Showing posts with label chaos. Show all posts
Showing posts with label chaos. Show all posts

No posts for a wee while

Mike's Notes

I was on holiday for the last few weeks and am back now. There will be no blog posts, newsletters or meetings until Pipi Core is back up and running.

Update 27/05/2026

Lots of surprises. Making rapid progress. The peace and quiet are bliss.

Update 31/05/2026

The problem and solution are how things are named. Pipi auto-generates thousands of code names using multiple pattern languages, and all the naming conventions require many minor fixes for several unexpected reasons after migrating from a developer laptop to a production server environment. Everything else is absolutely fine.

Other naming problems are also being solved now, including:

  • The rapid development of Boxlang by Ortus has brought forward another challenge. Pipi 10 will be migrated to run on top of Boxlang in 2027 to support multiple languages, including C++, CFML, COBOL, Go, Java, JavaScript, PHP, Python, Rust, etc.
  • Future integration with cloud-based LLMs.
  • Future integrations with Office365, Google Workspace, Zoho, LibreOffice, etc.

The common solution is to create standardised naming systems that are simple, stable, robust, schema-based, versioned, self-documenting, and extensible to meet unanticipated future needs.

This is done by replacing code-based naming rules with database-driven ones that can be easily edited in the future via an admin UI.

90% of these names are internal, hidden in the closed core, and how they work and what they are will not be discussed here. The rest will be publicly and fully documented as part of the open-source workspaces for developers to work with.

Update 02/06/2026

I'm changing the disclosure boundary between the Pipi closed-core and open-source workspaces. Previously, "disclose everything unless there is a security reason not to". This is now changed to "disclose on the basis of need to know".

Closed-core accounts for 90% and open-source workspaces for 10% of lines of code, databases, etc.

This will reduce the documentation burden, given Pipi's vast scale. So, the open-source workspaces will be fully shared and documented on GitHub, etc, without restriction. This includes;

  • Standards schema
  • Ontologies
  • Parameters
  • Laws of physics
  • HTML + CSS
  • Algorithms
  • Module DDD models
  • Workflow diagrams
  • Documentation
  • API schema
  • UI code
  • etc

This also means some existing technical documentation about the closed-core will become hidden and only available internally.

Update 07/06/2026

Pipi Core is the IDE used to edit Pipi Core (AKA: which came first, the chicken or the egg?). Temporary UIs have been created and are being used across multiple engines to edit the names in use. This is much faster than directly editing data, which had to be done initially. The next step will be turning auto-generation back on. Once that's done, temporary UIs will be used to build permanent UIs. More automation will then be enabled via the UIs, and so on, as Pipi Core builds itself with a human in the loop.

Update 08/06/2026

The list of code cases available to use now for auto-generated naming, I/O translation, etc with examples, includes;

  • camelCase: userProfilePicture
  • kebab-case: user-profile-picture
  • PascalCase: UserProfilePicture
  • snake_case: user_profile_picture
  • SCREAMING_SNAKE_CASE: USER_PROFILE_PICTURE
  • Train-Case: User-Profile-Picture
  • flatcase: userprofilepicture
  • UPPER-CASE-KEBAB-CASE: USER-PROFILE-PICTURE
  • Sentence case: User profile picture
  • Title Case: User Profile Picture
  • middot·case: user·profile·picture
  • dot.case: user.profile.picture
  • UPPER CASE: USER PROFILE PICTURE
  • lowercase: user profile picture

Update 12/06/20026

Checking that these changes to variable names and internal messaging do not clash with the Gödel Machine.

Update 17/06/2026

The DevOps Engine (dvp) has unexpectedly proven to be critical to solving this puzzle. Mostly fixed last night. Watching the rather excellent live Google talk, Beyond the GPU: Maximising goodput with self-healing AI infrastructure, this morning has given me valuable insights into how to fix the remaining issues by reviewing Google HPC YAML files. 😎😎 Sometimes insights come from the strangest places.

Update 01/07/2026

The main work now is rapidly configuring Pipi for production and full autonomous automation. Using Google Search AI Mode (Gemini) and then Grammarly Pro makes the work easier and 100x faster.

  • I have decided to have Pipi re-render the many Ajabbi draft public websites with the new and missing developer information. (20K pages)
  • The website's .robot.txt file will then be unlocked to enable search engines.
  • The HTML will be updated to make it easier for AI to read.
  • This blog will be imported into Pipi, cleaned up, re-exported from Pipi, and published to Blogger via the API.
  • The new posts created in Pipi will return to A Sandy Beach to discuss something already built rather than being built.

Update 02/07/2026

The DevOps and IaC engines are getting rapid data model overhauls. The IaC engine is a great test for the variable names. I'm building a capability into Pipi to autonomously and automatically run OpenTofu and Ansible, initially targeting the Pipi Data Centre, then GCP and AWS for deployments. It's going very well and making rapid progress.

Update 05/07/2026

Pipi will initially run the open-source enterprise applications on Google Cloud Run and Google Cloud Storage (GCS). The code is complete and will be very low-cost to run, giving Ajabbi, a bootstrapping-purpose startup, a very long runway.

Update 18/07/2026

The job has now shifted to configuring, networking and deploying many physical servers. Installing software, including Pipi, labelling cables and rack gear, throwing out junk, tidying, etc., leaving nothing to chance. Shipping delays are holding up part deliveries.

Update 28/07/2026

Most of the equipment has arrived, and the small data centre setup is coming together. More deliveries later this week. It's already running a lot better and is much more productive.

