
I’m thinking about divine love.
There’s this ancient idea called agape love — a kind of unconditional, divine love. I first came across it while reading Plato’s Symposium, but it didn’t really come alive for me until I experienced it myself.
Through living. Through witnessing.
Let me take you back.
When I was a Peace Corps volunteer in Zambia, I saw something that changed me.
In the village, everyone had a role:
Everyone was making a daily sacrifice.
And at the center of it all? A well.
Every morning, the whole community gathered at that well. It wasn’t just about water — it was life. It was unity. It was love in motion.
Looking at the flowing fountain here in Philly, I remembered that well.
In order to construct a well, you must connect to the source.
You don’t just find water on the surface. You dig. Through dirt. Through pain. Through suffering. Through hell itself.
But once you hit the source, it flows endlessly.
And now — anyone can come to drink. The whole community thrives.
That’s divine love.
That’s what I believe we’re here to do.
And I’m pretty sure — I gotta double-check this — but the original plan for City Hall included a fountain in the center, a place for people to come and drink.
The symbolic heart of the city.
Because a city, just like a village, needs a well —
A source where life begins.
A center of nourishment.
A place of gathering.
And for me, that center… is love.
If your love comes from above — from God, from the source — then you don’t need love from the world.
You’re not out here seeking validation.
You’re not begging for someone to love you.
You’re not striving for approval.
You shine in the light of Christ.
You love effortlessly.
You give endlessly.
You expect nothing in return.
Because you’re full.
Back in Zambia, every weekend — every day, really — people gathered at the church.
They went to learn about Christ.
To embody Christ.
To love their neighbor.
And when one person in a tribe walks that path — when they embody divine love — the entire tribe benefits.
Everyone is nourished.
And it’s not just about food or water.
The real nourishment is from the spirit.
You don’t thrive by consuming the world.
You thrive by embodying love.
That’s the key. You’re not out here chasing love.
You’re not trying to possess it through relationships, titles, or success.
You are love.
Because God is love, and you were made in His image.
That’s where the power lies.
Your real strength isn’t in your biceps or your bone density.
It’s not in your grind or your bank account.
The real power is in the spirit.
And that’s what’s missing in modern society.
We’re addicted to pleasure, chasing wealth, faking love.
The antidote?
Be love.
Spread love.
Stop needing it from others.
When you’re tapped into God, nothing can break you.
You’re like a Spartan with Hoplite armor.
You’re like level 99 in Skyrim with Daedric armor and full defense stats.
Unstoppable.
Because you’re powered by something eternal.
And that’s divine love — the well that never runs dry.
When you connect to that source, you become the well.
Others can come to drink.
You nourish the world, not by taking, but by overflowing.
So tap in. Dig deep. Connect to the source.
That’s where life begins.
—Dante

https://bitcoin.org/bitcoin.pdf
Satoshi Nakamoto
satoshin@gmx.com
www.bitcoin.org
A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution. Digital signatures provide part of the solution, but the main benefits are lost if a trusted third party is still required to prevent double-spending. This paper proposes a solution to the double-spending problem using a peer-to-peer network. The network timestamps transactions by hashing them into an ongoing chain of hash-based proof-of-work, forming a record that cannot be changed without redoing the proof-of-work. The longest chain serves as proof of the sequence of events witnessed and proof that it came from the largest pool of CPU power. As long as a majority of CPU power is controlled by nodes that are not cooperating to attack the network, they will generate the longest chain and outpace attackers. The network itself requires minimal structure. Messages are broadcast on a best effort basis, and nodes can leave and rejoin the network at will, accepting the longest proof-of-work chain as proof of what happened while they were gone.
Commerce on the Internet has come to rely almost exclusively on financial institutions serving as trusted third parties to process electronic payments. While the system works well enough for most transactions, it still suffers from the inherent weaknesses of the trust-based model. Completely non-reversible transactions are not really possible, since financial institutions cannot avoid mediating disputes. The cost of mediation increases transaction costs, limiting the minimum practical transaction size and cutting off the possibility for small casual transactions. There is a broader cost in the loss of the ability to make non-reversible payments for non-reversible services. With the possibility of reversal, the need for trust spreads. Merchants must be wary of their customers, hassling them for more information than they would otherwise need. A certain percentage of fraud is accepted as unavoidable. These costs and payment uncertainties can be avoided in person by using physical currency, but no mechanism exists to make payments over a communications channel without a trusted party.
