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I use AI to research things I care about, because I care about too many things to spend huge amounts of time on it. If you're paying for bleeding edge models they are more efficient at researching basically anything, given that you are steering them.
We are now habitually using it in cooking, gardening, personal finance, developing policy proposals for local political groups, many things.
It's kind of like online dating back in the days before tinder, when you had to pretend you didn't have a profile because it was considered something for desperate losers. Ironically online dating created way more desperate losers than were on it back then when it was uncool. Probably something similar will happen when the bubble pops and the economics make the current delivery models unsustainable, when OpenAI embraces the ad model and Anthropic gets acquired by Google and subsumed by it's ad services.
Like the internet itself this productivity window might be brief, supported by venture capital in order to overcome behavioural barriers only to sucker punch the consumer like social media.
ChatGPT says:
Understanding arXiv:2601.04835Understanding arXiv:2601.04835
Yes — I can read and understand the paper you linked.
Below is a clear, structured summary of arXiv:2601.04835, explaining what the paper is about and why it matters.
📄 Title📄 Title
A Mathematical Theory of Payment Channel Networks
Author: René Pickhardt
Date: January 2026
arXiv ID: 2601.04835
📌 What the Paper Is About📌 What the Paper Is About
This paper develops a rigorous mathematical theory of payment channel networks (PCNs) — the class of systems that includes the Bitcoin Lightning Network.
Rather than proposing a new protocol, the paper:
- Formalizes how liquidity behaves
- Describes when off-chain payments are feasible
- Explains why payment failures occur
- Quantifies the relationship between on-chain settlement and off-chain throughput
The approach is geometric and network-theoretic, using polytopes, flows, and conservation laws.
🧠 Core Ideas🧠 Core Ideas
1. Payment Channel Networks (PCNs)1. Payment Channel Networks (PCNs)
A payment channel network allows users to route payments through intermediaries without touching the blockchain, as long as:
- Channels have sufficient capacity
- Liquidity is positioned correctly along the route
Each channel has:
- A fixed total capacity
- A distribution of liquidity between participants
2. Liquidity as Geometry2. Liquidity as Geometry
🔷 Liquidity State Space (LG)🔷 Liquidity State Space (LG)
- Every possible allocation of liquidity across channels corresponds to a point in a high-dimensional space
- Conservation constraints make this space a polytope
Formally, for each channel ( e = (u, v) ):
[
\lambda(e,u) + \lambda(e,v) = \text{cap}(e)
]
Only allocations satisfying all such constraints are valid.
🔶 Wealth Distribution Polytope (WG)🔶 Wealth Distribution Polytope (WG)
Instead of tracking per-channel balances, the paper projects LG into:
- Total wealth per participant
This projection yields another polytope (WG), which turns out to be central.
Key Result:
A payment is feasible if and only if the resulting wealth distribution lies inside WG.
This provides a global criterion for payment feasibility.
3. Payment Feasibility & the Throughput Law3. Payment Feasibility & the Throughput Law
When a payment cannot be routed off-chain, the system must fall back to on-chain transactions.
The paper derives a simple but powerful relationship:
[
S = \frac{\zeta}{\rho}
]
Where:
- ( S ) = sustainable off-chain throughput
- ( \zeta ) = on-chain settlement bandwidth
- ( \rho ) = fraction of off-chain payments that fail
This formalizes a fundamental trade-off:
Off-chain scalability is ultimately limited by on-chain capacity.
4. Multi-Party Channels4. Multi-Party Channels
The theory extends beyond two-party channels:
- k-party channels enlarge the feasible wealth polytope
- Larger polytopes → fewer infeasible payments
- This mathematically explains why constructs like:
- channel factories
- coinpools
- multi-party escrow
are more capital-efficient
5. Fees and Channel Depletion5. Fees and Channel Depletion
The paper also analyzes how routing fees affect liquidity:
- Asymmetric or linear fees cause liquidity to drift
- Over time, channels collapse to boundary states
- The effective network degrades into a spanning forest
This explains empirically observed issues in Lightning:
- channel exhaustion
- rebalancing pressure
- reduced routing reliability
🧩 Why This Paper Matters🧩 Why This Paper Matters
This work provides:
- A foundational theory of payment channel networks
- A precise explanation of why payments fail
- A way to compare network designs mathematically
- Insight into how protocol choices affect scalability
It bridges:
- Cryptography
- Network flow theory
- Economic incentives
- Real-world Lightning Network behavior
🧠 In Plain English🧠 In Plain English
- The paper treats liquidity like a geometric object.
- Payments succeed only if they keep the system inside a valid region.
- When too many payments fail, the blockchain becomes the bottleneck.
- Multi-party channels fundamentally scale better.
- Many Lightning problems are structural, not implementation bugs.
If you’d like, I can also:
- Walk through specific theorems step-by-step
- Explain the math without equations
- Compare this theory to existing Lightning heuristics
- Critically evaluate the assumptions
Just tell me how deep you want to go.
Medium of exchange is held down by the incumbent system. Store of Value is the adaptation that makes bitcoin resilient in the face of a defensive response. Medium of exchange is coming, but you have to be patient.
Saylor isn't going to sell that pitch though, why would he? Why does he need to? Why do you think you know his private thoughts? For all you know Saylor is here on SN.
They don't have to discuss the benefits to label Bitcoiners as tax dodgers, scammers, greater fools, parasites of various stripes. That will give them cover in a crisis to bring in punitive laws that target the UX of self-custody and herd people custodial solutions where they can be 6102'd.
I think Bitcoiners will be vilified and scapegoated, which will put downward pressure on price for a while.
The question is of those defaulting what percentage are women?
And whose money is being seized, people who are working enough to pay the loans? Obviously you can't pay if you are broke.
And of those who are in default while earning what percentage are women?
And a fraction of the bitcoin.
I say this with only a small amount of joy, and a lot of unease. Sovereign money is the end of feminism.
Co-opting Christianity works, but purist Bitcoiners are allergic to the idea.
Christ was the living manifestation of the discovery of proof of work:
- the greatest sacrifice (work)
- the highest hill (proof)
- as a carpenter's son (the every man), with the criminals (for my enemies)
- alone (sovereignty, responsibility, NYKNYC)
That is the closest thing to what I will call God, my north star.
Vanishingly few albums can claim to be perfect.
Justin Vernon managed to capture a near universal human experience; it's a Mona Lisa, a Caravaggio, a Van Gough, a Kafka.
I was on the upper deck of a bus that was about to stop. As I got up I told my 2yo and 4yo to hold on, but my 2yo got up anyway and walked to the stairs in front of me while I collected our bulky bag from the seat.
Then the bus stopped, and my son went head first towards the stairs.
Hands full and unable to reach him, I stuck out a leg and caught him by the belly with my foot, which gave me enough time to grab the back of his shirt.
Averted a trip to the hospital.