Update 31/07/2026

Work on Pipi has reached a tipping point or system phase change as Pipi takes over tasks using autonomous automation. Pipi now has deadlines, not me. Soon it will set the deadlines. It's now a downhill run; daily posts from me resume tomorrow, and much more will come.

In hindsight. This whole project has been systematic trial and error, spending 10 years learning how to crack a hard problem.

Resources

References

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Last Updated

31/07/2026

No posts for a wee while

By: Mike Peters
On a Sandy Beach: 15/05/2026

Mike is the inventor and architect of Pipi and the founder of Ajabbi.

I was on a no-coding holiday for the last few weeks to clear my mind, and it has been great. I am back on the job today.

Suspended

Until the closed-source Pipi Core is back up and running 100% on autopilot, 10x faster, the following are suspended.

  • New posts "On a Sandy Beach
  • All newsletters, including the weekly Friday Report and the monthly Ajabbi Research Newsletter.
  • The fortnightly online Open R&D meeting.

Rapid refocus

  • A new developer area with five coding screens, designed to be more productive for hypervisual learners.
  • A better library has been set up for my A4 drawings in ring binders, the many reference books I use, and more bookshelves are on the way.
  • The server rack has been moved to a better location.
  • The light levels have been adjusted.
  • A big office tidy is almost done. An office-work-only desk has yet to be set up with a cat bed included.
  • A separate area with no screens for the happy cat, coffee, music, reading and drawing.

Less is more

Minimise screen time to be more productive at work. The new setup is also much less tiring.

Get the job done

The good thing is that, with a holiday and lots of drawing, I now have mental clarity about what needs fixing and how to fix it. Mainly, quite delicate changes here and there, organised into a list of steps. Now, I need to concentrate on one thing only: go as fast as possible, without meetings, post-deadlines, phone calls, or other distractions.

How

1. Use an AI workforce

Be the architect, and AI fills in the dots to make it happen.

Use Google Search AI mode (Gemini) to generate 99% of the code in one-page chunks (including references) to copy and paste, then manually change the variable names and SQL. Careful, test everything, resulting in 100x faster progress. Know how everything works and rapidly raise personal skill level.

2. Then build a cathedral

Make a wooden scale model of a cathedral for the builders. Google Search AI mode (Gemini) makes each brick, and Pipi Core assembles the bricks into floors, arches, walls, and vaults...

Speed is king

With the 100x coding productivity gains from Google Search AI mode (Gemini), plus the 10x10x10x speedup of Pipi Core currently underway over the next few months, what previously took a year will be done in hours and better.

Phase transitions

Once these initial migration issues from laptop to server are resolved, further transitions can be anticipated as the number of engines rapidly increases beyond 20. Increasing the number of engines slowly changes the whole system's behaviour from deterministic to probabilistic and adaptive.

Here is a partial list of transitions expected as the number of engines increases from 0 to 200. The actual numbers are a bit of a guess.

  • 20 engines enable Pipi 9 Core in a simple, deterministic structure.
  • 40 engines enable a workspace with a UI for administering Pipi Core.
  • 60 engines enable self-generation of user documentation.
  • 80 engines enable REPL and IAC (infrastructure-as-code).
  • 100 engines enable Workspaces for different user accounts.
  • Different Pipi 9 editions are made with the same engines, which recombine differently in response to the external environment.
  • And so on until...
  • 200 engines self-organise into a multi-layered complex fluid structure with probabilistic behaviour and emergent properties, as engines also act as agents.
  • 200+ engines enable Pipi 10 to interact with externally cloud-hosted LLMs, combining the very different strengths of both.
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The Coming Crisis: Fingers of Instability

Mike's Notes

An excellent description of what a critical state is, using a real-world example: the global financial system. Critical state also applies to other phenomena, including earthquakes.

I have read the two excellent books by Buchanan and Taleb in the references. I must also read Sornette's book.

I think everything in the universe has its time in the sun, with a birth, existence, and death, often followed by a transformation into its opposite. The laws of science apply to everything, including social systems like capitalism, trees, planets, schools of music, cars, etc.

Resources

References

  • Why Catastrophes Happen, by Mark Buchanan.
  • Antifragility, by Nassim Taleb.
  • Why Stock Markets Crash, by Didier Sornette.

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Last Updated

02/03/2026

The Coming Crisis: Fingers of Instability

By: John Maudlin
Thoughts from the Frontline: 28/02/2026

John Maudlin is Co-Founder, Mauldin Economics.

This letter is a little different. I am indeed working on my book about what I believe is a coming crisis by reviewing five different cycle theories. They all arrive at a similar scenario from  different points of view, but they all suggest a crisis occurring sometime around the end of this decade or perhaps shortly thereafter. And all for different reasons. One background element ties them together, which is the subject of today’s letter.

This is essentially a shortened first chapter. To long time readers, that background connection is our old friend: sandpiles and fingers of instability. but with a lot of edits and additions. Jumping in…

Ubiquity, Complexity Theory, and Sandpiles

With five different views about the coming crisis, which one is right? Do they conflict or reinforce each other? The correct answer is they’re all connected, but not in obvious ways. And in the end, it makes no difference which one is “more” right. The results will be the same. Understanding this below-the-radar connection is key to making sure you, your family, community and country all get through this to what will be the inevitable positive conclusion, even if it is a very bumpy ride.

We are going to start our exploration with excerpts from an important book by Mark Buchanan, called Why Catastrophes Happen. I HIGHLY recommend it to those of you who, like me, are trying to understand the complexity of the markets, economy and politics/society. The book is about chaos theory, complexity theory and critical states. It is written in layman’s terms. There are no equations, just easy-to-grasp, well-written stories and analogies. But it gives us an essential framework to understand the coming storms.