What is needed is an electronic payment system based on cryptographic proof instead of trust, allowing any two willing parties to transact directly with each other without the need for a trusted third party. Transactions that are computationally impractical to reverse would protect sellers from fraud, and routine escrow mechanisms could easily be implemented to protect buyers. In this paper, we propose a solution to the double-spending problem using a peer-to-peer distributed timestamp server to generate computational proof of the chronological order of transactions. The system is secure as long as honest nodes collectively control more CPU power than any cooperating group of attacker nodes.
We define an electronic coin as a chain of digital signatures. Each owner transfers the coin to the next by digitally signing a hash of the previous transaction and the public key of the next owner and adding these to the end of the coin. A payee can verify the signatures to verify the chain of ownership.
The problem, of course, is that the payee can’t verify that one of the owners did not double-spend the coin. A common solution is to introduce a trusted central authority, or mint, that checks every transaction for double spending. After each transaction, the coin must be returned to the mint to issue a new coin, and only coins issued directly from the mint are trusted not to be double-spent. The problem with this solution is that the fate of the entire money system depends on the company running the mint, with every transaction having to go through them, just like a bank.
We need a way for the payee to know that the previous owners did not sign any earlier transactions. For our purposes, the earliest transaction is the one that counts, so we don’t care about later attempts to double-spend. The only way to confirm the absence of a transaction is to be aware of all transactions. In the mint-based model, the mint was aware of all transactions and decided which arrived first. To accomplish this without a trusted party, transactions must be publicly announced [1], and we need a system for participants to agree on a single history of the order in which they were received. The payee needs proof that at the time of each transaction, the majority of nodes agreed it was the first received.
The solution we propose begins with a timestamp server. A timestamp server works by taking a hash of a block of items to be timestamped and widely publishing the hash, such as in a newspaper or Usenet post [2-5]. The timestamp proves that the data must have existed at the time, obviously, in order to get into the hash. Each timestamp includes the previous timestamp in its hash, forming a chain, with each additional timestamp reinforcing the ones before it.
To implement a distributed timestamp server on a peer-to-peer basis, we will need to use a proof-of-work system similar to Adam Back’s Hashcash [6], rather than newspaper or Usenet posts. The proof-of-work involves scanning for a value that, when hashed, such as with SHA-256, the hash begins with a number of zero bits. The average work required is exponential in the number of zero bits required and can be verified by executing a single hash.
For our timestamp network, we implement the proof-of-work by incrementing a nonce in the block until a value is found that gives the block’s hash the required zero bits. Once the CPU effort has been expended to make it satisfy the proof-of-work, the block cannot be changed without redoing the work. As later blocks are chained after it, the work to change the block would include redoing all the blocks after it.
The proof-of-work also solves the problem of determining representation in majority decision-making. If the majority were based on one-IP-address-one-vote, it could be subverted by anyone able to allocate many IPs. Proof-of-work is essentially one-CPU-one-vote. The majority decision is represented by the longest chain, which has the greatest proof-of-work effort invested in it. If a majority of CPU power is controlled by honest nodes, the honest chain will grow the fastest and outpace any competing chains. To modify a past block, an attacker would have to redo the proof-of-work of the block and all blocks after it and then catch up with and surpass the work of the honest nodes. We will show later that the probability of a slower attacker catching up diminishes exponentially as subsequent blocks are added.
To compensate for increasing hardware speed and varying interest in running nodes over time, the proof-of-work difficulty is determined by a moving average targeting an average number of blocks per hour. If they’re generated too fast, the difficulty increases.
The steps to run the network are as follows:
Nodes always consider the longest chain to be the correct one and will keep working on extending it. If two nodes broadcast different versions of the next block simultaneously, some nodes may receive one or the other first. In that case, they work on the first one they received, but save the other branch in case it becomes longer. The tie will be broken when the next proof-of-work is found and one branch becomes longer; the nodes that were working on the other branch will then switch to the longer one.
New transaction broadcasts do not necessarily need to reach all nodes. As long as they reach many nodes, they will get into a block before long. Block broadcasts are also tolerant of dropped messages. If a node does not receive a block, it will request it when it receives the next block and realizes it missed one.
By convention, the first transaction in a block is a special transaction that starts a new coin owned by the creator of the block. This adds an incentive for nodes to support the network, and provides a way to initially distribute coins into circulation, since there is no central authority to issue them. The steady addition of a constant amount of new coins is analogous to gold miners expending resources to add gold to circulation. In our case, it is CPU time and electricity that is expended.
The incentive can also be funded with transaction fees. If the output value of a transaction is less than its input value, the difference is a transaction fee that is added to the incentive value of the block containing the transaction. Once a predetermined number of coins have entered circulation, the incentive can transition entirely to transaction fees and be completely inflation free.
