Derived, Borrowed, Learned
The question this track follows
Every useful computation depends on structure the problem already has. The three essays here are about three different places that structure can come from, and about what happens at the edge where it runs out.
Derived. From Smooth to JaggedEssayFrom Smooth to JaggedHow mathematics grew a new tool each time reality posed a problem the existing math could not solve, from Taylor series to the Fourier transform, stochastic calculus, fractal geometry, and rough paths, ending with a personal encounter with Benoit Mandelbrot and its bearing on reading markets. follows mathematics growing a new tool each time reality proved rougher than the last one could describe: Taylor series, Fourier analysis, Itô calculus, fractal geometry, rough paths. Structure is derived from first principles, and the frontier is still moving.
Borrowed. The Resolution of RealityEssayThe Resolution of RealityA primer on quantum computing: how physical law replaces brute-force search, why the advantage depends on how native the problem is to the machine, and where hybrid classical-quantum architectures fit. is a primer on quantum computing, and its central claim is that a quantum computer's advantage depends on how native the problem is to the machine. At the native end, the machine and the molecule obey the same physics and the modeling step nearly disappears. Structure is borrowed from nature.
Learned. From Prediction to AgencyEssayFrom Prediction to AgencyA draft essay on frontier language models: each stage in their development answered a limit hit by the previous one, and each answer changed what a model is. It ends with the question the companion essays set up: whether a learned model of human behavior could become a substrate for markets, and why reflexivity puts a ceiling on that., on frontier language models, follows a different arc: each stage in their development answered a limit hit by the previous one, and the models now learn structure from data where nobody can derive it and nothing in nature computes it. It ends with the question the first two essays set up: whether a learned model of human behavior could ever become a borrowed substrate for markets, and why ReflexivityNoteSeedReflexivityA model of a market changes the market. Soros's reflexivity, the Lucas critique, and Goodhart's law are three statements of the same limit, and it is the one way markets are permanently harder than chemistry: molecules do not read the paper. puts a ceiling on that hope.
Where the track currently stops
The third essay is a draft. Its notes on individual labs carry last_reviewed dates and will go stale; the concept notes will not. Read the essays for the argument and the notes for the parts of the argument I am still learning.
How to read it
The reading order below alternates essays and notes. Each essay is complete on its own; the notes fill in the concepts an essay mentions but does not have room to develop. The The Structure GradientNoteSeedThe Structure GradientComputational advantage depends on how much structure a problem has and where the structure comes from: isomorphic (the problem is physics), structural (hidden mathematical regularity), or generic (nothing to exploit). Derived, borrowed, or learned. note is the shared vocabulary and is worth reading first.
Reading order
Essays are the long-form stops; notes fill in the concepts between them.
- The Structure GradientComputational advantage depends on how much structure a problem has and where the structure comes from: isomorphic (the problem is physics), structural (hidden mathematical regularity), or generic (nothing to exploit). Derived, borrowed, or learned.
- From Smooth to JaggedHow mathematics grew a new tool each time reality posed a problem the existing math could not solve, from Taylor series to the Fourier transform, stochastic calculus, fractal geometry, and rough paths, ending with a personal encounter with Benoit Mandelbrot and its bearing on reading markets.
- Taylor SeriesReconstruct a smooth function from its derivatives at one point. The first tool mathematics built for prediction, and the one that assumes the world is smooth.
- Fourier TransformAny physically realistic signal is a sum of sines and cosines; the transform recovers which ones and how strongly. Built to solve the heat equation, it is also the engine of Shor's algorithm.
- Brownian Motion and Itô CalculusA path continuous everywhere and differentiable nowhere broke ordinary calculus. Itô's lemma repaired it with one extra term, tuned exactly to Brownian roughness.
- Hurst Exponent and Fractal DimensionTwo measurements of roughness that are one measurement: for a self-affine path, D = 2 − H. A random walk has H = 0.5 and D = 1.5; anything else is a path of another roughness.
- Rough Path TheoryLyons (1998): a rough path alone does not determine the integrals along it, so enrich the path with its iterated integrals. Once you do, differential equations driven by paths of any roughness become well defined, with no probability required.
- Rough VolatilityGatheral, Jaisson, and Rosenbaum (2018) measured the Hurst exponent of realized volatility across many markets and found values near 0.1. Volatility is far rougher than a Brownian path, and the standard models are the wrong calculus for it.
- The Resolution of RealityA primer on quantum computing: how physical law replaces brute-force search, why the advantage depends on how native the problem is to the machine, and where hybrid classical-quantum architectures fit.
- Superposition and InterferenceA qubit before measurement holds amplitudes for both outcomes; computation is the controlled interference of those amplitudes. The double slit is the physical demonstration.
- Entanglement and Bell's TheoremWhy entangled particles are correlated more strongly than any pre-written script allows, shown through the CHSH game, the three-polarizer experiment, and the 2√2 bound. Originally Appendix B of The Resolution of Reality.
- Shor's Algorithm and the Quantum Fourier TransformShor's algorithm is period finding: prepare a superposition, apply the periodic function, and read off the period with a Fourier transform. It is the bridge between the quantum primer and the mathematics essay.
- Quantum SimulationFeynman's 1982 proposal and Lloyd's 1996 proof: a quantum computer can efficiently simulate any local quantum system. The case where the machine and the problem share the same physics.
- Analog Computation and IsomorphismAn analog computer is one that is analogous to the problem: the wind tunnel to flight, the tide machine to the sea. When the analogy is exact the word is isomorphism, and a quantum simulator is the deepest example.
- From Prediction to AgencyA draft essay on frontier language models: each stage in their development answered a limit hit by the previous one, and each answer changed what a model is. It ends with the question the companion essays set up: whether a learned model of human behavior could become a substrate for markets, and why reflexivity puts a ceiling on that.
- The frontier lab landscape, September 2026A map of who is shipping frontier-class models as of 2026-09-08: four US closed-weight labs, Meta re-entering with a closed model, and a group of Chinese labs shipping open weights a few months behind. Links out to one note per lab.
- World models: two meanings of one termThe phrase 'world model' now covers two different research programs: a learned dynamics model an agent plans inside, and a generative model of visual or 3D space. This note separates them and explains why the distinction matters.
- ReflexivityA model of a market changes the market. Soros's reflexivity, the Lucas critique, and Goodhart's law are three statements of the same limit, and it is the one way markets are permanently harder than chemistry: molecules do not read the paper.