As kids, we all had the fun of going to the beach and playing in the sand. Remember taking your plastic buckets and making sand piles? Slowly pouring the sand into an ever-bigger pile, until one side of the pile started an avalanche?

Imagine, Buchanan says, dropping one grain of sand after another onto a table. A pile soon develops. Eventually, just one grain starts an avalanche. Usually it’s a small one, but sometimes it builds on itself and seems like a side of the pile collapses. Why?

Well, in 1987 three physicists named Per Bak, Chao Tang, and Kurt Weisenfeld began to play the sandpile game in their lab at Brookhaven National Laboratory in New York. Now, piling one grain of sand at a time is a slow process, so they wrote a computer program to do it. Not as much fun, but a whole lot faster. Not that they really cared about sandpiles. They were interested in what are called nonequilibrium systems.

They learned some interesting things. What is the typical size of an avalanche? After a huge number of tests with millions of grains of sand, they found there is no typical size. "Some involved a single grain; others, ten, a hundred or a thousand. Still others were pile-wide cataclysms involving millions that brought nearly the whole mountain down. At any time, literally anything, it seemed, might be just about to occur." The piles were chaotic in their unpredictability.

Now, let’s read this next paragraph from Buchanan slowly. It is important, as it creates a mental image that may help us understand the organization of financial markets, the world economy and society (emphasis mine).

"To find out why (such unpredictability) should show up in their sandpile game, Bak and colleagues next played a trick with their computer. Imagine peering down on the pile from above, and coloring it in according to its steepness. Where it is relatively flat and stable, color it green; where steep and, in avalanche terms, ‘ready to go,’ color it red. What do you see? They found that at the outset the pile looked mostly green, but that, as the pile grew, the green became infiltrated with ever more red. With more grains, the scattering of red danger spots grew until a dense skeleton of instability ran through the pile. Here then was a clue to its peculiar

The Critical State

Something only a math nerd could love? Scientists refer to this as a “critical state.” The term can mean the point at which water goes to ice or steam, or the moment that critical mass induces a nuclear reaction, etc. It is the point at which something triggers a change in the basic nature or character of the object or group. Thus (and very casually for all you physicists), we refer to something being in a critical state (or use the term critical mass) when there is the opportunity for significant change.

"But to physicists, [the critical state] has always been seen as a kind of theoretical freak sideshow, a devilishly unstable and unusual condition that arises only under the most exceptional circumstances [in highly controlled experiments]… In the sandpile game, however, a critical state seemed to arise naturally through the mindless sprinkling of grains."

Thus, they asked themselves, could this phenomenon show up elsewhere? In the earth’s crust, triggering earthquakes, or as wholesale changes in an ecosystem – or as a stock market crash?

"Could the special organization of the critical state explain why the world at large seems so susceptible to unpredictable upheavals?" Could it help us understand not just earthquakes, but why cartoons in a third-rate paper in Denmark could cause world-wide riots?

Buchanan concludes in his opening chapter:

"There are many subtleties and twists in the story … but the basic message, roughly speaking, is simple: The peculiar and exceptionally unstable organization of the critical state does indeed seem to be ubiquitous in our world. Researchers in the past few years have found its mathematical fingerprints in the workings of all the upheavals I’ve mentioned so far [earthquakes, eco-disasters, market crashes], as well as in the spreading of epidemics, the flaring of traffic jams, the patterns by which instructions trickle down from managers to workers in the office, and in many other things. At the heart of our story, then, lies the discovery that networks of things of all kinds – atoms, molecules, species, people, and even ideas – have a marked tendency to organize themselves along similar lines. On the basis of this insight, scientists are finally beginning to fathom what lies behind tumultuous events of all sorts, and to see patterns at work where they have never seen them before."

Going back to the sandpile game, you find that as you double the number of grains of sand involved in an avalanche, the probability of an avalanche becomes 2.14 times more likely. We find something similar in earthquakes. In terms of energy, the data indicate that earthquakes become four times less likely each time you double the energy they release. Mathematicians refer to this as a "power law," a special mathematical pattern that stands out in contrast to the overall complexity of the earthquake process.

Fingers of Instability

So, what happens in our game?

"…after the pile evolves into a critical state, many grains rest just on the verge of tumbling, and these grains link up into ‘fingers of instability’ of all possible lengths. While many are short, others slice through the pile from one end to the other. The chain reaction triggered by a single grain might lead to an avalanche of any size whatsoever, depending on whether that grain fell on a short, intermediate or long finger of instability."

Now, we come to a critical point in our discussion of the critical state. Again, read this with not just markets but our entire society in mind:

"In this simplified setting of the sandpile, the power law also points to something else: the surprising conclusion that even the greatest of events have no special or exceptional causes. After all, every avalanche, large or small, starts out the same way, when a single grain falls and makes the pile just slightly too steep at one point. What makes one avalanche much larger than another has nothing to do with its original cause, and nothing to do with some special situation in the pile just before it starts. Rather, it has to do with the perpetually unstable organization of the critical state, which makes it always possible for the next grain to trigger an avalanche of any size."

This concept applies to not just financial markets, but to how we organize our political systems, generational differences, geopolitics and war, the over-production of elites and even how information is interpreted. They ALL connect. The Great Recession was a financial crisis. COVID-19 was a health crisis with a financial crisis and added political crises which further divided a fractious world.

We all see pressures building up in many different aspects of society. They each create their own fingers of instability. But in the sandpile of life, they are connected. 