The incentive may help encourage nodes to stay honest. If a greedy attacker is able to assemble more CPU power than all the honest nodes, he would have to choose between using it to defraud people by stealing back his payments or using it to generate new coins. He ought to find it more profitable to play by the rules, such rules that favour him with more new coins than everyone else combined, than to undermine the system and the validity of his own wealth.
Once the latest transaction in a coin is buried under enough blocks, the spent transactions before it can be discarded to save disk space. To facilitate this without breaking the block’s hash, transactions are hashed in a Merkle Tree [7][2][5], with only the root included in the block’s hash. Old blocks can then be compacted by stubbing off branches of the tree. The interior hashes do not need to be stored.
A block header with no transactions would be about 80 bytes. If we suppose blocks are generated every 10 minutes, 80 bytes * 6 * 24 * 365 = 4.2MB per year. With computer systems typically selling with 2GB of RAM as of 2008, and Moore’s Law predicting current growth of 1.2GB per year, storage should not be a problem even if the block headers must be kept in memory.
It is possible to verify payments without running a full network node. A user only needs to keep a copy of the block headers of the longest proof-of-work chain, which he can get by querying network nodes until he’s convinced he has the longest chain, and obtain the Merkle branch linking the transaction to the block it’s timestamped in. He can’t check the transaction for himself, but by linking it to a place in the chain, he can see that a network node has accepted it, and blocks added after it further confirm the network has accepted it.
As such, the verification is reliable as long as honest nodes control the network, but is more vulnerable if the network is overpowered by an attacker. While network nodes can verify transactions for themselves, the simplified method can be fooled by an attacker’s fabricated transactions for as long as the attacker can continue to overpower the network. One strategy to protect against this would be to accept alerts from network nodes when they detect an invalid block, prompting the user’s software to download the full block and alerted transactions to confirm the inconsistency. Businesses that receive frequent payments will probably still want to run their own nodes for more independent security and quicker verification.
Although it would be possible to handle coins individually, it would be unwieldy to make a separate transaction for every cent in a transfer. To allow value to be split and combined, transactions contain multiple inputs and outputs. Normally there will be either a single input from a larger previous transaction or multiple inputs combining smaller amounts, and at most two outputs: one for the payment, and one returning the change, if any, back to the sender.
It should be noted that fan-out, where a transaction depends on several transactions, and those transactions depend on many more, is not a problem here. There is never the need to extract a complete standalone copy of a transaction’s history.
The traditional banking model achieves a level of privacy by limiting access to information to the parties involved and the trusted third party. The necessity to announce all transactions publicly precludes this method, but privacy can still be maintained by breaking the flow of information in another place: by keeping public keys anonymous. The public can see that someone is sending an amount to someone else, but without information linking the transaction to anyone. This is similar to the level of information released by stock exchanges, where the time and size of individual trades, the “tape”, is made public, but without telling who the parties were.
As an additional firewall, a new key pair should be used for each transaction to keep them from being linked to a common owner. Some linking is still unavoidable with multi-input transactions, which necessarily reveal that their inputs were owned by the same owner. The risk is that if the owner of a key is revealed, linking could reveal other transactions that belonged to the same owner.
We consider the scenario of an attacker trying to generate an alternate chain faster than the honest chain. Even if this is accomplished, it does not throw the system open to arbitrary changes, such as creating value out of thin air or taking money that never belonged to the attacker. Nodes are not going to accept an invalid transaction as payment, and honest nodes will never accept a block containing them. An attacker can only try to change one of his own transactions to take back money he recently spent.
The race between the honest chain and an attacker chain can be characterized as a Binomial Random Walk. The success event is the honest chain being extended by one block, increasing its lead by +1, and the failure event is the attacker’s chain being extended by one block, reducing the gap by -1.
The probability of an attacker catching up from a given deficit is analogous to a Gambler’s Ruin problem. Suppose a gambler with unlimited credit starts at a deficit and plays potentially an infinite number of trials to try to reach breakeven. We can calculate the probability he ever reaches breakeven, or that an attacker ever catches up with the honest chain, as follows [8]:
qz = { 1 if p ≤ q
(q/p)^z if p > q }
Given our assumption that p > q, the probability drops exponentially as the number of blocks the attacker has to catch up with increases. With the odds against him, if he doesn’t make a lucky lunge forward early on, his chances become vanishingly small as he falls further behind.