Now, let’s couple this idea with a few other concepts. First, Hyman Minsky (who should have been a Nobel laureate) points out that stability leads to instability. The more comfortable we get with a given condition or trend, the longer it will persist and then when the trend fails, the more dramatic the correction.

The problem with long term macroeconomic stability is that it tends to produce unstable financial arrangements. Just as long term geopolitical or social stability will eventually produce a critical state. If we believe that tomorrow and next year will be the same as last week and last year, we are more willing to add debt or postpone savings in favor of current consumption. Or ignore any of a number of societal crises. Thus, says Minsky, the longer the period of stability, the higher the potential risk for even greater instability when market participants or a country’s citizens must change their behavior.

Relating this to our sandpile, the longer a critical state builds up in an economy, or in other words, the more "fingers of instability" are allowed to develop connections to other fingers of instability, the greater the potential for a serious "avalanche."

Therefore (and ironically), the longer a crisis takes to come about, the bigger the repercussions. One of the conclusions at the end of the book will be that we simply don’t know when the avalanche will be triggered. The US is such a large and wealthy country, and many of the rest of the shirts in the global laundry are just as (or even more) dirty, that global money might come to the US as a safe haven, thus prolonging our “stability” as the sandpile grows to an ever more critical state.

We Are Managing Uncertainty

Or, maybe, a series of smaller shocks lessens the long reach of the fingers of instability, giving a paradoxical rise to even more apparent stability. This is the thrust of Nassim Taleb’s book, Antifragility.

“People often think that the opposite of fragility is durability. If something is fragile, that means it’s easily broken. Therefore, if something isn’t easily broken, logically that should mean it’s the opposite of fragile. However, there’s another step beyond. Since there isn’t an established English word for such a thing, [Nassim] calls it antifragility—not just the lack of fragility, but its true opposite.

“We live in an unpredictable world. The models and theories we use to try to predict the future invariably fall apart as unforeseen events prove them wrong and, in turn, destroy the plans we made based on those models. Clearly, systems based on such flawed models are bound to be fragile—easily broken.

“The solution to this problem is antifragility. Instead of a never-ending search for more accurate models and better predictions, all we need to do is make sure that we’re in a position to benefit from uncertainty and volatility instead of being harmed by it.

“This is hardly a new concept; nature exhibits antifragility in almost everything she creates. An organism can strengthen itself through minor damage in the form of exercise. In a similar sense, a species can strengthen itself through minor damage in the form of natural selection, which leads to evolution.

“However, unlike nature, humans try to control the world through models and rules. We think we can perfectly predict the future and avoid any shocks that would cause our fragile systems to fall apart. We think we can outsmart millions of years of evolution and antifragility, and we’re almost invariably wrong.

“Instead of trying to predict the future, we should assume that there will be major events we can’t see coming—because, sooner or later, there will be. If we’re prepared for them, using the methods and practices explained in this book, we can make sure that such events work to our advantage instead of hurting us. By avoiding fragility and embracing antifragility wherever possible, we can set ourselves up to thrive in an uncertain world.”

Another way to think about it is the way Didier Sornette, a French geophysicist, has described financial crashes in his wonderful book, Why Stock Markets Crash (the math, though, was far beyond me!). He wrote:

"[T]he specific manner by which prices collapsed is not the most important problem: a crash occurs because the market has entered an unstable phase and any small disturbance or process may have triggered the instability. Think of a ruler held up vertically on your  the instantaneous cause of the collapse is secondary."

When things are unstable, it isn’t the last grain of sand that causes the pile to collapse or the slight breeze that causes the ruler on your fingertip to fall. Those are the "proximate" causes. They’re the closest reasons at hand for the collapse. The real reason, though, is the "remote" cause, the farthest reason. The farthest reason is the underlying instability of the system itself.

This is one reason we get "fat tails" in financial markets. In theory, returns on investment should look like a smooth bell curve, with the ends tapering off into nothing. According to the theoretical distribution, events that deviate from the mean by five or more standard deviations ("5-sigma events") are extremely rare, with 10 or more sigma being practically impossible – at least in theory.

However, under certain circumstances, such events are more common than expected; 15-sigma or even rarer events have happened in the world of investing. Examples include Long Term Capital in the late 1990s and any of a dozen bubbles in history. Because the real-world commonality of high-sigma events is much greater than in theory, the distribution is "fatter" at the extremes ("tails") than one would expect.

This holds true in geopolitics, too. The unthinkable sometimes happens. Before World War I began, no one thought it would come to war. Peace had been the rule for 40 years. Surely, mankind had evolved. Until…

Thus, the build-up of critical states, those fingers of instability, is perpetuated even as, and precisely because, we hedge risks. We try to "stabilize" the risks we see, shoring them up with derivatives, emergency plans, insurance, treaties, alliances, political change and all manner of risk-control procedures. And by doing so, the economic and social systems can absorb body blows that would have been severe only a few decades ago. We distribute the risks, and their effects, throughout the system.

Yet as we reduce the known risks, we sow the seeds for the next 10-sigma event. It is the improbable, unseen risks that will create the next real crisis. It is not that the fingers of instability have been removed from the equation, it is that they lurk in different places, not yet visible.

A Stable Disequilibrium

We end up in a critical state that Paul McCulley calls "stable disequilibrium." It has "players" all over the world, tied inextricably together in a vast dance through investment, debt, derivatives, trade, globalization, international business and finance. Each player works hard to maximize their own personal outcome and reduce their exposure to "fingers of instability."

The longer we go on, asserts Minsky, the more likely and violent any "avalanche" is. The more the fingers of instability can build, the more that state of stable disequilibrium can go critical on us.

It's all connected. We are building an unstable sandpile and it will come crashing down at some point. Then we will have to dig our way out.

The good news is we have seen this movie before. And after the crisis, a new period of stability and growth follows, for at least another 50-80 years. In my upcoming book we will look for ways to get through to that happier future.

Scottsdale, Houston, Los Angeles, West Palm Beach, Boston and New York

Next week I fly to Houston where I am on an economic advisory board for the Rice University economics department. Then I will be in LA meeting with the Inner Circle, exploring several companies that are literally changing the technology landscape of defense and energy. We will be opening clinics in West Palm Beach and the DC area, hopefully in early April. Construction has begun. Then NYC and Boston.

I finish this from Scottsdale where Dr. Roizen and I are attending the 2026 Functional Longevity Summit, along with 3-400 doctors. The organizers have asked us to talk about Therapeutic Plasma Exchange. For those interested in staying healthy for longer, Mike and many experts now believe the first part of your journey should begin with therapeutic plasma exchange. Seriously. You can learn more at Lifespan-Edge.com (note the dash). If you haven’t, you really need to read our main research report. The research and other information can make a real difference in your life. You can set up a discovery call to talk with our doctors about the procedure and see if it is right for you. As well as look at a lot more research.

And with that, I will hit the send button. Have a great week.

Your thinking how to make my body antifragile analyst,

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Mathematicians Crack a Fractal Conjecture on Chaos

Mike's Notes

Very useful.

Here is a summary generated by Google.

"Harmonic analysis of Gaussian Multiplicative Chaos (GMC) on the circle involves studying its Fourier coefficients, revealing key properties like its near-certainty of being a Rajchman measure, meaning coefficients vanish at high frequencies, and connecting to random matrix theory (Circular-β-ensembles) and spectral theory, with recent work proving exact Fourier dimensions and describing convergence laws for scaled coefficients. This field investigates how random multiplicative structures on the circle behave under Fourier transformation, revealing spectral properties and connections to number theory and random matrix models, with significant recent advances in understanding critical and subcritical phases.

Key Concepts & Findings:

    • Rajchman Property: GMC on the circle almost surely becomes a Rajchman measure, meaning its Fourier coefficients 𝜇̂(𝑛) tend to zero as the frequency 𝑛 → ∞.
    • Fourier Dimensions: Researchers establish precise Fourier dimensions (how fast coefficients decay) for various GMC models, confirming conjectures and linking to spectral properties.
    • Connection to Random Matrices: There's a deep link between GMC on the circle and the Circular-β-Ensemble (CBE) from random matrix theory, with holomorphic analogues (HMC) also studied.
    • Spectral Approach: Using random orthogonal polynomials and spectral methods allows for efficient analysis of GMC properties, including its support's Hausdorff dimension.
    • Convergence Laws: Normalized Fourier coefficients are shown to converge in distribution to specific random variables (e.g., complex normal) in certain phases (L1-phase).
    • Critical & Subcritical Phases: Significant focus is on subcritical 𝛾 = √2 and critical 𝛾 = √2 regimes, with new methods improving understanding, especially for the unit interval and circle.

Recent Research Directions:

    • Proving exact Fourier decay rates and dimensions for various GMC models (e.g., critical GMC).
    • Developing unified approaches for classical multiplicative chaos measures.
    • Studying the holomorphic analogue (HMC) and its convergence properties.
    • Exploring connections to number theory and large random matrices.

In essence, researchers use harmonic analysis tools (like Fourier coefficients) to understand the random, fractal-like structures of GMC on the circle, revealing underlying deterministic patterns and links to other mathematical areas."

- Gemini 3

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Last Updated

16/12/2025

Mathematicians Crack a Fractal Conjecture on Chaos

By: Lyndie Chiou
Scientific American: 9/12/2025

Lyndie Chiou is a scientist, a science writer and founder of ZeroDivZero, a science conference website. Her writing has also appeared in Sky & Telescope. Follow her on X @lyndie_chiou.

A type of chaos found in everything from prime numbers to turbulence can unify a pair of unrelated ideas, revealing a mysterious, deep connection that disappears without randomness.

The world may seem orderly, but randomness and chaos shape everything in the universe, from enormous galaxies all the way down to subatomic particles. Take a chilly window sheeting over with ice: even one oddly shaped snowflake can exert an influence on the final frosty pattern.

Understanding how random fluctuations can ripple out to produce global effects is what French mathematician Vincent Vargas of the University of Geneva in Switzerland set out to do more than 10 years ago. His earliest ideas for simple geometries appeared in a decade-old paper, but it wasn’t until 2023, while he was working with Christophe Garban of the University of Lyon in France, that the concept finally crystallized into what is now known as the Garban-Vargas conjecture. Now mathematicians have proved the conjecture using an insightful technique that should open the door for understanding much more complex systems.

The conjecture involves the behavior of a form of randomness found in a huge range of fields, from quantum chaos to Brownian motion to air turbulence. Mathematicians use a mathematical “measuring tape” called Gaussian multiplicative chaos, or GMC, to pick out subtle patterns hidden inside an otherwise impenetrable sea of randomness. GMC has even been used to find patterns in the prime numbers. The topic is one of the most important and fundamental ideas in probability theory today.

French mathematician Jean-Pierre Kahane is credited with first developing GMC in 1985, although his pioneering work was quickly forgotten. “I was one of the people who revived his work,” Vargas says. “I met him many times, and he said he was amazed how important the topic [had] become. Everywhere on the planet, people are working on something related to Gaussian chaos.”