We now consider how long the recipient of a new transaction needs to wait before being sufficiently certain the sender can’t change the transaction. We assume the sender is an attacker who wants to make the recipient believe he paid him for a while, then switch it to pay back to himself after some time has passed. The receiver will be alerted when that happens, but the sender hopes it will be too late.
The receiver generates a new key pair and gives the public key to the sender shortly before signing. This prevents the sender from preparing a chain of blocks ahead of time by working on it continuously until he is lucky enough to get far enough ahead, then executing the transaction at that moment. Once the transaction is sent, the dishonest sender starts working in secret on a parallel chain containing an alternate version of his transaction.
The recipient waits until the transaction has been added to a block and z blocks have been linked after it. He doesn’t know the exact amount of progress the attacker has made, but assuming the honest blocks took the average expected time per block, the attacker’s potential progress will be a Poisson distribution with expected value:
λ = z * (q/p)
To get the probability the attacker could still catch up now, we multiply the Poisson density for each amount of progress he could have made by the probability he could catch up from that point:
Σ from k=0 to ∞ λ^k * e^-λ / k! * { (q/p)^(z-k) if k ≤ z
1 if k > z }
Rearranging to avoid summing the infinite tail of the distribution…
1 – Σ from k=0 to z λ^k * e^-λ / k! * (1 – (q/p)^(z-k))
Converting to C code…
“`c
double AttackerSuccessProbability(double q, int z)
{
double p = 1.0 – q;
double lambda = z * (q / p);
double sum = 1.0;
int i, k;
for (k = 0; k <= z; k++)
{
double poisson = exp(-lambda);
for (i = 1; i <= k; i++)
poisson *= lambda / i;
sum -= poisson * (1 – pow(q / p, z – k));
}
return sum;
}
Running some results, we can see the probability drop off exponentially with z.
q=0.1
z=0 P=1.0000000
z=1 P=0.2045873
z=2 P=0.0509779
z=3 P=0.0131722
z=4 P=0.0034552
z=5 P=0.0009137
z=6 P=0.0002428
z=7 P=0.0000647
z=8 P=0.0000173
z=9 P=0.0000046
z=10 P=0.0000012
q=0.3
z=0 P=1.0000000
z=5 P=0.1773523
z=10 P=0.0416605
z=15 P=0.0101008
z=20 P=0.0024804
z=25 P=0.0006132
z=30 P=0.0001522
z=35 P=0.0000379
z=40 P=0.0000095
z=45 P=0.0000024
z=50 P=0.0000006
Solving for P less than 0.1%…
P < 0.001
q=0.10 z=5
q=0.15 z=8
q=0.20 z=11
q=0.25 z=15
q=0.30 z=24
q=0.35 z=41
q=0.40 z=89
q=0.45 z=340
We have proposed a system for electronic transactions without relying on trust. We started with the usual framework of coins made from digital signatures, which provides strong control of ownership, but is incomplete without a way to prevent double-spending. To solve this, we proposed a peer-to-peer network using proof-of-work to record a public history of transactions that quickly becomes computationally impractical for an attacker to change if honest nodes control a majority of CPU power.
The network is robust in its unstructured simplicity. Nodes work all at once with little coordination. They do not need to be identified, since messages are not routed to any particular place and only need to be delivered on a best effort basis. Nodes can leave and rejoin the network at will, accepting the proof-of-work chain as proof of what happened while they were gone. They vote with their CPU power, expressing their acceptance of valid blocks by working on extending them and rejecting invalid blocks by refusing to work on them. Any needed rules and incentives can be enforced with this consensus mechanism.

The Lightning Network was proposed by Joseph Poon and Thaddeus Dryja (2016) as a solution to Bitcoin’s scalability limits.
Bitcoin can only process ~7 transactions per second, far below the capacity of Visa (~47,000 tps). Increasing block size would lead to centralization, high fees, and reduced security. The Lightning Network addresses this by moving most transactions off-chain while retaining Bitcoin’s trustless, decentralized security model oai_citation:0‡lightning-network-paper.pdf.
Instead of broadcasting every transaction globally, two parties can open a micropayment channel:
This allows billions of off-chain payments per day with minimal fees.
Channels aren’t just one-way:
A single channel only connects two users. But a network of interconnected channels allows payments to be routed across multiple hops:
HTLCs enable multi-hop routing:
The Lightning Network is not a separate coin or trusted overlay. It is Bitcoin transactions, enforced by Bitcoin’s blockchain, but conducted off-chain until necessary.
It allows:
Lightning represents the path for Bitcoin to handle global financial volume, from large settlements to tiny micropayments, without compromising its principles oai_citation:1‡lightning-network-paper.pdf.