Vargas first encountered the measure while studying turbulence and finance. He then came across it again in a project on conformal field theory, which is used to study patterns that remain constant as you zoom in or out. Lately he has focused on investigating its fundamental mathematical nature.

To understand GMC, imagine a turbulent fluid full of swirling eddies at many different scales. Enormous eddies randomly break apart into smaller ones, which themselves break into even smaller eddies, in a vast, nested hierarchy of randomness. GMC serves as a mathematical model that measures this kind of multiscale randomness—it captures random fluctuations that persist across every scale of the observation. Because of this, it is often referred to as a fractal measure.

Mathematicians have uncovered surprising behaviors in the types of randomness governed by GMC. For instance, events at the smallest scales can govern the entire system; the powerful tendrils of fractal structure shape chaos at every level. As a result, these systems cannot be understood by looking at averages. Instead the rules of GMC produce a universal picture that applies to every scale.

But this fascinating picture only holds up to a critical threshold. If the underlying randomness becomes too strong, the GMC measure collapses. Or, in the language of eddies, once enough randomness infuses the swirls, they become unstable, losing all their hidden order. Like ice transitioning to a liquid, this breakdown marks an important phase transition for chaos.

In 2023 Garban and Vargas introduced a new lens for studying GMC chaos. It came from a field of mathematics called harmonic analysis. Instead of looking at eddies directly, they examined the frequencies of patterns hidden in the eddies, much like analyzing a complex sound by breaking it into pure tones.

Then an idea came to them. If they could match two completely different physical descriptions—complexity and harmonics—they might learn something new. Mathematicians refer to this idea of matching unrelated physical descriptions as matching “dimensions.”

As an example, consider snowflakes falling to the ground. As the snow gently lands, two possible dimensions might be how many patterns appear in the distribution of the snowflakes and how many clumpy piles form across different scales. But is there a formula that can relate the two dimensions of patterns (harmonics) and clumpiness (correlations)?

“The key word is dimension,” Vargas says. “That’s the name of the game. You have lots of natural dimensions, but when do they coincide?”

After studying systems governed by GMC on a circle, the duo conjectured an extraordinarily elegant equation that matched a GMC system’s correlation dimension to its harmonic dimension.

Unfortunately they couldn’t prove their formula, even for a simple geometry. In 2023 they posted their conjecture to the preprint server arXiv.org, and it subsequently became a major open problem.

In 2024 mathematicians Zhaofeng Lin and Yanqi Qiu of the Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, and Mingjie Tan of Wuhan University resolved the conjecture. Their research, which was posted as a preprint to arXiv.org and has not yet been peer-reviewed, not only confirmed the formula but also revealed why it works.

Mathematically, they likened GMC to a “fair betting game,” in which the expected winnings remain constant no matter the size of the game. When applied to fractal fluctuations, this means that the system remains balanced as you zoom in and out, and each smaller scale contributes randomness in a way that conserves energy.

Mathematicians call a process that exhibits this type of fair, scale-by-scale behavior a martingale. Unlike normal betting games, however, chaos “games” are much more complex, requiring higher-dimensional martingales.

“I heard about this conjecture during an online math workshop,” Qiu says. “I had focused on martingales for my Ph.D. thesis a few years back, and I had a hunch they would be the right tool here.”

The group used its higher-dimensional martingale structure to carefully track the accumulation of randomness at every scale. And sure enough, by conserving energy, numerous tiny “fair games” combined to give the same formula for the decay that Garban and Vargas had conjectured.

Qiu and his colleagues’ proof not only settled the conjecture but also paved the way for further proofs on more complex fractal models. The roadway to a complete theory isn’t entirely free of barriers, though. Even the new method fails when randomness forces the system to its critical phase-transition point. This phase transition itself is a rich and intriguing topic with its own set of deep questions, mathematicians say. But “to go further,” Qiu says, “we need new ideas.”

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This Is How the AI Bubble Will Pop

Mike's Notes

This greed-driven frenzy over AI has built a massive speculative bubble. It confirms what I have read from Ray Dalio, Jim Miller, John Maudlin, Jack Barnes, Dr Doom, et al.

The bubble is already 4x the size of the one in 2007. When it bursts, the middle-class hysteria will blame all Jews, especially in the US, where the implosion will be the greatest.

I think AI is useful and has a great future after a lot more R&D, but not in the hands of these idiots.

When Ajabbi get financial, it will hold gold.

I discovered this article via Amazing CTO.

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21/11/2025

This Is How the AI Bubble Will Pop

By: Derek Thompson
Derek Thompson: 02/10/2025

The AI infrastructure boom is the most important economic story in the world. But the numbers just don't add up.

Some people think artificial intelligence will be the most important technology of the 21st century. Others insist that it is an obvious economic bubble. I believe both sides are right. Like the 19th century railroads and the 20th century broadband Internet build-out, AI will rise first, crash second, and eventually change the world.

The numbers just don’t make sense. Tech companies are projected to spend about $400 billion this year on infrastructure to train and operate AI models. By nominal dollar sums, that is more than any group of firms has ever spent to do just about anything. The Apollo program allocated about $300 billion in inflation-adjusted dollars to get America to the moon between the early 1960s and the early 1970s. The AI buildout requires companies to collectively fund a new Apollo program, not every 10 years, but every 10 months.

Exponential View

It’s not clear that firms are prepared to earn back the investment, and yet by their own testimony, they’re just going to keep spending, anyway. Total AI capital expenditures in the U.S. are projected to exceed $500 billion in 2026 and 2027—roughly the annual GDP of Singapore. But the Wall Street Journal has reported that American consumers spend only $12 billion a year on AI services. That’s roughly the GDP of Somalia. If you can grok the economic difference between Singapore and Somalia, you get a sense of the economic chasm between vision and reality in AI-Land. Some reports indicate that AI usage is actually declining at large companies that are still trying to figure out how large language models can save them money.

Every financial bubble has moments where, looking back, one thinks: How did any sentient person miss the signs? Today’s omens abound. Thinking Machines, an AI startup helmed by former Open AI executive Mira Murati, just raised the largest seed round in history: $2 billion in funding at a $10 billion valuation. The company has not released a product and has refused to tell investors what they’re even trying to build. “It was the most absurd pitch meeting,” one investor who met with Murati said. “She was like, ‘So we’re doing an AI company with the best AI people, but we can’t answer any questions.” Meanwhile, a recent analysis of stock market trends found that none of the typical rules for sensible investing can explain what’s going on with stock prices right now. Whereas equity prices have historically followed earnings fundamentals, today’s market is driven overwhelmingly by momentum, as retail investors pile into meme stocks and AI companies because they think everybody else is piling into meme stocks and AI companies.

Every economic bubble also has tell-tale signs of financial over-engineering, like the collateralized debt obligations and subprime mortgage-backed securities that blew up during the mid-2000s housing bubble. Ominously, AI appears to be entering its own phase of financial wizardry. As the Economist has pointed out, the AI hyperscalers—that is, the largest spenders on AI—are using accounting tricks to depress their reported infrastructure spending, which has the effect of inflating their profits1. As the investor and author Paul Kedrosky told me on my podcast Plain English, the big AI firms are also shifting huge amounts of AI spending off their books into SPVs, or special purpose vehicles, that disguise the cost of the AI build-out.

My interview with Kedrosky received the most enthusiastic and complimentary feedback of any show I’ve done in a while. His level of insight-per-minute was off the charts, touching on:

  • How AI capital expenditures break down
  • Why the AI build-out is different from past infrastructure projects, like the railroad and dot-com build-outs
  • How AI spending is creating a black hole of capital that’s sucking resources away from other parts of the economy
  • How ordinary investors might be able to sense the popping of the bubble just before it happens
  • Why the entire financial system is balancing on big chip-makers like Nvidia
  • If the bubble pops, what surprising industries will face a reckoning

Below is a polished transcript of our conversation, organized by topic area and adorned with charts and graphs to visualize his points. I hope you learn as much from his commentary as much as I did. From a sheer economic perspective, I don’t think there’s a more important story in the world.

AI SPENDING: 101

Derek Thompson: How big is the AI infrastructure build-out?

Paul Kedrosky: There’s a huge amount of money being deployed and it’s going to a very narrow set of recipients and some really small geographies, like Northern Virginia. So it’s an incredibly concentrated pool of capital that’s also large enough to affect GDP. I did the math and found out that in the first half of this year, the data-center related spending—these giant buildings full of GPUs [graphical processing units] and racks and servers that are used by the large AI firms to generate responses and train models—probably accounted for half of GDP growth in the first half of the year. Which is absolutely bananas. This spending is huge.


JP Morgan chart showing the rising contribution to GDP growth from tech capex

Thompson: Where is all this money going?

Kedrosky: For the biggest companies—Meta and Google and Amazon—a little more than half the cost of a data center is the GPU chips that are going in. About 60 percent. The rest is a combination of cooling and energy. And then a relatively small component is the actual construction of the data center: the frame of the building, the concrete pad, the real estate.

HOW AI IS ALREADY WARPING THE 2025 ECONOMY

Thompson: How do you see AI spending already warping the 2025 economy?

Kedrosky: Looking back, the analogy I draw is this: massive capital spending in one narrow slice of the economy during the 1990s caused a diversion of capital away from manufacturing in the United States. This starved small manufacturers of capital and made it difficult for them to raise money cheaply. Their cost of capital increased, meaning their margins had to be higher. During that time, China had entered the World Trade Organization and tariffs were dropping. We’ve made it very difficult for domestic manufacturers to compete against China, in large part because of the rising cost of capital. It all got sucked into this “death star” of telecom.

So in a weird way, we can trace some of the loss of manufacturing jobs in the 1990s to what happened in telecom because it was the great sucking sound that sucked all the capital out of everywhere else in the economy.

The exact same thing is happening now. If I’m a large private equity firm, there is no reward for spending money anywhere else but in data centers. So it’s the same phenomenon. If I’m a small manufacturer and I’m hoping to benefit from the on-shoring of manufacturing as a result of tariffs, I go out trying to raise money with that as my thesis. The hurdle rate just got a lot higher, meaning that I have to generate much higher returns because they’re comparing me to this other part of the economy that will accept giant amounts of money. And it looks like the returns are going to be tremendous because look at what’s happening in AI and the massive uptake of OpenAI. So I end up inadvertently starving a huge slice of the economy yet again, much like what we did in the 1990s.

Thompson: That’s so interesting. The story I’m used to telling about manufacturing is that China took our jobs. “The China shock,” as economists like David Autor call it, essentially took manufacturing to China and production in Shenzhen replaced production in Ohio, and that’s what hollowed out the Rust Belt. You’re adding that telecom absorbed the capital.

And now you fast-forward to the 2020s. Trump is trying to reverse the China shock with the tariffs. But we’re recreating the capital shock with AI as the new telecom, the new death star that’s taking capital that might at the margin go to manufacturing.

Kedrosky: It’s even more insidious than that. Let’s say you’re Derek’s Giant Private Equity Firm and you control $500 billion. You do not want to allocate that money one $5 million check at a time to a bunch of manufacturers. All I see is a nightmare of having to keep track of all of these little companies doing who knows what.

What I’d like to do is to write 30 separate $50 billion checks. I’d like to write a small number of huge checks. And this is a dynamic in private equity that people don’t understand. Capital can be allocated in lots of different ways, but the partners at these firms do not want to write a bunch of small checks to a bunch of small manufacturers, even if the hurdle rate is competitive. I’m a human, I don’t want to sit on 40 boards. And so you have this other perverse dynamic that even if everything else is equal, it’s not equal. So we’ve put manufacturers who might otherwise benefit from the onshoring phenomenon at an even worse position in part because of the internal dynamics of capital.

Thompson: What about the energy piece of this? Electricity prices rising. Data centers are incredibly energy thirsty. I think consumers will revolt against the construction of local data centers, but the data centers have enormous political power of their own. How is this going to play out?

Kedrosky: So I think you’re going to rapidly see an offshoring of data centers. That will be the response. It’ll increasingly be that it’s happening in India, it’s happening in the Middle East, where massive allocations are being made to new data centers. It’s happening all over the world. The focus will be to move offshore for exactly this reason. Bloomberg had a great story the other day about an exurb in Northern Virginia that’s essentially surrounded now by data centers. This was previously a rural area and everything around them, all the farms sold out, and people in this area were like, wait a minute, who do I sue? I never signed up for this. This is the beginnings of the NIMBY phenomenon because it’s become visceral and emotional for people. It’s not just about prices. It’s also about: If you’ve got a six acre building beside you that’s making noise all the time, that is not what you signed up for.

A very specific prediction for how and why AI bubble will pop

> The rest of the article is for Derek Thompson paying subscribers. <

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Chaos in the machine: How foundation models can make accurate predictions in time-series data

Mike's Notes

The GitHub link contains the experiment. Yuanzhao has done research into reservoir computing, a future enhancement for Pipi 11.

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08/06/2025

Chaos in the machine: How foundation models can make accurate predictions in time-series data

By: Santa Fe Institute
Santa Fe Institute: 19/05/2025

Yuanzhao Zhang, Complexity Postdoctoral Fellow, Omidyar Fellow, Santa Fe Institute

Yuanzhao was born and raised in a small coastal city in China. His research focuses on collective dynamics on complex networks. In particular, he is interested in how individual differences shape collective behaviors and how complex dynamical patterns emerge from decentralized interactions.

Yuanzhao received a B.Sc. in Mathematics from Zhejiang University in 2014, an M.Sc. in Applied Mathematics in 2015 and his Ph.D.in Physics from Northwestern University.

Until recently, using machine learning for a specific task meant training the system on vast amounts of relevant data. The same was true for data representing a system that changes over time, says SFI Complexity Postdoctoral Fellow Yuanzhao Zhang. “The traditional paradigm in forecasting dynamical systems has always been that you need to train on the system you want to predict,” he says. If you want to forecast the weather in Santa Fe, start by training your model on the area’s historical weather data. 

But the advent of foundation models — a term coined in 2021 to describe the architecture at the heart of today’s AI systems — has changed the game. These models, like previous systems, train on large datasets. But unlike earlier, specialized deep-learning models, they’re designed to carry out a wide range of tasks. “They work right out of the box,” Zhang says. Notably, they can complete new tasks that weren’t included in their training data. For large language models, those include tasks like generating computer code or translating between languages. Reports of this behavior, called “zero-shot learning,” ignited a global race to build models that can similarly make zero-shot predictions for time-series data.

Zhang wanted to understand whether existing foundation models could predict chaotic systems and, if so, how they do it. In a recent analysis, Zhang and William Gilpin, a physicist at the University of Texas at Austin, reported that a foundation model called Chronos could generate predictions of chaotic dynamical systems at least as accurately as models trained on relevant data. Their paper was accepted to the Thirteenth International Conference on Learning Representations, which focuses on deep learning approaches in AI and was held in Singapore in April 2025.

Zhang says the paper represents the first test of zero-shot learning in forecasting chaotic systems, such as the weather and financial markets, which are governed by mathematical equations and extremely sensitive to small changes in initial conditions. Zhang and Gilpin tested their idea by using Chronos to predict how 135 chaotic systems would change over time. They tested each system using 20 distinct initial conditions. They compared the short- and long-term predictions of the model to deep learning models specifically trained using chaotic data. 

“We wanted to compare this zero-shot paradigm with the old paradigm and see if the foundation model can outperform the traditional models,” Zhang says. 

The promising results show that foundation models can make accurate predictions after training on data from any time series — not just data from the system or task that a user wants to predict. Forecasting the weather in Santa Fe may not require historical data, just other time-series behaviors in which the model could identify patterns. 

The study raises interesting ideas about what kind of training is required to accurately perform time-series tasks. “There’s this question: Do you actually need to learn chaos to have a good forecasting performance for chaotic systems?” Zhang asks. “I think the answer is no.” 

Zhang and Gilpin’s current work only looks at one-dimensional data; in future work, Zhang says he hopes to expand that to more complicated, multidimensional data. He’d also like to determine how the system carries out these tasks. “Is it, in some sense, learning the dynamics?” he asks. “Is it using anything more sophisticated than parroting?” 

The new study offers a step forward in answering those larger, deeper questions, he says. 

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